Fractional Calculus: Integral and Differential Equations of Fractional Order

CISM LECTURE NOTES In ternational Cen tre for Mec hanical Sciences P alazzo del T orso, Piazza Garibaldi, Udine, Italy FRA CTIONAL CALCULUS : In tegral and Diﬀerential Equations of F ractional Order Rudolf GORENFLO and F rancesco MAINARDI Departmen t of Mathematics and Informatics Departmen t of Ph ysics F ree Univer sity of Berlin Univ ersity of Bo logna Arnimallee 3 Via Irnerio 46 D-14195 Berlin, German y I-40126 Bologna, Italy gorenfl o@mi.fu-ber lin.de frances co.mainardi @unibo.it URL: www.fracalmo.org FRA CALMO PRE-PRINT 54 pages : pp. 223-276 ABSTRA CT . . . . . . . . . . . . . . . . . . . . . . . . . . p. 223 1. INTR ODUCTION TO FRACTIONAL CALCULUS . . . . . . . . p. 224 2. FRA CTIONAL IN TEGRAL EQUA TIONS . . . . . . . . . . . . p. 235 3. FRA CTIONAL DIFFERENT IA L EQUA TIONS: 1-st P A R T . . . . p. 241 4. FRA CTIONAL DIFFERENT IA L EQUA TIONS: 2-nd P AR T . . . . p. 253 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . p. 261 APPENDIX : THE MITT AG-L EFFLER TYPE FUNCTIONS . . . p. 263 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . p. 271 The pap er i s based on the lectures deliv ered b y the authors at the CISM Course Sc aling L aws and F r ac tality in Continuum Me chanics: A Survey of the Metho ds b ase d on R enormalization Gr oup and F r actional Calculus , held at the se at of CISM, Udine, from 23 to 27 Septem b er 1 996, under the direction of Professors A. Carpin teri and F. Mai nardi. This T E X pre-prin t is a revised version (Decem ber 2 000) of the chapter published in A. Carpinteri and F. M a inardi ( Edi tors): F ractals and F ractional Calculu s in Con tinuum Mechanics, Springer V erlag, Wien an d N ew Y ork 1997, pp. 223-276. Suc h b o ok is the v olume No . 378 of the series CISM COURSES AND LECTURES [ISBN 3-211-82913-X] i c  1997, 2000 Prof. Rudolf Gorenﬂo - Berlin - German y c  1997, 2000 Pro f. F rancesco Mainardi - B ologna - Ita ly ii R. Gor enﬂo and F. Maina r di 223 FRA CTIONAL CALCULUS : In tegral and D i ﬀerential Equations of F ractional Order Rudolf GORENFLO and F rances co MAINARDI Departmen t of Mathematics and Informatics Departmen t of Ph ysics F ree Univer sity of Berlin Univ ersity of B o logna Arnimallee 3 Via Irnerio 46 D-14195 Berlin, German y I-40126 Bologna, Italy gorenfl o@math.fu-b erlin.de frances co.mainardi @unibo.it URL: www.fracalmo.org ABSTRA CT In these lectures we introduce the linear op erato rs of fractional in tegratio n a nd frac- tional diﬀeren t iation in t he framework of the Riemann-Liouville fractional calculus. P articular atten tion is dev oted to the tec hnique of Laplace transforms for treati ng these op erators in a wa y accessible t o a pplied scien tists, av oiding unpro ductive gen- eralities and excessiv e mathematical rigor. By applying this tec hnique w e shall deriv e the analytical solutions of the most simple linear in tegral and diﬀeren tia l equations of fractional order. W e shall sho w the fundamen tal role of the Mittag-Leﬄer function, whose prop erties are rep orted in an ad ho c A pp endix. The topics discussed here will b e: (a) essen t ials of Riemann-Liouville fractional calculus with basic form ulas of Laplace transforms, (b) A b el t yp e integral equati o ns of ﬁrst and second kind, (c) relaxation a nd oscill ation type diﬀeren tial equations of fractional order. 2000 Math. Subj. Class. : 26 A33, 33 E 12, 33E2 0 , 44A20 , 4 5E10, 45 J05. This researc h was partially su pp orted b y Rese arch Gran ts of the F ree Unive rsity of Berlin and the U niv ersit y of Bol o gna. The authors also appreciate the supp ort given b y the National Researc h Councils of It a ly (CNR-GNFM) and b y the In ternational Cen tre of Mec hanical Sciences (CISM). 224 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der 1. INTR ODUCTION TO FRACTIONAL CALCULUS 1.1 Hi storic a l F or ewor d F ractional calculus is the ﬁeld of mathematical analysis which deals with the in v estigatio n and applications of integrals and deriv ati v es of arbitrary order. The term f r actional is a misnomer, but it is retai ned follo wing the prev ailing use. The fractional calculus ma y b e considered an old and y et novel topic. It is an old topic since, starting from some speculati o ns of G.W. Leibniz (1695, 1697 ) and L. E uler (1730), it has b een dev elop ed up to no w ada ys. A list of mathematicia ns, who ha v e pro vided i mp ortan t contributions up to t he mi ddle of our cen tury , includes P .S. Laplace (1812), J.B . J. F ourier ( 1 822), N.H. Ab el (1823-1826 ), J. Liouvi lle (1832- 1873), B. Riemann (1847), H. Holmgren (186 5 -67), A.K. Gr ¨ un w ald (1867-1872), A . V. Letnik o v (1868-1872), H. Lauren t (18 84), P .A . Nekrasso v (18 8 8), A. Krug (189 0), J. Hadamard (1892), O. Hea viside (1892-1912) , S. Pi nc herle (1902), G.H. Hardy and J.E. Littlewoo d ( 1917-1928), H . W eyl (1917), P . L ´ evy (1923), A . Marc haud (1927), H.T. Da vis (1924-1936 ) , A. Zygmund ( 1 935-1945), E.R. Lov e (1938-1996 ) , A. Erd ´ elyi (1939-1965), H . Kob er (1 9 40), D.V. Widder (1941), M. Riesz (1949). Ho w ever, it may b e considered a novel topic as well, since only from a litt le more than tw en ty years it has b een ob ject of sp ecialized conferenc es and treatises. F or the ﬁrst conference the merit is ascrib ed to B. Ross who organized the Firs t Confer enc e on F r a c tional Calculus and i ts Applic atio ns at the Univ ersity of New H av en in June 1974, and edited the pro ceedings, see [1]. F or the ﬁrst monograph the merit is ascrib ed to K.B. Oldham and J. Spanier, see [2], who, after a join t collab orat ion started in 19 6 8, published a b o ok devoted to fractional calculus in 1 974. No wada ys, the list of texts a nd pro ceedings dev oted solely or partly to fractional cal culus and its applications includes a b out a dozen of titles [1-14] , among which the encyclopaedic treatise b y Samk o, Kilbas & Mari chev [5] is the most prominen t. F urthermore, w e recall t he attention to the treatises by Davis [15], Erd ´ elyi [16], Gel’fand & Shilov [17], Djrbashian [18, 22], Caputo [19], Bab enko [20], Gorenﬂo & V essella [2 1 ], whic h con tain a detailed analysis of some mathematical asp ects and/or ph ysical applicati o ns of fractional calculus, although without explicit men tion in their titles. In recen t years considerable in terest in fractional calculus has been stim ulated b y the applicati ons that this calculus ﬁnds in numerical analysis and diﬀeren t areas of ph ysics and engineering, p ossibly including fractal phenomena. In this resp ect A. Carpin teri and F. Mainardi ha v e edited the presen t b o ok of lecture notes and en titled it as F r actals and F r actional Calculus i n Continuum Me chanics . F or the topic of fractional calculus, in addition to this join t article of in tro duction, we ha v e con tributed also with t w o single arti cles, one by Gorenﬂo [23], dev oted to n umerical metho ds, and one by Mainardi [24], concerning applicati ons in mec hanics. R. Gor enﬂo and F. Maina r di 225 1.2 The F r actional Inte gr al According to the Riemann-Liouville approac h to fractional calculus the noti o n o f fractional i n tegral of order α ( α > 0 ) i s a natural consequence of the well kno wn form ula (usually attributed to Cauc hy), that reduce s the calculati on of the n − fold primitive of a function f ( t ) to a singl e i n tegral of conv olution t yp e. In our notation the Cauc h y form ula reads J n f ( t ) := f n ( t ) = 1 ( n − 1)! Z t 0 ( t − τ ) n − 1 f ( τ ) dτ , t > 0 , n ∈ I N , (1 . 1) where I N is t he set of positive integers. F rom this de ﬁnition w e note that f n ( t ) v anishes a t t = 0 with its deriv a t iv es o f order 1 , 2 , . . . , n − 1 . F or conv ention we require that f ( t ) and henceforth f n ( t ) b e a c ausal function, i.e. iden tically v anishing for t < 0 . In a natural w ay one is l ed to extend t he ab ov e form ula from p ositiv e in teger v alues of the index to an y p ositiv e real v alues b y using t he Gamma function. Indeed, noting that ( n − 1 ) ! = Γ( n ) , and introducing t he arbitrary p ositive real n um b er α , one deﬁnes the F r actional Inte gr al of or der α > 0 : J α f ( t ) := 1 Γ( α ) Z t 0 ( t − τ ) α − 1 f ( τ ) dτ , t > 0 α ∈ I R + , (1 . 2) where I R + is the set of p ositive real n um b ers. F or complemen tation w e deﬁne J 0 := I (Iden tit y op erator) , i .e. w e mean J 0 f ( t ) = f ( t ) . F urthermore, b y J α f (0 + ) w e mean the limit ( i f it exists) of J α f ( t ) for t → 0 + ; t his limit may b e inﬁnite. Remark 1 : Here, and in al l our foll o wing treatment, t he integrals are in tended i n the ge ne r alize d Riemann sense, so that an y function is required to b e lo c al ly absolutely i n tegrable in I R + . Ho w ever, w e wil l not b ot her to give descriptions of sets of admissible func- tions and will not hesitate, when necessary , to use formal expressions with generalized functions ( distributions ), whic h, as far as p ossible, wi l l b e re-in terpreted in the frame- w ork of classical functions. The reader interes ted i n the strict mat hemati cal ri gor is referred to [5], where the fractional calculus is t reat ed in the framew ork of Leb esgue spaces of summable functions and Sob o lev spaces of generalized functions. Remark 2 : In order to remain in accordance with the standard notation I for the Iden tity op er- ator we use t he character J for the in tegral op erator and its p ow er J α . If one likes to denote by I α the integral op erators, he would adopt a diﬀeren t notati on for the Iden tit y , e.g. I I , t o av oid a p ossible confusion. 226 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e note t he semigr oup pr op erty J α J β = J α + β , α , β ≥ 0 , (1 . 3) whic h implies the c ommutative pr op erty J β J α = J α J β , and the eﬀect of our o p era- tors J α on the p ow er functions J α t γ = Γ( γ + 1) Γ( γ + 1 + α ) t γ + α , α > 0 , γ > − 1 , t > 0 . (1 . 4) The prop erties (1.3-4) are of course a natural generalization of those known when the order is a p ositive in teger. The pro o fs, see e. g. [ 2 ], [5 ] or [10], a re based on the prop erties of the tw o Eulerian in tegrals, i.e. the Gamma and Beta functions, Γ( z ) := Z ∞ 0 e − u u z − 1 du , Γ( z + 1) = z Γ( z ) , Re { z } > 0 , (1 . 5) B ( p, q ) := Z 1 0 (1 − u ) p − 1 u q − 1 du = Γ( p ) Γ( q ) Γ( p + q ) = B ( q , p ) , Re { p , q } > 0 . (1 . 6) It may b e conv enient to in tro duce the following causal function Φ α ( t ) : = t α − 1 + Γ( α ) , α > 0 , (1 . 7) where the suﬃx + is just denoting that the function is v anishing for t < 0 . Being α > 0 , this function turns out to b e lo c al ly a bsolutely in tegrable in I R + . Let us now recall the notion of L aplac e c onvolution , i. e. the conv o lution in tegral with tw o causal functions, whic h reads in a standard notation f ( t ) ∗ g ( t ) := R t 0 f ( t − τ ) g ( τ ) dτ = g ( t ) ∗ f ( t ) . Then w e note from (1.2) and (1.7 ) that the fractional in tegral of order α > 0 can b e considered as the Laplace con v olution b etw een Φ α ( t ) and f ( t ) , i.e. J α f ( t ) = Φ α ( t ) ∗ f ( t ) , α > 0 . (1 . 8) F urthermore, based on the Eulerian in tegrals, one prov es the c omp osition rule Φ α ( t ) ∗ Φ β ( t ) = Φ α + β ( t ) , α , β > 0 , (1 . 9) whic h can b e used to re-obtain (1.3) and (1.4). In tro ducing the Laplace transform by the notation L { f ( t ) } := R ∞ 0 e − st f ( t ) dt = e f ( s ) , s ∈ C , and using the sig n ÷ to denote a Laplace transform pair, i. e . f ( t ) ÷ e f ( s ) , w e note t he foll owing rule for the Laplace transform of the fractional in tegral, J α f ( t ) ÷ e f ( s ) s α , α > 0 , (1 . 10) whic h is the straig htforw ard generalization of t he case with an n -fold rep eated in tegral ( α = n ). F or the pro of it is suﬃcien t to recall the conv oluti on theorem for Laplace transforms a nd note the pair Φ α ( t ) ÷ 1 /s α , wi th α > 0 , see e.g. Do etsc h [25]. R. Gor enﬂo and F. Maina r di 227 1.3 The F r actional Derivative After the notion of fractional in tegral, that of fractional deriv at i v e of order α ( α > 0 ) becomes a natural requiremen t and one is attempted to substitute α with − α in the ab o ve form ulas. Ho wev er, this generalization needs some care i n order to guaran tee the con v ergence of the in tegrals and preserv e the w ell known prop erties of the ordinary deriv ative of in teger order. Denoting b y D n with n ∈ I N , the op erator of the deriv ative of order n , we ﬁrst note that D n J n = I , J n D n 6 = I , n ∈ I N , (1 . 11) i.e. D n is left-inv erse (and not rig ht-in verse) t o the corresp onding in tegral op erator J n . In fact we easil y recognize from (1.1) that J n D n f ( t ) = f ( t ) − n − 1 X k =0 f ( k ) (0 + ) t k k ! , t > 0 . (1 . 12) As a consequence we exp ect that D α is deﬁned as left-inv erse to J α . F or this purpo se, in tro ducing the p ositiv e in teger m suc h that m − 1 < α ≤ m , one deﬁnes the F r ac ti onal Derivati v e of or de r α > 0 : D α f ( t ) := D m J m − α f ( t ) , namely D α f ( t ) :=          d m dt m  1 Γ( m − α ) Z t 0 f ( τ ) ( t − τ ) α +1 − m dτ  , m − 1 < α < m , d m dt m f ( t ) , α = m . (1 . 13) Deﬁning for complemen tati o n D 0 = J 0 = I , then w e easily recognize that D α J α = I , α ≥ 0 , (1 . 14) and D α t γ = Γ( γ + 1) Γ( γ + 1 − α ) t γ − α , α > 0 , γ > − 1 , t > 0 . (1 . 15) Of course, the prop erties (1.14-15) are a natural generalization of those known when the order is a p ositive in teger. Since i n (1.15) the a rgumen t of the Gamma function in the denominator can b e negative , w e need to consider the analytical con tinu ation of Γ( z ) in (1.5) to the left half-plane, see e.g. Henrici [26]. Note the remark able fact t hat the fractional deriv ative D α f is not zero for the constan t function f ( t ) ≡ 1 if α 6∈ I N . In fact, (1. 15) with γ = 0 teac hes us that D α 1 = t − α Γ(1 − α ) , α ≥ 0 , t > 0 . (1 . 16) This, of course, is ≡ 0 for α ∈ I N, due t o the pol es of the gamma function in t he p oin ts 0 , − 1 , − 2 , . . . . 228 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e now observ e that an al ternative deﬁnition of f ractional d eriv ative, orig- inally in tro duced b y Caputo [19], [27 ] in the late sixties and adopted by Ca- puto and Mainardi [28] in the framew ork of t he theory of Line ar Visc o elasticity (see a review i n [24]), is the so-called Caputo F r actional Deriva ti ve of or der α > 0 : D α ∗ f ( t ) := J m − α D m f ( t ) with m − 1 < α ≤ m , namely D α ∗ f ( t ) :=          1 Γ( m − α ) Z t 0 f ( m ) ( τ ) ( t − τ ) α +1 − m dτ , m − 1 < α < m , d m dt m f ( t ) , α = m . (1 . 17) This deﬁnition is of course more restrictive than (1.1 3 ), i n that requires the absolute in tegrability of the deriv ativ e of order m . Whenev er we use the op erator D α ∗ w e (tacitly) assume t hat this condition is met. W e easil y recognize that in g eneral D α f ( t ) := D m J m − α f ( t ) 6 = J m − α D m f ( t ) := D α ∗ f ( t ) , (1 . 18) unless the function f ( t ) along with its ﬁrst m − 1 deriv atives v anishes at t = 0 + . In fact, assuming t hat the passage of the m -deriv ative under the i n tegral i s legitimate, one recognizes that, for m − 1 < α < m and t > 0 , D α f ( t ) = D α ∗ f ( t ) + m − 1 X k =0 t k − α Γ( k − α + 1) f ( k ) (0 + ) , (1 . 19) and therefore, recalli ng the fractional deriv ative of the p ow er functions (1.15), D α f ( t ) − m − 1 X k =0 t k k ! f ( k ) (0 + ) ! = D α ∗ f ( t ) . (1 . 20) The alternativ e deﬁn ition (1.1 7) for the fractional deriv ativ e th us incorpora tes the initial v a l ues of the function and of its integer deriv atives of low er order. The sub- traction o f the T aylor p olynomial of degree m − 1 at t = 0 + from f ( t ) means a sort of regularization of t he fractional deriv ative. In particular, a ccording to this deﬁnition, the relev ant prop ert y for whic h the fractional deriv ative of a constant is still zero, i.e. D α ∗ 1 ≡ 0 , α > 0 . (1 . 21) can b e easily recognized. W e no w explore the most relev an t diﬀerences b et we en the t w o fractional deriv a- tive s (1.13 ) and (1. 17). W e agree to denote (1. 17) as the Caputo fr actional derivative to distinguish i t from t he sta ndard Riemann-Liouville fractional deriv ative (1.13). W e observ e, again by lo oking at (1.15) , that D α t α − 1 ≡ 0 , α > 0 , t > 0 . (1 . 22) R. Gor enﬂo and F. Maina r di 229 F rom (1.22) and ( 1 .21) w e thus recognize the follo wing statemen ts ab out functions whic h for t > 0 admit the same fractional deriv ative of order α , with m − 1 < α ≤ m , m ∈ I N , D α f ( t ) = D α g ( t ) ⇐ ⇒ f ( t ) = g ( t ) + m X j =1 c j t α − j , (1 . 23) D α ∗ f ( t ) = D α ∗ g ( t ) ⇐ ⇒ f ( t ) = g ( t ) + m X j =1 c j t m − j . (1 . 24) In these formulas the co eﬃcien ts c j are arbitra ry constan ts. Inciden t a lly , w e note that (1. 2 2) pro vides an instructiv e example to show ho w D α is not right-in verse to J α , since J α D α t α − 1 ≡ 0 , but D α J α t α − 1 = t α − 1 , α > 0 , t > 0 , . ( 1 . 25) F or the t w o deﬁnitions w e also note a diﬀerence with resp ect to the formal li mit as α → ( m − 1) + . F rom (1.13) and (1.17 ) w e obtain resp ectiv ely , α → ( m − 1) + = ⇒ D α f ( t ) → D m J f ( t ) = D m − 1 f ( t ) ; (1 . 26) α → ( m − 1) + = ⇒ D α ∗ f ( t ) → J D m f ( t ) = D m − 1 f ( t ) − f ( m − 1) (0 + ) . (1 . 27) W e now consider the L aplac e tr ansform of the t wo fractional deriv atives. F or t he standard fractional deriv ative D α the Laplace tra nsform, assumed to exist, requires the kno wledge of the (b ounded) initial v alues o f the fractional in tegral J m − α and of its integer deriv ativ es of order k = 1 , 2 , m − 1 , a s w e learn from [2] , [5], [10]. The correspo nding rule reads, in our notation, D α f ( t ) ÷ s α e f ( s ) − m − 1 X k =0 D k J ( m − α ) f (0 + ) s m − 1 − k , m − 1 < α ≤ m . (1 . 28) The Caputo fr acti onal derivative app ears more suitable to b e treated b y the Laplace transform tech nique in that it requires the kno wledge of t he (b ounded) ini- tial v al ues of the function and of its integer deriv atives of order k = 1 , 2 , m − 1 , in analogy wi th the case when α = m . In fact, by using (1.10) and noting t hat J α D α ∗ f ( t ) = J α J m − α D m f ( t ) = J m D m f ( t ) = f ( t ) − m − 1 X k =0 f ( k ) (0 + ) t k k ! . (1 . 29) w e easily prov e the foll owing rule for the Laplace transform, D α ∗ f ( t ) ÷ s α e f ( s ) − m − 1 X k =0 f ( k ) (0 + ) s α − 1 − k , m − 1 < α ≤ m , (1 . 30) Indeed, the result (1.30), ﬁrst stated by Caputo [19] by using the F ubini-T onelli theorem, appears a s the most ”natural” generalizatio n of the corresponding result w ell kno wn for α = m . 230 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e no w show how b oth the deﬁnitions (1.13) and (1.17) for the fractional deriv a- tive of f ( t ) can b e deriv ed, at least formal ly , b y the con v olution of Φ − α ( t ) with f ( t ) , in a sort of analogy with (1. 8) for the fracti o nal in tegral. F or this purp ose we need to recall from t he treati se on g eneralized functions by Gel’fand and Shilov [16] that (with prop er in terpretation of the quotient as a limi t if t = 0) Φ − n ( t ) := t − n − 1 + Γ( − n ) = δ ( n ) ( t ) , n = 0 , 1 , . . . (1 . 31) where δ ( n ) ( t ) denotes the generalized deriv ati ve of order n o f the Dirac delta distribu- tion. Here, w e assume that the reader has some minimal k no wledge concerning these distributions, suﬃcien t for handling classical problems in physics and engineering. The equation (1.31) pro v ides an in teresting (not so well known ) represen tation of δ ( n ) ( t ) , which is useful in our follo wing treatmen t of fractional deriv atives. In fact, w e note that the deriv at iv e of order n of a causal function f ( t ) can b e obtained formal ly b y the (generalized) con volution b etw een Φ − n and f , d n dt n f ( t ) = f ( n ) ( t ) = Φ − n ( t ) ∗ f ( t ) = Z t + 0 − f ( τ ) δ ( n ) ( t − τ ) dτ , t > 0 , ( 1 . 32) based on the w ell known prop erties Z t + 0 − f ( τ ) δ ( n ) ( τ − t ) dτ = ( − 1) n f ( n ) ( t ) , δ ( n ) ( t − τ ) = ( − 1) n δ ( n ) ( τ − t ) . (1 . 33) According t o a usual con v en tion, i n (1.32-33) the limits of in tegration are extended to take i n to accoun t for the p ossibilit y of impulse functions cen tred at the extremes. Then, a formal deﬁnition of the fractional deriv a tive o f order α could b e Φ − α ( t ) ∗ f ( t ) = 1 Γ( − α ) Z t + 0 − f ( τ ) ( t − τ ) 1+ α dτ , α ∈ I R + . The formal c haracter is eviden t in that the k ernel Φ − α ( t ) turns out to b e not lo cally absolutely integrable and consequen tly the integral i s in general div ergen t. In order to obtain a deﬁnition that is still v alid for classical functions, we need to r e gularize t he div ergen t integral in some wa y . F o r t his purp ose let us consider the integer m ∈ I N suc h that m − 1 < α < m and write − α = − m + ( m − α ) or − α = ( m − α ) − m . W e then o btain [Φ − m ( t ) ∗ Φ m − α ( t )] ∗ f ( t ) = Φ − m ( t ) ∗ [Φ m − α ( t ) ∗ f ( t )] = D m J m − α f ( t ) , ( 1 . 34) or [Φ m − α ( t ) ∗ Φ − m ( t )] ∗ f ( t ) = Φ m − α ( t ) ∗ [Φ − m ( t ) ∗ f ( t )] = J m − α D m f ( t ) . ( 1 . 35) As a consequence we deriv e tw o alternative deﬁnitions for the fractional deriv ati v e, correspo nding to (1.13) a nd (1.17), resp ectively . The si ngular b eha viour of Φ − m ( t ) is reﬂected in the non-comm utativity of con v olution in these form ulas. R. Gor enﬂo and F. Maina r di 231 1.4 Other Deﬁnitions and Notations Up to no w w e ha ve considered the approac h to fractional calculus usually re- ferred to Riemann and Liouvil le. Ho w ever, while Riemann (1847) had generalized the i n tegral Cauc h y form ula with starting p oint t = 0 as rep ort ed i n (1.1), originally Liouville (1 832) had ch osen t = −∞ . In this case w e deﬁne J α −∞ f ( t ) := 1 Γ( α ) Z t −∞ ( t − τ ) α − 1 f ( τ ) dτ , α ∈ I R + , (1 . 36) and consequen tl y , for m − 1 < α ≤ m , m ∈ I N , D α −∞ f ( t ) := D m J m − a −∞ f ( t ) , namely D α −∞ f ( t ) :=        d m dt m  1 Γ( m − α ) Z t −∞ f ( τ ) dτ ( t − τ ) α +1 − m  , m − 1 < α < m , d m dt m f ( t ) , α = m . (1 . 37) In this case, assuming f ( t ) to v anish as t → −∞ along with its ﬁrst m − 1 deriv atives, w e ha ve t he iden tit y D m J m − α −∞ f ( t ) = J m − α −∞ D m f ( t ) , (1 . 38) in contrast with (1.18) . While for the fractional integral (1.2 ) a suﬃcien t condition that the integral con- v erge i s that f ( t ) = O  t ǫ − 1  , ǫ > 0 , t → 0 + , (1 . 39) a suﬃcien t condition t hat (1 .36) con ve rge is that f ( t ) = O  | t | − α − ǫ  , ǫ > 0 , t → −∞ . (1 . 40) In tegrable functions satisfying the properties (1.39 ) and (1.40 ) are sometimes referred to as functions o f Riemann class and Liouville class, resp ectiv ely [10]. F or example p o we r functions t γ with γ > − 1 and t > 0 (and hence also constants) are of Riemann class, while | t | − δ with δ > α > 0 and t < 0 and exp ( ct ) with c > 0 are of Liouvil le class. F or the ab ov e functions w e obtain (as real v ersions of the formu las given in [10]) J α −∞ | t | − δ = Γ( δ − α ) Γ( δ ) | t | − δ + α , D α −∞ | t | − δ = Γ( δ + α ) Γ( δ ) | t | − δ − α , (1 . 41) and J α −∞ e ct = c − α e ct , D α −∞ e ct = c α e ct . (1 . 42) 232 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der Causal functions can b e considered in the ab ov e in tegrals with the due care. In fact, in view of the p ossible jump discon tinuities of the in tegrands at t = 0 , in this case i t is worth while to write Z t −∞ ( . . . ) dτ = Z t 0 − ( . . . ) dτ . As an exa mple we consider for 0 < α < 1 t he c hain of iden tities 1 Γ(1 − α ) Z t 0 − f ′ ( τ ) ( t − τ ) α dτ = t − α Γ(1 − α ) f (0 + ) + 1 Γ(1 − α ) Z t 0 f ′ ( τ ) ( t − τ ) α dτ = t − α Γ(1 − α ) f (0 + ) + D α ∗ f ( t ) = D α f ( t ) , (1 . 43) where we hav e used (1.19) with m = 1 . In recen t years it has b ecome customary to use in place of (1.36) t he W ey l frac- tional integral W α ∞ f ( t ) := 1 Γ( α ) Z ∞ t ( τ − t ) α − 1 f ( τ ) dτ , α ∈ I R + , (1 . 44) based on a deﬁnition o f W eyl (1917). F or t > 0 it is a sort of complemen tary integral with resp ect to the usual Riemann-Liouville integral (1.2). The relation b et w een (1.36) and ( 1 .44) can b e readily obtained b y noti ng that, see e.g. [10], J α −∞ f ( t ) = 1 Γ( α ) Z t −∞ ( t − τ ) α − 1 f ( τ ) dτ = − 1 Γ( α ) Z − t ∞ ( t + τ ′ ) α − 1 f ( − τ ′ ) dτ ′ = 1 Γ( α ) Z ∞ t ′ ( τ ′ − t ′ ) α − 1 f ( − τ ′ ) dτ ′ = W α ∞ g ( t ′ ) , (1 . 45) with g ( t ′ ) = f ( − t ′ ) , t ′ = − t . In the ab o v e passages w e ha v e carried out the changes of v ariable τ → τ ′ = − τ and t → t ′ = − t . F or conv enience of the reader, let us recall that exhaustiv e tables of Riemann- Liouville and W eyl fractional integrals are a v ailable in the second v olume of the Bateman Pro ject collection of Integral T ransforms [16 ] , i n t he c hapter X I I I devoted to fr actional inte gr als . Last but not the least, let us consider the question of notat ion. The presen t authors opp ose to the use of the notation D − α for denoting the fractional integral, since it is misleading, ev en i f it is used in distinguished treatises as [2], [10], [ 15]. It is w ell known that deriv ation and integration op erators are not inv erse to eac h other, ev en if their order is in teger, and therefore suc h uniﬁcation of symbols, presen t only in the framework of the fractional calculus, app ears not justiﬁed. F urthermore, w e ha ve to keep in mind that for fractional order the deriv ative is yet an inte gr al op erator, so that, p erhaps, i t would b e less disturbing to denote our D α as J − α , than our J α as D − α . R. Gor enﬂo and F. Maina r di 233 1.5 The L aw of Exp onents In the ordinary calculus the prop erties of the op erators of in tegration and diﬀer- en tiation wi t h resp ect to the laws of commutation and additivit y of their ( i n teger) exp onen ts are w ell kno wn. Using our notat i on, t he (triv ial) la ws J m J n = J n J m = J m + n , D m D n = D n D m = D m + n , (1 . 46) where m, n = 0 , 1 , 2 , . . . , can b e referred to as the L aw of Exp onents for the op erators of in tegration a nd diﬀeren tiat ion of i n teger order, respectively . Of course, for an y p ositive in teger o rder, t he op erators D m and J n do not commute, see (1.11-12 ). In the fractional calculus the L aw of Exp onents is kno wn to b e generally true for the op erators of fr ac ti onal i nte gr ation thanks to their semigroup prop erty (1.3 ) . In general, b oth the op erators of fractional diﬀeren tiation, D α and D α ∗ , do not satisfy either the semigroup prop ert y , or the (weak er) comm utative prop ert y . T o show ho w the L aw of Exp onents do es not necessarily hold for t he standard fractional deriv ative, w e pro vide t w o simple examples (wi th p o w er functions) for whic h ( (a) D α D β f ( t ) = D β D α f ( t ) 6 = D α + β f ( t ) , (b) D α D β g ( t ) 6 = D β D α g ( t ) = D α + β g ( t ) . (1 . 47) In the example ( a ) let us take f ( t ) = t − 1 / 2 and α = β = 1 / 2 . Then, using (1.15) , w e get D 1 / 2 f ( t ) ≡ 0 , D 1 / 2 D 1 / 2 f ( t ) ≡ 0 , but D 1 / 2+1 / 2 f ( t ) = D f ( t ) = − t − 3 / 2 / 2 . In the example (b) let us tak e g ( t ) = t 1 / 2 and α = 1 / 2 , β = 3 / 2 . Then, again using (1 . 15), we get D 1 / 2 g ( t ) = √ π / 2 , D 3 / 2 g ( t ) ≡ 0 , but D 1 / 2 D 3 / 2 g ( t ) ≡ 0 , D 3 / 2 D 1 / 2 g ( t ) = − t 3 / 2 / 4 and D 1 / 2+3 / 2 g ( t ) = D 2 g ( t ) = − t 3 / 2 / 4 . Although mo dern mathemati cia ns w ould seek the conditions to justify the L aw of Exp onents when the order of diﬀeren t iation and integration are comp osed together, w e resist the temptati on to div e into the delicate details of the matter, but rat her refer the intere sted reader to § IV.6 (”The La w of Exp onen ts”) in the b o ok b y Mil l er and Ross [10]. Let us, ho w ev er, extract (in our notatio n, writing J α in place of D − α for α > 0) three imp ortant cases, con tained i n their Theorem 3: If f ( t ) = t λ η ( t ) or f ( t ) = t λ ln t η ( t ) , wher e λ > − 1 and η ( t ) = P ∞ n =0 a n t n having a p ositive r adi us R of c onver ge nc e, then for 0 ≤ t < R , the fol lowing thr e e f ormulas ar e valid:      µ ≥ 0 and 0 ≤ ν ≤ µ = ⇒ D ν J µ f ( t ) = J µ − ν f ( t ) , µ ≥ 0 and ν > µ = ⇒ D ν J µ f ( t ) = D µ − ν f ( t ) , 0 ≤ µ < λ + 1 and ν ≥ 0 = ⇒ D ν D µ f ( t ) = D µ + ν f ( t ) . (1 . 48) A t least in the case of f ( t ) without the factor ln t , the pro of of these formulas is straigh tforw ard. Use the deﬁnitions (1.2) and (1.13) of fractional in tegration and 234 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der diﬀeren tiation, the semigroup prop erty (1.3) of fractional in tegration, and apply the form ulas (1.4) and (1.15) term wise to the inﬁnite series you meet in the course of calculations. O f course, the condition that the function η ( t ) be analytic can be considerably relaxed; i t only need b e ”suﬃcien tly” smo oth.” The lack of comm utativ i t y and the non-v alidity of the law of exp onen ts has led to the notio n of se quential fr actional diﬀer entiation in whic h the order in whic h fractional diﬀeren tiation op erators D α 1 , D α 2 , . . . , D α k are concatenated is crucial. F or this and t he rela t ed ﬁeld of fractional diﬀeren tial equations w e refer a g ain to Miller a nd Ross [1 0 ]. F urthermore, P o dlubn y [29] has also giv en form ulas for the Laplace transforms of sequen tial fractional deriv atives. In order to giv e an impression on t he strange eﬀects to b e exp ected in use of sequen t ial fractional deriv atives we consider for a function f ( t ) contin uous for t ≥ 0 and for p ositive num b ers α and β with α + β = 1 the three problems ( a ) , ( b ), ( c ) with the respective general solutions u , v , w in the set o f l o cally integrable functions,      ( a ) D α D β u ( t ) = f ( t ) ⇒ u ( t ) = J f ( t ) + a 1 + a 2 t β − 1 , ( b ) D β D α v ( t ) = f ( t ) ⇒ v ( t ) = J f ( t ) + b 1 + b 2 t α − 1 , ( c ) D u ( t ) = f ( t ) ⇒ w ( t ) = J f ( t ) + c , (1 . 49) where a 1 , a 2 , b 1 , b 2 , c are arbitrary constan ts. Whereas the result for ( c ) is o bvious, in order to obtain the ﬁnal results for ( a ) [or ( b )] w e need to apply ﬁrst the op erat or J α [or J β ] and then the op erato r J β [or J α ]. The a dditional terms mu st b e taken in to accoun t b ecause D γ t γ − 1 ≡ 0 , J 1 − γ t γ − 1 = Γ( γ ) , γ = α, β . W e observe that, whereas the general solution of ( c ) contains one a rbit rary constan t, that of ( a ) and like wise of ( b ) con tains two arbitra ry constan ts, even though α + β = 1 . In case α 6 = β the singular b ehaviour of u ( t ) at t = 0 + is distinct from that of v ( t ) . F rom ab o v e w e can conclude in ro ugh w ords: suﬃcien tly ﬁne sequen tial ization increases the n um b er of free constan ts i n the general sol ution of a fractional diﬀer- en tial equation, hence the n um b er of conditions that m ust b e imp osed to make the solution unique. F or an example see Bagley ’s treatment of a comp osite fractional os- cillation equation [30]; there the highest order of deriv ative is 2, but four conditions are required to ach ieve uniqueness. In the presen t l ectures we shall av oi d t he ab ov e troubles since we shall consider only diﬀeren tial equations con taining single fractional deriv atives. F urthermore w e shall adopt the Caputo fr actional derivative in order to meet the usual physical re- quiremen ts for which the initial conditions a re expressed in terms of a gi ven n um b er of b ounded v alues assumed b y the ﬁeld v aria ble a nd it s deriv ativ es of integer order, see ( 1 .24) and (1. 30). R. Gor enﬂo and F. Maina r di 235 2. FRA CTIONAL IN TEGRAL EQUA TIONS In this section w e shall consider the most simple in tegral equations of fractional order, namely the A b el in tegral equations of the ﬁrst and the second k i nd. The former in v estigatio ns on suc h equations are due to Ab el ( 1823-26), after whom they are named, for t he ﬁrst kind, and to Hille and T amarkin (1930) for the second kind. The in terested reader is referred to [ 5], [ 2 1] and [ 31-33] for historical notes and detailed analysis with applicati ons. Here w e l imit ourselv es to put some emphasis on t he metho d of the Laplace transforms, that makes easier and more comprehensible t he treatmen t o f suc h fractional integral equations, and pro vide some applicatio ns. 2.1 Ab el inte gr al e quation of the ﬁrst kind Let us consider the Ab el integral equation of the ﬁrst kind 1 Γ( α ) Z t 0 u ( τ ) ( t − τ ) 1 − α dτ = f ( t ) , 0 < α < 1 , (2 . 1) where f ( t ) is a given func tion. W e easily recognize that this equati on can b e expressed in terms of a fractional in tegral, i.e. J α u ( t ) = f ( t ) , (2 . 2) and consequen tly solve d in terms of a fractional deriv ative , according to u ( t ) = D α f ( t ) . (2 . 3) T o this end w e need to recall the deﬁnition (1.2) and the prop ert y (1.14) D α J α = I . Let us no w solv e (2. 1 ) using the Laplace transform. Noting from (1.7 -8) and (1.10) that J α u ( t ) = Φ α ( t ) ∗ u ( t ) ÷ ˜ u ( s ) /s α , we then obtain ˜ u ( s ) s α = ˜ f ( s ) = ⇒ ˜ u ( s ) = s α ˜ f ( s ) . (2 . 4) No w we can choose t w o diﬀeren t wa ys to get the in vers e Laplace transform from (2.4), according to the standard rules. W riting (2.4) a s ˜ u ( s ) = s " ˜ f ( s ) s 1 − α # , (2 . 4 a ) w e obtain u ( t ) = 1 Γ(1 − α ) d dt Z t 0 f ( τ ) ( t − τ ) α dτ . (2 . 5 a ) On t he ot her hand, writing ( 2 .4) as ˜ u ( s ) = 1 s 1 − α [ s ˜ f ( s ) − f (0 + )] + f (0 + ) s 1 − α , (2 . 4 b ) w e obtain u ( t ) = 1 Γ(1 − α ) Z t 0 f ′ ( τ ) ( t − τ ) α dτ + f (0 + ) t − α Γ(1 − a ) . (2 . 5 b ) 236 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der Th us, the soluti ons (2.5a) and (2.5b) are expressed in terms of the fractional deriv a- tive s D α and D α ∗ , resp ectively , according to (1.1 3), (1.17) and (1.19) with m = 1 . The w a y b ) requires that f ( t ) b e diﬀeren tiable with L -transformable deriv ativ e; consequen tl y 0 ≤ | f (0 + ) | < ∞ . Then it turns out from (2.5b) that u (0 + ) can b e inﬁnite i f f (0 + ) 6 = 0 , b eing u ( t ) = O ( t − α ) , a s t → 0 + . The wa y a ) requires w eake r conditions in that the in tegral at t he R.H.S. of (2. 5a) m ust v anish as t → 0 + ; conse- quen tly f ( 0 + ) could b e inﬁnite but wi th f ( t ) = O ( t − ν ) , 0 < ν < 1 − α as t → 0 + . T o this end keep i n mind that Φ 1 − α ∗ Φ 1 − ν = Φ 2 − α − ν . Then it turns o ut from (2. 5a) that u ( 0 + ) can b e inﬁnite if f (0 + ) i s inﬁnite, b eing u ( t ) = O ( t − ( α + ν ) ) , as t → 0 + . Finally , let us remark that we can analog ously treat the case of equation (2.1) with 0 < α < 1 replaced by α > 0 . If m − 1 < α ≤ m with m ∈ I N , then again we ha v e (2.2), no w with D α f ( t ) given b y the formula (1.13) whic h can also b e obtained b y the Lapla ce tra nsform metho d. 2.2 Ab el inte gr al e quation of the se c ond k ind Let us now consider the Ab el equation o f t he second kind u ( t ) + λ Γ( α ) Z t 0 u ( τ ) ( t − τ ) 1 − α dτ = f ( t ) , α > 0 , λ ∈ C . (2 . 6) In terms of the fractional in tegral op erator suc h eq uation reads (1 + λ J α ) u ( t ) = f ( t ) , (2 . 7) and consequen tly can b e formal ly solved as follows : u ( t ) = (1 + λJ α ) − 1 f ( t ) = 1 + ∞ X n =1 ( − λ ) n J αn ! f ( t ) . (2 . 8) Noting by (1. 7-8) that J αn f ( t ) = Φ αn ( t ) ∗ f ( t ) = t αn − 1 + Γ( αn ) ∗ f ( t ) the formal soluti on reads u ( t ) = f ( t ) + ∞ X n =1 ( − λ ) n t αn − 1 + Γ( αn ) ! ∗ f ( t ) . (2 . 9) Recalling from the App endix the deﬁnition of the function, e α ( t ; λ ) := E α ( − λ t α ) = ∞ X n =0 ( − λ t α ) n Γ( αn + 1 ) , t > 0 , α > 0 , λ ∈ C , (2 . 10) where E α denotes the Mitta g-Leﬄer function of order α , w e note that ∞ X n =1 ( − λ ) n t αn − 1 + Γ( αn ) = d dt E α ( − λt α ) = e ′ α ( t ; λ ) , t > 0 . (2 . 11) R. Gor enﬂo and F. Maina r di 237 Finally , the sol ution reads u ( t ) = f ( t ) + e ′ α ( t ; λ ) ∗ f ( t ) . (2 . 12) Of course the ab ov e formal pro of can b e made ri gorous. Simply observ e that b ecause of the rapid gro wth of the gamma function the inﬁnite series i n (2. 9) and (2.11) are uniformly con v ergen t i n ev ery b ounded interv al of the v ariable t so that term-wise in tegrations and diﬀeren tiati o ns are allow ed. How ev er, w e prefer to use the alternati ve t ec hnique of Laplace transforms, whic h will allow us to obtain the solution i n diﬀeren t forms, including the result (2.12). Applying the Laplace transform to (2.6) w e obtain  1 + λ s α  ˜ u ( s ) = ˜ f ( s ) = ⇒ ˜ u ( s ) = s α s α + λ ˜ f ( s ) . (2 . 13) No w, let us pro ceed to obtai n the i nv erse Laplace transform of (2.13) using the follo wing Laplace transform pair (see App endix) e α ( t ; λ ) := E α ( − λ t α ) ÷ s α − 1 s α + λ . (2 . 14) As for the Ab el equatio n o f t he ﬁrst k i nd, w e can ch o o se tw o diﬀeren t wa y s t o get the i n v erse Laplace transforms from (2.13), according to the standard rules. W riting (2.13) as ˜ u ( s ) = s  s α − 1 s α + λ ˜ f ( s )  , (2 . 13 a ) w e obtain u ( t ) = d dt Z t 0 f ( t − τ ) e α ( τ ; λ ) dτ . (2 . 15 a ) If w e writ e (2. 13) as ˜ u ( s ) = s α − 1 s α + λ [ s ˜ f ( s ) − f (0 + )] + f (0 + ) s α − 1 s α + λ , (2 . 13 b ) w e obtain u ( t ) = Z t 0 f ′ ( t − τ ) e α ( τ ; λ ) dτ + f (0 + ) e α ( t ; λ ) . (2 . 15 b ) W e also note that, e α ( t ; λ ) b eing a function diﬀer en tiable wit h respect to t with e α (0 + ; λ ) = E α (0 + ) = 1 , there exists anot her p ossibility to re-write (2.13), namely ˜ u ( s ) =  s s α − 1 s α + λ − 1  ˜ f ( s ) + ˜ f ( s ) . (2 . 13 c ) Then we obtain u ( t ) = Z t 0 f ( t − τ ) e ′ α ( τ ; λ ) dτ + f ( t ) , (2 . 15 c ) in agreemen t wit h (2.12 ) . W e see t hat the w ay b ) i s more restrictive than the wa y s a ) and c ) since it requires t hat f ( t ) b e diﬀeren tiable with L -transformable deriv ative. 238 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der 2.3 Some applic ations of A b el inte g r al e quations It is well known t hat N iels H enrik A b el w as led to his famous equation by the mec hanical problem of t he tauto chr one , that is by the problem of determining the shap e of a curv e in the vertical plane suc h that the ti me required for a particle t o slide do wn the curv e to its low est p oi n t is equal to a given function of its initial heigh t (whic h is considered as a v ariable i n an in t erv al [0 , H ]). After appropriate c hanges of v ariables he obtai ned his famous in tegral equation of ﬁrst kind wit h α = 1 / 2 . He did, ho w ev er, solve the general case 0 < α < 1 . See T ama rk in’s t ra nslation 1) of and commen ts to Ab el’s short pap er 2) . As a sp ecial case Ab el discussed the problem of the iso chr one , in which it is required that the time of sliding do wn is independen t of the i nitial heigh t. Al ready in his earl ier publication 3) he recognized the solution as deriv ative of non-in teger order. W e p oi n t out that integral equati ons of Ab el typ e, including the simplest (2.1) and (2.6), hav e found so man y applications in diverse ﬁelds that it is al most imp ossible to pro vide an exhaustive list of t hem. Ab el in tegral equations o ccur in man y situations where ph y sical measuremen ts are to b e ev aluat ed. In many of these the independen t v ariable is the radius of a circle or a sphere and only after a c hange of v ariables the in tegral o p erator has the form J α , usually with α = 1 / 2 , and the equation i s of ﬁrst kind. Applications are, e.g. , in ev al uat i on of sp ectroscopic measuremen ts of cylindrical gas disch arges, t he study of the solar or a planetary a tmosphere, the in v estigation of star densities in a globular cluster, the in ve rsion of tra ve l times of seismic w a ve s for determination of t errestrial sub-surface structure, spherical stereology . Descriptions and analy sis of sev eral problems of this kind can b e found in the b o oks by Gorenﬂo and V essella [21] and by Craig and Bro wn [31 ] , see also [32] . Equati ons of ﬁrst and of second k ind, depending on the arrangemen t of the measuremen t s, arise in spherical stereology . See [3 3 ] where an analysis o f the basic problems and man y refere nces to previous literature are given. 1) Abel, N. H.: Solution of a mec hanical pr oblem. T ranslated from the German. In: D. E. Smith, editor: A Source B o ok in Mathematics, pp. 6 56-66 2. Dov er Publications, New Y ork, 1959 . 2) Abel, N. H.: Auﬂo esung einer mec hanischen Aufgab e. Journal f ¨ ur die reine und angewandte Mathematik (Crelle), V ol. I (18 26), pp. 1 53-15 7. 3) Abel, N. H.: So lution de quelques probl` emes ` a l’aide d’int ´ egrales d´ eﬁnies. T rans la ted from the Norwegian original, published in Maga zin for Naturvidensk a ber ne. Aarg ang 1 , Bind 2, Christiana 1823. F rench T ra nslation in Oeuvr es Compl` etes, V o l I, pp. 11-18. Nouvelle ` edition par L. Sylow et S. Lie, 1 881. R. Gor enﬂo and F. Maina r di 239 Another ﬁeld in w hich Ab el integral equations or in tegral equations with more general we akl y singular k ernels are imp ortant is that of inverse b oundary value pr ob- lems in partial diﬀerential equations, i n particular parab olic o nes in whic h naturally the indep enden t v ariable has the meaning of time. W e a re going to describe in detail the o ccurrence o f Ab el in tegral equati ons of ﬁrst and of second k ind in the problem of heating (or co ol ing) of a semi-inﬁnite ro d b y inﬂux (or eﬄux) of heat across t he b oundary in to (or from) i t s interior. Consider the e quation of he at ﬂow u t − u xx = 0 , u = u ( x, t ) , (2 . 16) in the semi-inﬁnite in terv als 0 < x < ∞ and 0 < t < ∞ of space and t i me, re- sp ectiv ely . In this dimensionless equati on u = u ( x, t ) means temperature. Assume v anishing initial temp erature, i. e. u ( x, 0) = 0 for 0 < x < ∞ and given inﬂux across the b oundary x = 0 from x < 0 to x > 0 , − u x (0 , t ) = p ( t ) . (2 . 17) Then, under appropriate regularit y conditions, u ( x, t ) is given b y the formula, see e.g. [ 34], u ( x, t ) = 1 √ π Z t 0 p ( τ ) √ t − τ e − x 2 / [4( t − τ ) ] dτ , x > 0 , t > 0 . (2 . 18) W e turn our sp ecial i n terest t o the i n terior b oundary temp erature φ ( t ) := u (0 + , t ) , t > 0 , whic h b y (2.18) i s represen ted as 1 √ π Z t 0 p ( τ ) √ t − τ dτ = J 1 / 2 p ( t ) = φ ( t ) , t > 0 . (2 . 19) W e recognize ( 2.19) as an Ab el integral equati on of ﬁrst kind for de termina ti on of an unknown inﬂux p ( t ) if the interior b oundary temp er atur e φ ( t ) is given b y measure- men ts, or inten ded to b e achiev ed b y controlling the inﬂux. Its solution is given b y form ula (1.13) wit h m = 1 , α = 1 / 2 , as p ( t ) = D 1 / 2 φ ( t ) = 1 √ π d dt Z t 0 φ ( τ ) √ t − τ dτ . (2 . 20) It may b e il luminating to consider the follo wing special cases,      ( i ) φ ( t ) = t = ⇒ p ( t ) = 1 2 √ π t , ( ii ) φ ( t ) = 1 = ⇒ p ( t ) = 1 √ π t , (2 . 21) where w e ha v e used formu la (1.15). So, for linear increase of interior b oundary temp erature the required inﬂux is contin uous a nd increasing from 0 tow ards ∞ ( w i th un b ounded deriv ative at t = 0 + ), whereas for instantaneous jump-like increase from 0 t o 1 the required i nﬂux decreases from ∞ at t = 0 + to 0 as t → ∞ . 240 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e now mo dify our problem to obtain an Ab el in tegral equatio n of se c ond k i nd . Assume that the ro d x > 0 is b ordered at x = 0 by a bath of liquid in x < 0 with con trolled ex terior b oundary temp erature u (0 − , t ) := ψ ( t ) . Assuming Newton’s radiat ion la w we ha v e an inﬂux of heat from 0 − to 0 + pro- p ortional to t he diﬀerence of exterior and i nterior temp erature, p ( t ) = λ [ ψ ( t ) − φ ( t )] , λ > 0 . (2 . 22) Inserting (2.22) i n to (2.19) w e obtain φ ( t ) = λ √ π Z t 0 ψ ( τ ) − φ ( τ ) √ t − τ dτ , namely , in op erator notat i on,  1 + λ J 1 / 2  φ ( t ) = λ J 1 / 2 ψ ( t ) . (2 . 23) If we now assume the exterior b oundary temp e r atur e ψ ( t ) as given and the evolution in time of the interior b oundary temp er atur e φ ( t ) as unknown , then (2 .23) is an A b el in tegral equation of se c ond kind for determination of φ ( t ) . With α = 1 / 2 the equation (2.23) is of the form (2.7), and b y (2.8) its solution is φ ( t ) = λ  1 + λJ 1 / 2  − 1 J 1 / 2 ψ ( t ) = − ∞ X m =0 ( − λ ) m +1 J ( m +1) / 2 ψ ( t ) . (2 . 24) Let us i nv estigate the v ery special case of constan t exterior b oundary temperat ure ψ ( t ) = 1 . (2 . 25) Then, b y (1 .4) wit h γ = 0 , J ( m +1) / 2 ψ ( t ) = t ( m +1) / 2 Γ [( m + 1) / 2 + 1] , hence φ ( t ) = − ∞ X m =0 ( − λ ) m +1 t ( m +1) / 2 Γ [( m + 1) / 2 + 1] = 1 − ∞ X n =0 ( − λ ) n t n/ 2 Γ ( n/ 2 + 1) , so that φ ( t ) = 1 − E 1 / 2  − λt 1 / 2  = 1 − e 1 / 2 ( t ; λ ) . (2 . 26) Observ e that φ ( t ) is creep function, increasing strictly monoto nical ly from 0 tow ards 1 a s t runs from 0 to ∞ . F or more or less distinct treatmen ts of this problem of ”Newtonian heating” the reader may consult [21 ], and [35-37]. In [37 ] a formulation in terms of fractional dif- feren ti al equations is derive d and, furthermore, t he analogous problem of ”N ewtonian co oling” is discussed. R. Gor enﬂo and F. Maina r di 241 3. FRA CTIONAL DIFFERENT IA L EQUA TIONS: 1-st P A R T W e no w analyse the most simple diﬀeren tial equations of fractional order. F or this purpo se, following our recen t w orks [37-42], we c ho ose the examples whic h, by means of fractional deriv at iv es, generalize the well-kno wn ordinary diﬀeren tial equations related to relaxat i on and oscilla t ion phenomena. In this section we treat t he simplest t yp es, whic h w e refer to as the simple fr actional r elaxation and oscil lation e quations . In the next section we shall consider the types, somewhat mo re cum b ersome, whic h w e refer t o a s the c omp osite fr actional r elaxation and osc i l lation e quations . 3.1 The s i mple fr actional r elaxation and oscil lation e quations The classical phenomena of relaxa t ion and oscillatio ns in their simplest form are kno wn to b e gov erned by l inear ordinary diﬀeren ti al equations, of order one and t w o respecti vely , that hereafter we recall with the corresponding solutions. Let us denote b y u = u ( t ) the ﬁeld v aria ble and b y q ( t ) a given con tin uous function, with t ≥ 0 . The r elaxation diﬀeren tial equation reads u ′ ( t ) = − u ( t ) + q ( t ) , (3 . 1) whose sol ution, under the initial condition u ( 0 + ) = c 0 , i s u ( t ) = c 0 e − t + Z t 0 q ( t − τ ) e − τ dτ . (3 . 2) The oscil lation diﬀeren t i al equation reads u ′′ ( t ) = − u ( t ) + q ( t ) , (3 . 3) whose sol ution, under the initial conditions u (0 + ) = c 0 and u ′ (0 + ) = c 1 , i s u ( t ) = c 0 cos t + c 1 sin t + Z t 0 q ( t − τ ) sin τ dτ . (3 . 4) F rom the p oin t of view of the fractional calculus a natural generalization o f eq ua- tions (3.1) and (3.3 ) is obtained b y replacing the ordinary deriv ative wi t h a fractional one o f order α . In order to preserv e the typ e of initi a l conditions required in the clas- sical phenomena, w e agree to replace the ﬁrst and second deriv ative in (3 .1) and (3.3 ) with a Caputo fractional deriv ative of order α with 0 < α < 1 and 1 < α < 2 , re- sp ectiv ely . W e agree to refer to the corresp onding equatio ns as the s imple fr actional r elaxation e quation and the simple fr actional os c il lation e quation . 242 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der Generally sp eaking, we consider the follo wing diﬀeren tia l equation of fractional order α > 0 , D α ∗ u ( t ) = D α u ( t ) − m − 1 X k =0 t k k ! u ( k ) (0 + ) ! = − u ( t ) + q ( t ) , t > 0 . (3 . 5) Here m is a p ositive in teger uniquely deﬁned b y m − 1 < α ≤ m , which pro vides the n um b er of the prescrib ed initial v alues u ( k ) (0 + ) = c k , k = 0 , 1 , 2 , . . . , m − 1 . Impli cit in the form of (3.5) is our desire to obtai n solutio ns u ( t ) for whic h the u ( k ) ( t ) are con tin uous for t ≥ 0 , k = 0 , 1 , . . . , m − 1 . In particular, the cases of fr actional r elaxation and fr acti onal oscil lation are obtai ned for m = 1 and m = 2 , resp ectively W e note that when α is the integer m the equation (3.5) reduces to an ordinary diﬀeren tial equat i on whose solution can b e expressed i n terms of m li nearly inde- p enden t solutions of t he homo ge ne ous equation and of o ne particular solution of the inhomo gene ous equation. W e summarize the w ell-kno wn result as follo ws u ( t ) = m − 1 X k =0 c k u k ( t ) + Z t 0 q ( t − τ ) u δ ( τ ) dτ . (3 . 6) u k ( t ) = J k u 0 ( t ) , u ( h ) k (0 + ) = δ k h , h, k = 0 , 1 , . . . , m − 1 , (3 . 7) u δ ( t ) = − u ′ 0 ( t ) . (3 . 8) Th us, the m functions u k ( t ) represen t the fundamental solutions of the diﬀeren tial equation of order m , namely those linearly indep enden t solutions of t he homo ge- ne ous equation whic h satisfy the initial conditions in (3.7) . The fun ction u δ ( t ) , with which t he free term q ( t ) app ears conv ol uted, represen ts the so called i mpulse- r esp onse solution , namely the particular solution of the inhomo gene ous equation wi t h all c k ≡ 0 , k = 0 , 1 , . . . , m − 1 , and with q ( t ) = δ ( t ) . In the cases of ordinary re- laxation and oscillat ion w e recognize that u 0 ( t ) = e − t = u δ ( t ) and u 0 ( t ) = cos t , u 1 ( t ) = J u 0 ( t ) = sin t = cos ( t − π / 2) = u δ ( t ) , resp ectiv ely . Remark 1 : The mo re general equation D α u ( t ) − m − 1 X k =0 t k k ! u ( k ) (0 + ) ! = − ρ α u ( t ) + q ( t ) , ρ > 0 , t > 0 , (3 . 5 ′ ) can b e reduced to (3.5) by a change of scale t → t/ρ . W e prefer, for ease of notation, to discuss t he ”dimensionless” form (3.5). Let us no w solv e (3.5) by the metho d of Laplace transforms. F or this purp ose w e can use directly the Caputo form ula (1.30) or, alternative ly , reduce (3.5) with the prescrib ed initial conditions as an equiv alen t ( fractional) i n tegral equation and then treat the i n tegral equatio n by the Laplace transform metho d. Here w e prefer to foll ow the second w ay . Then, apply i ng the op erator J α to b oth sides of ( 3.5) we obtain R. Gor enﬂo and F. Maina r di 243 u ( t ) = m − 1 X k =0 c k t k k ! − J α u ( t ) + J α q ( t ) . (3 . 9) The application of the Laplace t ransform y ields ˜ u ( s ) = m − 1 X k =0 c k s k +1 − 1 s α ˜ u ( s ) + 1 s α ˜ q ( s ) , hence ˜ u ( s ) = m − 1 X k =0 c k s α − k − 1 s α + 1 + 1 s α + 1 ˜ q ( s ) . (3 . 10) In tro ducing the Mittag-Leﬄer t yp e functions e α ( t ) ≡ e α ( t ; 1) := E α ( − t α ) ÷ s α − 1 s α + 1 , (3 . 11 ) u k ( t ) := J k e α ( t ) ÷ s α − k − 1 s α + 1 , k = 0 , 1 , . . . , m − 1 , (3 . 12) w e ﬁnd, from inv ersion of the Laplace transforms in (3. 10), u ( t ) = m − 1 X k =0 c k u k ( t ) − Z t 0 q ( t − τ ) u ′ 0 ( τ ) dτ . (3 . 13) F or ﬁnding the la st t erm in the R.H.S. of (3.13), we ha v e used the well-kno wn rule for the Laplace transform of the deriv ative noting that u 0 (0 + ) = e α (0 + ) = 1 , and 1 s α + 1 = −  s s α − 1 s α + 1 − 1  ÷ − u ′ 0 ( t ) = − e ′ α ( t ) . (3 . 14) The form ula (3.13) encompasses the solutions (3.2) and (3.4) found for α = 1 , 2 , respecti vely . When α is not integer, namely for m − 1 < α < m , w e not e that m − 1 represen ts the i n teger part of α (usually denoted by [ α ]) and m the n um b er of i nitial conditions necessary and suﬃcien t to ensure the uniqueness of the solution u ( t ) . Thu s the m functions u k ( t ) = J k e α ( t ) w i th k = 0 , 1 , . . . , m − 1 represen t those particular solutions of the homo gene ous equation whic h satisfy the initial conditions u ( h ) k (0 + ) = δ k h , h, k = 0 , 1 , . . . , m − 1 , (3 . 15) and t herefore t hey represen t the fundamental solu tions o f the fractional equation (3.5), in analogy with the case α = m . F urthermore, the fun ction u δ ( t ) = − e ′ α ( t ) represen ts the impulse-r esp onse s o lution . Hereafter, we are going to compute and exhibit the fundamental s olutions and t he impulse-r esp onse solution for the cases (a) 0 < α < 1 and (b) 1 < α < 2 , p oin ting out the comparison w i th the corresp onding solutions obtai ned when α = 1 and α = 2 . 244 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der Remark 2 : The reader is in vited to v erify that the solution ( 3.13) has con tin uous deriv atives u ( k ) ( t ) for k = 0 , 1 , 2 , . . . , m − 1 , whic h fulﬁll the m i nit ial conditions u ( k ) (0 + ) = c k . In fact, lo oking bac k at (3.9), one mu st recognize the smo othing p o we r of the op erat o r J α . Ho w ev er, the so called impulse-r es p onse solution o f our equation (3.5), u δ ( t ) , is exp ected to b e not so regular like the ordinary solution (3.13) . In fact, from (3.10) and (3.13-14), one obtai ns u δ ( t ) = − u ′ 0 ( t ) ÷ 1 s α + 1 , (3 . 16) and therefore, using the limiting theorem for Laplace transforms, o ne can recognize that, b eing m − 1 < α < m , u ( h ) δ (0 + ) = 0 , h = 0 , 1 , . . . , m − 2 ; u ( m − 1) δ (0 + ) = ∞ . (3 . 17) W e no w prefer to derive the relev ant prop erties of the basic functions e α ( t ) directly from their represen tation as a Laplace in v erse in tegral e α ( t ) = 1 2 π i Z B r e st s α − 1 s α + 1 ds , (3 . 18) in detail for 0 < α ≤ 2 , without detouring on the general theory of Mittag-Leﬄer functions in the complex plane. In (3.18) B r denotes the Bromwic h path, i.e . a line Re { s } = σ wit h a v alue σ ≥ 1 , and Im { s } running from −∞ to + ∞ . F or transparency reasons, we separately discuss the cases (a) 0 < α < 1 and (b) 1 < α < 2 , recalling that in the li mi ting cases α = 1 , 2 , w e know e α ( t ) as elemen tary function, namely e 1 ( t ) = e − t and e 2 ( t ) = cos t . F or α not integer the p o w er function s α is uniquely deﬁned a s s α = | s | α e i arg s , with − π < a rg s < π , that is in the complex s -plane cut along the negativ e real a x is. The essen tial step consists in decomp osing e α ( t ) in to tw o parts according to e α ( t ) = f α ( t ) + g α ( t ) , as i ndicated b elo w. In case (a ) t he function f α ( t ) , in case (b) t he fun ction − f α ( t ) is c ompletely monotone ; in b oth cases f α ( t ) tends t o zero as t tends to inﬁnit y , from ab ov e in case (a), from b elow in case (b). The other part, g α ( t ) , is iden ticall y v anishing in case (a), but of osci l latory c haracter wi th exp onen tially decreasing amplitude i n case (b). In order to obtain the desired decomp osition of e α w e b end the Bromwic h path of in tegration B r into the equiv alen t Hankel path H a (1 + ), a l o op w hich starts from −∞ along the l ow er side of the negative real axis, encircles t he circular disc | s | = 1 in the p o sitive sense and ends a t −∞ along the upp er side of the negative real axis. R. Gor enﬂo and F. Maina r di 245 One obtains e α ( t ) = f α ( t ) + g α ( t ) , t ≥ 0 , (3 . 19) with f α ( t ) : = 1 2 π i Z H a ( ǫ ) e st s α − 1 s α + 1 ds , (3 . 20) where now the Hankel path H a ( ǫ ) denotes a lo op constituted by a small circle | s | = ǫ with ǫ → 0 and b y the t wo b orders of the cut negative real axis, and g α ( t ) : = X h e s ′ h t Res  s α − 1 s α + 1  s ′ h = 1 α X h e s ′ h t , (3 . 21) where s ′ h are the relev a nt p oles of s α − 1 / ( s α + 1). In fact the p oles turn out to be s h = exp [ i (2 h + 1) π /α ] wi th unitary mo dulus; they are all simple but relev ant are only t hose si t uated i n the main Riemann sheet, i .e. the p oles s ′ h with argument suc h that − π < arg s ′ h < π . If 0 < α < 1 , there are no suc h p oles, since for al l i n tegers h w e ha v e | arg s h | = | 2 h + 1 | π /α > π ; as a consequence, g α ( t ) ≡ 0 , hence e α ( t ) = f α ( t ) , i f 0 < α < 1 . (3 . 22) If 1 < α < 2 , then there exist precisely t w o relev ant p oles, namely s ′ 0 = exp( iπ /α ) and s ′ − 1 = exp( − iπ /α ) = s 0 ′ , whic h are lo cated in the left half plane. Then o ne obtains g α ( t ) = 2 α e t cos ( π /α ) cos h t sin  π α i , if 1 < α < 2 . (3 . 23) W e note that this function exhibits oscillati ons with circular frequency ω ( α ) = sin ( π /α ) and wi th an exp onen tiall y deca ying ampli tude with rat e λ ( α ) = | cos ( π /α ) | . Remark 3 : One easily recognizes that (3.23) i s v al id al so for 2 ≤ α < 3 . In the classical case α = 2 the t w o p ol es are purely imaginary ( coinciding with ± i ) so that w e reco v er the sinu soidal b ehaviour with unitary frequency . In the case 2 < α < 3 , ho w ev er, the tw o p oles are lo cat ed in the right half plane, so pro viding ampliﬁed oscillations. This instabili ty , whic h is common to t he case α = 3 , i s the reason why w e limit ourselv es to consider α in the range 0 < α ≤ 2 . It is no w an exercise i n complex analysis to sho w t hat the con tribution from the Hank el path H a ( ǫ ) as ǫ → 0 is pro vided b y f α ( t ) := Z ∞ 0 e − r t K α ( r ) dr , (3 . 24) with K α ( r ) = − 1 π Im  s α − 1 s α + 1     s = r e iπ  = 1 π r α − 1 sin ( απ ) r 2 α + 2 r α cos ( απ ) + 1 . (3 . 25) 246 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der This function K α ( r ) v anishes iden tically if α is an in teger, it is p ositive for all r if 0 < α < 1 , negative for al l r i f 1 < α < 2 . In fact in (3. 2 5) the denominator is, for α not in teger, alwa ys p ositive b eing > ( r α − 1) 2 ≥ 0 . Hence f α ( t ) has the afore- men tioned monotonicit y prop erties, decreas ing t ow ards zero in case (a), increasing to w ards zero in case (b). W e also note that, i n o rder to satisfy t he initial condition e α (0 + ) = 1 , we ﬁnd R ∞ 0 K α ( r ) dr = 1 if 0 < α < 1 , R ∞ 0 K α ( r ) dr = 1 − 2 /α if 1 < α < 2 . In Fig. 1 w e sho w the plots of the sp e ctr al functions K α ( r ) for some v alues of α in the in terv als ( a ) 0 < α < 1 , (b) 1 < α < 2 . 0 0.5 1 1.5 2 0.5 1 K α (r) α =0.25 α =0.50 α =0.75 α =0.90 r Fig. 1a – Plo ts of the b asic sp e ctr al function K α ( r ) for 0 < α < 1 0 0.5 1 1.5 2 0.5 −K α (r) α =1.25 α =1.50 α =1.75 α =1.90 r Fig. 1b – Plo ts of the b as ic sp e ctr a l function − K α ( r ) for 1 < α < 2 R. Gor enﬂo and F. Maina r di 247 In a ddition to the basic fundamen tal solutions, u 0 ( t ) = e α ( t ) w e need to compute the i mpulse-response solutions u δ ( t ) = − D 1 e α ( t ) for cases (a ) and ( b) a nd, only in case (b), the second fundamen tal solution u 1 ( t ) = J 1 e α ( t ) . F or this purp ose w e note that i n general it turns out that J k f α ( t ) = Z ∞ 0 e − r t K α,k ( r ) dr , (3 . 26) with K α,k ( r ) : = ( − 1) k r − k K α ( r ) = ( − 1) k π r α − 1 − k sin ( απ ) r 2 α + 2 r α cos ( απ ) + 1 , (3 . 27) where K α ( r ) = K α, 0 ( r ) , and J k g α ( t ) = 2 α e t cos ( π /α ) cos h t sin  π α  − k π α i . ( 3 . 27) This can b e done i n direct analogy to the computation of the fun ctions e α ( t ), the Laplace t ransform of J k e α ( t ) b eing give n b y (3 .12). F or the impulse-resp onse solutio n w e note that the eﬀect of t he diﬀeren t ial op erator D 1 is the same as that of the virtual op erator J − 1 . In conclusion w e can resume the solutions for the fractional rela x ation and oscil- lation equati ons as follows: (a) 0 < α < 1 , u ( t ) = c 0 u 0 ( t ) + Z t 0 q ( t − τ ) u δ ( τ ) dτ , (3 . 28 a ) where        u 0 ( t ) = Z ∞ 0 e − r t K α, 0 ( r ) dr , u δ ( t ) = − Z ∞ 0 e − r t K α, − 1 ( r ) dr , (3 . 29 a ) with u 0 (0 + ) = 1 , u δ (0 + ) = ∞ ; (b) 1 < α < 2 , u ( t ) = c 0 u 0 ( t ) + c 1 u 1 ( t ) + Z t 0 q ( t − τ ) u δ ( τ ) dτ , (3 . 28 b ) where                  u 0 ( t ) = Z ∞ 0 e − r t K α, 0 ( r ) dr + 2 α e t cos ( π /α ) cos h t sin  π α i , u 1 ( t ) = Z ∞ 0 e − r t K α, 1 ( r ) dr + 2 α e t cos ( π /α ) cos h t sin  π α  − π α i , u δ ( t ) = − Z ∞ 0 e − r t K α, − 1 ( r ) dr − 2 α e t cos ( π /α ) cos h t sin  π α  + π α i , (3 . 29 b ) with u 0 (0 + ) = 1 , u ′ 0 (0 + ) = 0 , u 1 (0 + ) = 0 , u ′ 1 (0 + ) = 1 , u δ (0 + ) = 0 , u ′ δ (0 + ) = + ∞ . 248 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der 0 5 10 15 0.5 1 e α (t)=E α (−t α ) α =0.25 α =0.50 α =0.75 α =1 t Fig. 2a – Plo ts of the b asic fundamental s olution u 0 ( t ) = e α ( t ) for 0 < α ≤ 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 t 5 10 e α (t)=E α (−t α ) α =1.25 α =1.5 α =1.75 α =2 15 Fig. 2b – Plo ts of the b as ic fundamental solution u 0 ( t ) = e α ( t ) for 1 < α ≤ 2 : R. Gor enﬂo and F. Maina r di 249 In Fig. 2 we quote the plots of t he basic fundamen tal solution for the following cases : (a) α = 0 . 2 5 , 0 . 50 , 0 . 75 , 1 , and (b) α = 1 . 25 , 1 . 5 0 , 1 . 75 , 2 , obtained from the ﬁrst form ula in (3.29a) and (3.29b), resp ectively . W e ha ve veriﬁed t hat our presen t results conﬁrm those obtained b y Bl ank [43] b y a n umerical treatmen t and those obta i ned b y Mainardi [39] b y an analyti cal treatmen t, v alid when α i s a rational n um b er, see § A 2 of the App endix. Of particular in terest i s the case α = 1 / 2 where w e reco ve r a well-kno wn form ula of the Laplace transform theory , see ( A .34), e 1 / 2 ( t ) := E 1 / 2 ( − √ t ) = e t erfc( √ t ) ÷ 1 s 1 / 2 ( s 1 / 2 + 1) , (3 . 30) where erfc den otes the c omplementary err or function. W e no w desire to p oin t out that in b oth the cases (a) and (b) (in whic h α i s just not in teger) i.e. for f r actional r elaxation and fr acti onal osc il lation , all the fun damen tal and impulse-resp onse sol uti ons exhibit an algebr aic de c ay a s t → ∞ , as discussed b elo w. Let us start with the asymptotic b eha viour of u 0 ( t ) . T o this purp ose we ﬁrst deriv e an asymptotic series for the function f α ( t ), v al id for t → ∞ . Using the iden tity 1 s α + 1 = 1 − s α + s 2 α − s 3 α + . . . + ( − 1) N − 1 s ( N − 1) α + ( − 1) N s N α s α + 1 , in form ula (3.2 0 ) and the Hank el represen tat i on of the recipro cal Gamma function, w e (formally) obtai n the asymptotic expansion ( for α non in teger) f α ( t ) = N X n =1 ( − 1) n − 1 t − nα Γ(1 − nα ) + O  t − ( N +1) α  , as t → ∞ . (3 . 31) The v alidity of this asymptoti c expansion can b e established rigorously using the (generalized) W atson lemma, see [44]. W e also can start from t he spectral represen- tation ( 3 .24-25) and expand the sp ectral function for small r . Then the (ordinary) W atson lemma yields (3.31 ) . W e note that this asymptoti c expansion coincides with that for u 0 ( t ) = e α ( t ), ha ving assumed 0 < α < 2 ( α 6 = 1). In fact t he contribution of g α ( t ) is i den tically zero if 0 < α < 1 and exp onen t ially small as t → ∞ if 1 < α < 2 . The asymptotic expansions of the solutions u 1 ( t ) and u δ ( t ) are obtained from (3.31) integrating or diﬀeren tiat ing term by term with respect to t . In particular, taking the leading term in ( 3 .31), we obtai n the asymptotic represe n tati ons u 0 ( t ) ∼ t − α Γ(1 − α ) , u 1 ( t ) ∼ t 1 − α Γ(2 − α ) , u δ ( t ) ∼ − t − α − 1 Γ( − α ) , as t → ∞ , (3 . 32 ) that p oint o ut the algebraic decay of the fundamen tal and impulse-response solutions. In Fig. 3 we sho w some plots of the b asic fundamental solution u 0 ( t ) = e α ( t ) for α = 1 . 25 , 1 . 50 , 1 . 75 . Here t he alg ebraic deca y of t he fractional oscillatio n can b e recognized and compared with the t wo contributions pro vided by f α (monotonic b eha viour ) and g α ( t ) (exp onen tially damp ed oscillation). 250 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der −0.2 −0.1 0 0.1 0.2 5 g α (t) f α (t) e α (t) α =1.25 t 10 0 Fig. 3a – Decay of the b asi c fundamental s olution u 0 ( t ) = e α ( t ) for α = 1 . 25 −0.05 0 0.05 e α (t) f α (t) g α (t) 10 α =1.50 5 15 t Fig. 3b – Decay of the b asi c fundamental s olution u 0 ( t ) = e α ( t ) for α = 1 . 50 −1 −0.5 0 0.5 1 e α (t) f α (t) g α (t) 40 50 α =1.75 30 60 x 10 −3 t Fig. 3c – Deca y of t he b as ic f undamental solution u 0 ( t ) = e α ( t ) for α = 1 . 75 R. Gor enﬂo and F. Maina r di 251 3.2 The z e r os o f the solutions of the f r actional os cil lation e quation No w w e ﬁnd it inter esting to carry out some in v estigations ab o ut the zeros of the basic fundamen tal soluti o n u 0 ( t ) = e α ( t ) in the case (b) of fractional oscillat ions. F or the second fundamen t al solution and t he impulse-response solutio n t he analysis of the zeros can b e easily carried out anal o gously . Recalling the ﬁrst equation in (3 .29b), the required zeros of e α ( t ) are the sol uti ons of the equatio n e α ( t ) = f α ( t ) + 2 α e t cos ( π /α ) cos h t sin  π α i = 0 . (3 . 33) W e ﬁrst note that the function e α ( t ) exhibits an o dd n um b er of zeros, in that e α (0) = 1 , and, for suﬃcien tly large t , e α ( t ) turns o ut to b e p ermanen tly negativ e, as sho wn in (3 .32) b y the sign of Γ(1 − α ) . The smallest zero l i es in the ﬁrst p o sitivity in terv al of cos [ t sin ( π /α )] , hence in the in terv al 0 < t < π / [2 sin ( π /α )] ; all ot her zeros can only lie in the succeeding p o sitivity interv al s of cos [ t sin ( π /α )] , in eac h of these tw o zeros are presen t as long as 2 α e t cos ( π /α ) ≥ | f α ( t ) | . (3 . 34) When t is suﬃcien tly large the zeros are exp ected to b e found appro ximately from the equation 2 α e t cos ( π /α ) ≈ t − α | Γ(1 − α ) | , (3 . 35) obtained from (3.33) by ignoring the oscillation factor o f g α ( t ) [ see (3 . 23)] and taking the ﬁrst t erm i n the asymptotic ex pansion of f α ( t ) [see (3 . 31-32)]. As we hav e sho wn in [40 ] , such appro ximat e equation turns out to b e useful when α → 1 + and α → 2 − . F or α → 1 + , only one zero is presen t, which is exp ected to b e ve ry far from the origin i n view of the la rge p erio d of the function cos [ t sin ( π /α )] . In fact, since there is no zero for α = 1, and b y increasing α more and more zeros arise, w e are sure that only one zero exists for α suﬃcien tly close to 1. Putting α = 1 + ǫ the asymptotic p osition T ∗ of this zero can b e found from the rela tion (3.35) in the limit ǫ → 0 + . Assuming in this limit the ﬁrst-order approximation, w e get T ∗ ∼ log  2 ǫ  , (3 . 36) whic h sho ws that T ∗ tends to inﬁnit y slow er t han 1 /ǫ , as ǫ → 0 . F or details see [40]. 252 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der F or α → 2 − , there is an increasing n um b er of zeros up to inﬁnity since e 2 ( t ) = cos t has i nﬁnitely man y zeros [in t ∗ n = ( n + 1 / 2) π , n = 0 , 1 , . . . ]. Putting no w α = 2 − δ the a sy mpto t ic p osit ion T ∗ for t he l a rgest zero can b e found again from (3. 3 5) in the limit δ → 0 + . A ssuming in this limit the ﬁrst-order approximation, w e get T ∗ ∼ 12 π δ log  1 δ  . (3 . 37) F or detai l s see [40]. No w, for δ → 0 + the length of the p ositivi t y in terv als of g α ( t ) tends t o π and, as l ong a s t ≤ T ∗ , there are t w o zeros in eac h p ositivi ty in terv al. Hence, in the limit δ → 0 + , t here is in av erage one zero p er in terv al of length π , so w e exp ect that N ∗ ∼ T ∗ /π . Remark 4 : F or the ab ov e considerations w e got inspiration from an in teresting pap er b y Wiman [45] who at the b eginning of o ur cen tury , after having treated the Mittag- Leﬄer function in the complex plane, cons idered the p osition of the zeros of the function on the negati ve real a xis (wit hout pro viding an y de tail ) . Our expressions of T ∗ are in disagreemen t with those b y Wiman for numerical factors; how ever, the results of our n umerical studies carried out in [40] conﬁrm and ill ustrate the v alidity of our analy sis. Here, we ﬁnd it in t eresting to analyse the phenomenon of the transition of the (o dd) n um b er of zeros as 1 . 4 ≤ α ≤ 1 . 8 . F or this purp ose, in T able I we rep ort the in terv als of amplitude ∆ α = 0 . 01 where these transitions o ccur, and the lo cation T ∗ (ev aluated within a rela tive error of 0 . 1% ) of the larg est zeros found at the tw o extreme v alues of the a b o ve in terv als. W e recognize that the transiti on from 1 to 3 zeros o ccurs a s 1 . 40 ≤ α ≤ 1 . 41, that one from 3 to 5 zeros o ccurs as 1 . 56 ≤ α ≤ 1 . 57, and so on. T he last tra nsiti on in the considered range of α is from 15 to 17 zeros, and it just o ccurs as 1 . 79 ≤ α ≤ 1 . 80 . N ∗ α T ∗ 1 ÷ 3 1 . 40 ÷ 1 . 41 1 . 730 ÷ 5 . 726 3 ÷ 5 1 . 56 ÷ 1 . 57 8 . 366 ÷ 13 . 48 5 ÷ 7 1 . 64 ÷ 1 . 65 14 . 61 ÷ 20 . 00 7 ÷ 9 1 . 69 ÷ 1 . 70 20 . 80 ÷ 26 . 33 9 ÷ 11 1 . 72 ÷ 1 . 73 27 . 03 ÷ 32 . 83 11 ÷ 13 1 . 7 5 ÷ 1 . 76 33 . 11 ÷ 38 . 81 13 ÷ 15 1 . 7 8 ÷ 1 . 79 39 . 49 ÷ 45 . 51 15 ÷ 17 1 . 7 9 ÷ 1 . 80 45 . 51 ÷ 51 . 46 T able I N ∗ = n um b er of zeros, α = fractional order, T ∗ lo cation of the largest zero. R. Gor enﬂo and F. Maina r di 253 4. FRA CTIONAL DIFFERENT IA L EQUA TIONS: 2-nd P AR T In this section we shall consider the following fractional diﬀeren ti a l eq uations for t ≥ 0 , equipp ed with the necessary initia l conditions, du dt + a d α u dt α + u ( t ) = q ( t ) , u (0 + ) = c 0 , 0 < α < 1 , (4 . 1) d 2 v dt 2 + a d α v dt α + v ( t ) = q ( t ) , v ( 0 + ) = c 0 , v ′ (0 + ) = c 1 , 0 < α < 2 , (4 . 2) where a is a p ositive constan t. The unknown functions u ( t ) and v ( t ) (the ﬁeld v ari - ables) are required to b e suﬃcien t ly well b ehav ed to b e t reated wi t h their deriv at i v es u ′ ( t ) and v ′ ( t ) , v ′′ ( t ) b y the t ec hnique of Laplace tra nsform. The give n function q ( t ) is supposed to b e con tinuou s. In the ab ov e equations the fractional deriv a t iv e of order α is a ssumed to be pro vided b y the op erator D α ∗ , the Caputo deriv a ti ve , see (1.17), in agreemen t with our choice follow ed in the previous section. Note that in (4.2) we mus t distinguish the cases (a) 0 < α < 1 , (b) 1 < α < 2 and α = 1 . The equations (4.1) and(4.2) wi ll b e referred to as the c omp osite f r actional r elax- ation e quation and t he c omp osite fr actional oscil lation e quation , resp ectiv ely , to b e distinguished from the corresp onding simple fractional eq uat i ons treated in § 3. The fractional diﬀeren tial equation in ( 4 .1) with α = 1 / 2 corresp onds to the Basset pr oblem , a classical problem in ﬂuid dynamics concerning the unsteady motion of a parti cle accelerating in a v iscous ﬂuid under the action of the gravit y , see [24]. The fractional diﬀeren tial equation in (4. 2) with 0 < α < 2 mo dels an oscillation pro cess wit h fractional damping term. It w as formerly treated b y Ca puto [19], who pro vided a preliminary analysis b y the Lapla ce transform. The sp ecial cases α = 1 / 2 and α = 3 / 2 , but with the standard deﬁnition D α for t he fractional deriv ative, hav e b een discussed b y Bagley [30]. Recen tly , Beyer and Kempﬂe [46] discussed (4.2) for −∞ < t < + ∞ to in ve stigate the uniqueness and c ausality o f the solutions. As they let t running in all of I R , t hey used F o urier t ransforms and c haracterized the fractional deriv ative D α b y its properties in frequency space, thereb y requiring that for non- in teger α the principal branc h of ( iω ) α should b e tak en. Under the global condition that the solution is square summable, they show ed that the system described b y (4. 2 ) is c ausal iﬀ a > 0 . Also here we shall apply the metho d of Laplace transform to solve the frac- tional diﬀeren tial equations and get some insight into their fundamental and impulse- r esp onse solutions . Ho w ev er, in con trast with the previous section, w e no w ﬁnd it more con v enien t to apply directly the formu la (1.30) for t he Laplace transform o f frac- tional and in teger deriv atives, t han reduce the equations with the prescrib ed initi a l conditions as equiv al en t (fractional) integral equati ons t o b e t reat ed b y the Laplace transform. 254 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der 4.1 The c omp osite fr actional r elaxation e quation Let us apply the Laplace transform to the fractional relaxation equation (4.1). Using t he rule (1.30) w e are led to the transformed algebraic equati on e u ( s ) = c 0 1 + a s α − 1 w 1 ( s ) + e q ( s ) w 1 ( s ) , 0 < α < 1 , (4 . 3) where w 1 ( s ) := s + a s α + 1 , (4 . 4) and a > 0 . Putting u 0 ( t ) ÷ e u 0 ( s ) := 1 + a s α − 1 w 1 ( s ) , u δ ( t ) ÷ e u δ ( s ) := 1 w 1 ( s ) , (4 . 5) and recognizing that u 0 (0 + ) = l im s →∞ s e u 0 ( s ) = 1 , e u δ ( s ) = − [ s e u 0 ( s ) − 1] , (4 . 6) w e can conclude that u ( t ) = c 0 u 0 ( t ) + Z t 0 q ( t − τ ) u δ ( τ ) dτ , u δ ( t ) = − u ′ 0 ( t ) . (4 . 7) W e th us recognize that u 0 ( t ) and u δ ( t ) are the f undamental solution and impulse- r esp onse so lution for the eq uation ( 4.1), resp ectively . Let us ﬁrst consider the problem to get u 0 ( t ) as t he i n v erse Laplace transform of e u 0 ( s ) . W e easil y see that the function w 1 ( s ) has no zero i n t he main sheet of the Riemann surface including its b oundaries on the cut (simply sho w that Im { w 1 ( s ) } do es not v anish if s is not a real p ositive n um b er), so that the in v ersion of the Laplace transform e u 0 ( s ) can b e carried out b y deforming the original B rom wic h path into the Hank el path H a ( ǫ ) in tro duced in the previous section, i.e. in to the lo op constituted b y a small circle | s | = ǫ with ǫ → 0 and by the tw o b orders of the cut negative real axis. As a consequence w e write u 0 ( t ) = 1 2 π i Z H a ( ǫ ) e st 1 + as α − 1 s + a s α + 1 ds . (4 . 8) It is no w an exercise in complex analysis to show that t he con tribution from the Hank el path H a ( ǫ ) as ǫ → 0 is pro vided b y u 0 ( t ) = Z ∞ 0 e − r t H (1) α, 0 ( r ; a ) dr , (4 . 9) R. Gor enﬂo and F. Maina r di 255 with H (1) α, 0 ( r ; a ) = − 1 π Im  1 + as α − 1 w 1 ( s )     s = r e iπ  = 1 π a r α − 1 sin ( απ ) (1 − r ) 2 + a 2 r 2 α + 2 (1 − r ) a r α cos ( απ ) . (4 . 10) F or a > 0 and 0 < α < 1 the function H (1) α, 0 ( r ; a ) i s p osit i v e for all r > 0 since it has the sign of the n umerator; in fact in (4.10) the denominator is strictly p ositi v e b eing eq ual to | w 1 ( s ) | 2 as s = r e ± iπ . H ence, t he fundamental solution u 0 ( t ) has the p eculiar prop ert y to b e c ompletely monotone , and H (1) α, 0 ( r ; a ) is its s p e ctr al f unction . No w the determination of u δ ( t ) = − u ′ 0 ( t ) i s straigh tforw ard. W e see that also the impulse-r esp onse solution u δ ( t ) is c ompletely monotone since it can b e represen ted b y u δ ( t ) = Z ∞ 0 e − r t H (1) α, − 1 ( r ; a ) dr , (4 . 11) with s p e ctr al f unction H (1) α, − 1 ( r ; a ) = r H (1) α, 0 ( r ; a ) = 1 π a r α sin ( απ ) (1 − r ) 2 + a 2 r 2 α + 2 (1 − r ) a r α cos ( απ ) . (4 . 12) W e recognize that both the solutions u 0 ( t ) and u δ ( t ) turn out to b e strictly decreasing from 1 tow ards 0 a s t runs from 0 t o ∞ . Their b ehaviour as t → 0 + and t → ∞ can b e insp ected by means of a prop er asymptoti c analy sis. The b eha viour of the solutions as t → 0 + can b e determined from the b eha viour of their Laplace t ransforms as Re { s } → + ∞ as w ell k no wn from the theory of t he Laplace transform, see e.g. [25]. W e obtain as Re { s } → + ∞ , e u 0 ( s ) = s − 1 − s − 2 + O  s − 3+ α  , e u δ ( s ) = s − 1 − a s − (2 − α ) + O  s − 2  , (4 . 13) so that u 0 ( t ) = 1 − t + O  t 2 − α  , u δ ( t ) = 1 − a t 1 − α Γ(2 − α ) + O ( t ) , as t → 0 + . ( 4 . 14) The sp ectral represen tations (4.9) and ( 4.11) are suitable to obtain the asymptotic b eha viour of u 0 ( t ) and u δ ( t ) as t → + ∞ , b y using the W atson lemma. In fact, expanding the spectral functions for small r and taking the dominant term in the correspo nding asympto t ic series, we obtain u 0 ( t ) ∼ a t − α Γ(1 − α ) , u δ ( t ) ∼ − a t − α − 1 Γ( − α ) , as t → ∞ . (4 . 15) W e note that the limiting case α = 1 can b e easil y treated extending t he v alidity of eqs (4. 3-7) to α = 1 , as it is legitimate. In this case w e obtain u 0 ( t ) = e − t/ (1 + a ) , u δ ( t ) = 1 1 + a e − t/ (1 + a ) , α = 1 . (4 . 16) Of course, in the case a ≡ 0 we reco ve r the standard solutions u 0 ( t ) = u δ ( t ) = e − t . 256 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e conclude t his sub-section with some considerations on the sol uti ons when the order α i s just a rati o nal n um b er. If we take α = p/q , where p, q ∈ I N are assumed (for con venien ce) to b e relatively prime, a factorization in (4.4) is p ossible b y using the pro cedure indicated b y Miller and Ross [10]. In these cases t he soluti o ns can b e expressed i n terms of a linear comb ination of q Mittag-Leﬄer functions of fractional order 1 /q , whic h, on their turn can b e express ed in terms o f i ncomplete gamma functions, see (A.1 4) of the App endix. Here w e shall illustrate the factorization i n the simplest case α = 1 / 2 and pro vide the solutions u 0 ( t ) and u δ ( t ) in terms of the functions e α ( t ; λ ) (with α = 1 / 2), in tro duced in t he previous section. In this case, in view of the a pplication to the Basset pr oblem , see [24], t he equati on (4.1) deserv es a parti cular atten tion. F or α = 1 / 2 w e can wri te w 1 ( s ) = s + a s 1 / 2 + 1 = ( s 1 / 2 − λ + ) ( s 1 / 2 − λ − ) , λ ± = − a/ 2 ± ( a 2 / 4 − 1) 1 / 2 . (4 . 17) Here λ ± denote the t wo roots (real or conjugate complex) of the second degree p olynomial wit h p ositive coeﬃcients z 2 + a z + 1 , which , i n particular, satisfy the follo wing binary relations λ + · λ − = 1 , λ + + λ − = − a , λ + − λ − = 2( a 2 / 4 − 1 ) 1 / 2 = ( a 2 − 4) 1 / 2 . (4 . 18) W e recognize that we mus t treat separately the follo wing t w o cases i ) 0 < a < 2 , o r a > 2 , and ii ) a = 2 , whic h corresp ond to t w o disti nct ro ots ( λ + 6 = λ − ), or tw o coinciden t ro ots ( λ + ≡ λ − = − 1), resp ectively . F or t his purp ose, using the notat i on introduced in [24 ] , we write f M ( s ) := 1 + a s − 1 / 2 s + a s 1 / 2 + 1 =        i ) A − s 1 / 2 ( s 1 / 2 − λ + ) + A + s 1 / 2 ( s 1 / 2 − λ − ) , ii ) 1 ( s 1 / 2 + 1) 2 + 2 s 1 / 2 ( s 1 / 2 + 1) 2 , (4 . 19) and e N ( s ) := 1 s + a s 1 / 2 + 1 =        i ) A + s 1 / 2 ( s 1 / 2 − λ + ) + A − s 1 / 2 ( s 1 / 2 − λ − ) , ii ) 1 ( s 1 / 2 + 1) 2 , (4 . 20) where A ± = ± λ ± λ + − λ − . (4 . 21) Using ( 4 .18) we note that A + + A − = 1 , A + λ − + A − λ + = 0 , A + λ + + A − λ − = − a . (4 . 22) R. Gor enﬂo and F. Maina r di 257 Recalling the Laplace transform pairs (A.34 ), (A.36 ) and (A.37 ) in App endix, w e obtain u 0 ( t ) = M ( t ) := ( i ) A − E 1 / 2 ( λ + √ t ) + A + E 1 / 2 ( λ − √ t ) , ii ) (1 − 2 t ) E 1 / 2 ( − √ t ) + 2 p t/π , (4 . 23) and u δ ( t ) = N ( t ) := ( i ) A + E 1 / 2 ( λ + √ t ) + A − E 1 / 2 ( λ − √ t ) , ii ) (1 + 2 t ) E 1 / 2 ( − √ t ) − 2 p t/π . (4 . 24) W e th us recognize in (4 .23-24) the presence of the functions e 1 / 2 ( t ; − λ ± ) = E 1 / 2 ( λ ± √ t ) and e 1 / 2 ( t ) = e 1 / 2 ( t ; 1) = E 1 / 2 ( − √ t ) . In particular, the sol ution of the Basset p r oblem can b e easily obtained from (4.7) with q ( t ) = q 0 b y using (4. 23-24) and noting that R t 0 N ( τ ) d τ = 1 − M ( t ) . Denoting this soluti o n by u B ( t ) w e get u B ( t ) = q 0 − ( q 0 − c 0 ) M ( t ) . (4 . 25) When a ≡ 0, i.e. in the absence of term con taining the fractional deriv ative (due to the Basset force), w e reco ver the classical Stok es solution, that we denote b y u S ( t ) , u S ( t ) = q 0 − ( q 0 − c 0 ) e − t . In the particular case q 0 = c 0 , we get the steady-state solution u B ( t ) = u S ( t ) ≡ q 0 . F or v anishing initial condition c 0 = 0 , w e ha v e the creep-lik e solutions u B ( t ) = q 0 [1 − M ( t )] , u S ( t ) = q 0 h 1 − e − t i , that we compare i n the normalized plots of Fi g . 5 of [2 4 ]. In this case i t is instructiv e to compare the b eha viours of t he t w o sol utions as t → 0 + and t → ∞ . Recalling the general asy mptot ic expressions o f u 0 ( t ) = M ( t ) in (4.14 ) and (4 .15) with α = 1 / 2 , w e recognize that u B ( t ) = q 0 h t + O  t 3 / 2 i , u S ( t ) = q 0  t + O  t 2  , as t → 0 + , and u B ( t ) ∼ q 0 h 1 − a/ √ π t i , u S ( t ) ∼ q 0 [1 − E S T ] , as t → ∞ , where E S T denotes exp onential ly smal l terms . In particular we note that the nor- malized plot of u B ( t ) /q 0 remains under that of u S ( t ) /q 0 as t runs from 0 to ∞ . The reader is invited to con vince himself of the following fact. In t he general case 0 < α < 1 the solution u ( t ) has the part i cular prop ert y of b eing equal to 1 for all t ≥ 0 if q ( t ) has this prop erty and u (0 + ) = 1 , whereas q ( t ) = 1 for all t ≥ 0 and u (0 + ) = 0 implies that u ( t ) is a creep function tending to 1 as t → ∞ . 258 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der 4.2 The c omp osite fr actional oscil lation e quation Let us now apply the Laplace transform to the fractional oscillation equati o n (4.2). Using t he rule (1.30) w e are led to the transformed algebraic equati ons (a) e v ( s ) = c 0 s + a s α − 1 w 2 ( s ) + c 1 1 w 2 ( s ) + e q ( s ) w 2 ( s ) , 0 < α < 1 , (4 . 26 a ) or (b) e v ( s ) = c 0 s + a s α − 1 w 2 ( s ) + c 1 1 + a s α − 2 w 2 ( s ) + e q ( s ) w 2 ( s ) , 1 < α < 2 , (4 . 26 b ) where w 2 ( s ) := s 2 + a s α + 1 , (4 . 27) and a > 0 . Putting e v 0 ( s ) := s + a s α − 1 w 2 ( s ) , 0 < α < 2 , (4 . 28) w e recognize that v 0 (0 + ) = lim s →∞ s e v 0 ( s ) = 1 , 1 w 2 ( s ) = − [ s e v 0 ( s ) − 1] ÷ − v ′ 0 ( t ) , (4 . 29) and 1 + a s α − 2 w 2 ( s ) = e v 0 ( s ) s ÷ Z t 0 v 0 ( τ ) dτ . (4 . 30) Th us w e can conclude that (a) v ( t ) = c 0 v 0 ( t ) − c 1 v ′ 0 ( t ) − Z t 0 q ( t − τ ) v ′ 0 ( τ ) dτ , 0 < α < 1 , (4 . 31 a ) or (b) v ( t ) = c 0 v 0 ( t ) + c 1 Z t 0 v 0 ( τ ) dτ − Z t 0 q ( t − τ ) v ′ 0 ( τ ) dτ , 1 < α < 2 . (4 . 31 b ) In b oth of t he abov e equations the term − v ′ 0 ( t ) represen ts the impulse-r esp onse solution v δ ( t ) for the c omp osite fr ac tional oscil lation e quation (4.2), namely the particular sol ution of the inhomogeneous equation with c 0 = c 1 = 0 and with q ( t ) = δ ( t ) . F or the fundamental solutions of (4.2) w e recognize from eqs (4.31) that w e ha v e t w o distinct couples of solutions according to the case (a) and (b) whic h read (a) { v 0 ( t ) , v 1 a ( t ) = − v ′ 0 ( t ) } , (b) { v 0 ( t ) , v 1 b ( t ) = Z t 0 v 0 ( τ ) dτ } . (4 . 32) R. Gor enﬂo and F. Maina r di 259 W e ﬁrst consider t he parti cular case α = 1 for whic h the fundame n tal and impulse respo nse solutions are k no wn in terms of elementary functions. This limiting case can also b e t reat ed b y ext ending the v alidity of eqs (4 .26a) and (4 . 31a) to α = 1 , as it is legi t imate. F rom e v 0 ( s ) = s + a s 2 + a s + 1 = s + a / 2 ( s + a / 2) 2 + (1 − a 2 / 4) − a/ 2 ( s + a/ 2) 2 + (1 − a 2 / 4) , (4 . 3 3 ) w e obtain the b asic f undamental solution v 0 ( t ) =            e − at/ 2 h cos( ω t ) + a 2 ω sin( ω t ) i if 0 < a < 2 , e − t (1 − t ) if a = 2 , e − at/ 2  cosh( χt ) + a 2 χ sinh( χt )  if a > 2 , (4 . 34) where ω = p 1 − a 2 / 4 , χ = p a 2 / 4 − 1 . (4 . 35) By a diﬀeren tiation of ( 4.34) w e easily o bta i n the s e c ond fundamental solution v 1 a ( t ) and the i mpulse-r esp onse solution v δ ( t ) since v 1 a ( t ) = v δ ( t ) = − v ′ 0 ( t ) . W e p oin t out that a ll the solutions exhibit an exp onential de c ay as t → ∞ . Let us no w consider the problem to g et v 0 ( t ) as the i nv erse Laplace transform of e v 0 ( s ) , as giv en by ( 4 .26-27), v 0 ( t ) = 1 2 π i Z B r e st s + a s α − 1 w 2 ( s ) ds , (4 . 3 6) where B r denotes the usual Bromwic h path. Using a result b y Beyer and Kempﬂe [46] w e know that the function w 2 ( s ) (for a > 0 and 0 < α < 2 , α 6 = 1 ) has exactly two simple, c onjugate c omplex zer os on the princip al br anch i n the op en lef t half- plane , cut along the negativ e real a xis, sa y s + = ρ e + iγ and s − = ρ e − iγ with ρ > 0 and π / 2 < γ < π . This enables us t o rep eat the considerations carried out for the simple fractional oscillation equation to decomp ose the basic fundamen t al solution v 0 ( t ) into tw o parts according to v 0 ( t ) = f α ( t ; a ) + g α ( t ; a ) . In fact, t he ev al uation of the B rom wic h i n tegral (4.36) can b e achiev ed b y adding the contribution f α ( t ; a ) from the Hank el path H a ( ǫ ) a s ǫ → 0 , to the residual con tribution g α ( t ; a ) from the t w o p oles s ± . As an exercise in complex analysis w e obtain f α ( t ; a ) = Z ∞ 0 e − r t H (2) α, 0 ( r ; a ) dr , (4 . 37) 260 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der with s p e ctr al f unction H (2) α, 0 ( r ; a ) = − 1 π Im  s + a s α − 1 w 2 ( s )     s = r e iπ  = 1 π a r α − 1 sin ( απ ) ( r 2 + 1) 2 + a 2 r 2 α + 2 ( r 2 + 1) a r α cos ( απ ) . (4 . 38) Since in (4.38) the denominator is strictly p ositive b eing equal t o | w 2 ( s ) | 2 as s = r e ± iπ , the sp e ctr al function H (2) α, 0 ( r ; a ) turns out to b e p ositi v e for a ll r > 0 for 0 < α < 1 and negati v e for all r > 0 for 1 < α < 2 . Hence, in case (a) the function f α ( t ) , in case ( b) t he fun ction − f α ( t ) is c ompletely monotone ; in b o th cases f α ( t ) tends to zero as t → ∞ , from ab o ve in case (a), from b elow in case (b), according to the asymptotic b ehaviour f α ( t ; a ) ∼ a t − α Γ(1 − α ) , as t → ∞ , 0 < α < 1 , 1 < α < 2 , (4 . 39) as derived b y applying the W atson lemma in ( 4.37) and considering ( 4.38). The other part, g α ( t ; a ) , is obtained a s g α ( t ; a ) = e s + t Res  s + a s α − 1 w 2 ( s )  s + + conjugate complex = 2 Re ( s + + a s α − 1 + 2 s + + a α s α − 1 + e s + t ) . (4 . 40) Th us this t erm exhibits an o s cil latory c haracter wi th ex p onen tially decreasing am- plitude like exp ( − ρ t | cos γ | ) . Then w e recognize that the basic fundamen tal solution v 0 ( t ) exhibit s a ﬁnite n um b er of zeros and that, for suﬃcien tl y large t , it turns out to b e permanen tly p ositive if 0 < α < 1 and p ermanen tly negative if 1 < α < 2 with an algebr ai c de c ay pro vided b y (4.39). F or the second fundamen tal solutions v 1 a ( t ) , v 1 b ( t ) and for the impulse-response solution v δ ( t ) , the corresponding analy sis is straightforw ard in vi ew of their connec- tion wit h v 0 ( t ), p ointed out in (4 . 31-32). The algebr aic de c ay of al l the solutions as t → ∞ , for 0 < α < 1 and 1 < α < 2 , i s henceforth resumed in the relations v 0 ( t ) ∼ a t − α Γ(1 − α ) , v 1 a ( t ) = v δ ( t ) ∼ − a t − α − 1 Γ( − α ) , v 1 b ( t ) ∼ a t 1 − α Γ(2 − α ) . (4 . 41) In conclusion, except in the particular case α = 1 , all the presen t sol uti ons of the comp osite fractional oscillatio n eq uat i on exhibit similar ch aracteristics with the correspo nding solutions of the simple fractional oscillati on equati on, namely a ﬁnite numb er of damp e d os cil lations follow ed by a monotonic algebr aic de c ay as t → ∞ . R. Gor enﬂo and F. Maina r di 261 5. CONCLUSIONS Starting from the cl a ssical R iemann-Liouvil le deﬁnitions o f the fractio nal in tegra- tion op erator and its left-inv erse, the fractional diﬀeren t i ation op erator, and using the p o we rful to ol o f the Laplace transform metho d, w e ha v e describ ed the basic analytic al theory of fractional integral and diﬀeren tial equations. F or a numeric al treatment w e refer to Gorenﬂo [23] and to the references there quoted. F or the fr actional inte gr al e quations w e ha v e considered the basic examples pro- vided b y the linear A b el e quations of ﬁrst and se c ond kind . F or b ot h the k inds w e ha v e given the solution in diﬀeren t forms and discussed an interes ting application to in v erse heat conduction problems. Then w e ha v e analyzed in detail the scale of fr actiona l or dinary diﬀer ential e qua- tions ( F O D E ), see (3.5) , D α ∗ u ( t ) + u ( t ) = q ( t ) , t > 0 , 0 < α ≤ 2 , with a mo diﬁed fractional diﬀeren tiation D α ∗ , the Caputo f r actio nal de ri vative , that takes accoun t of give n initial v al ues u ( 0 + ) if 0 < α < 1 , the case of fr actional r elaxation , u (0 + ) and u ′ (0 + ) i f 1 < α < 2 , the case of fr actional osci l lation . W e ha ve in v estigated in depth the prop erties of the fundamental and t he impulse- r esp onse solutions . All these solutions can b e explicitly written down in terms of Mittag-L eﬄer functions . They tend t o zero like p ow ers t − β (with appropriate ch oices of β ), mo notonically if 0 < α < 1 , but exhibiting ﬁnitely many oscil l ations around zero if 1 < α < 2 (the more of these the nearer α is to the li miting v alue 2). If 1 < α < 2 t hese equations are able to mo del pro cesses in termediate betw een exp onen tial decay ( α = 1) and pure sinusoidal oscill ation ( α = 2). W e hav e found these qualita t iv e properti es essen tially by b ending the Brom wic h integration path of the Laplace in vers ion formula into the Hank el path, th us for eac h of these functions obtaining an integral represen tation as the Lap lace transform of a fun ction that no where c hanges its sign, augmen ted if 1 < α < 2 b y an oscillatory con tri bution resulting from a pair of conjugate complex p o l es lyi ng in the l eft half-plane. By quite a nalogous metho ds w e ha ve studied the c omp osite e quations , see (4.1-2), ( D + a D α ∗ + 1) u ( t ) = q ( t ) , 0 < α < 1 , and ( D 2 + aD α ∗ + 1) v ( t ) = q ( t ) , 0 < α < 2 , with a > 0 , whic h mo del pro cesses of relaxation and of oscillatio n, resp ectively . W e ha v e obtained similar prop erties of the fundamental and impulse-r e sp onse solutions with resp ect to monotonicity and oscillatory b eha viour. Let us stress the fact that our adoption of the Caputo fr actional derivative D α ∗ with m − 1 < α ≤ m , m ∈ I N , a nd the consequen t prescription o f the initi al v alues in analogy wit h t he ordinary diﬀe ren tial equations of in t eger order m , stands in con trast to the ma jority of the treatments of fractional diﬀeren tial equations, where the sta ndard fractional deriv ati v e D α is used, see e . g. [5] , [10]. As p oin ted in § 1.3 the adoption of D α requires the prescription of certain fractional i ntegrals as t → 0 + . 262 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der In our opinion, the diﬀeren t prescription of the the initial data p oin ts out the ma jor diﬀerence b etw een the tw o deﬁnitions for the fractional deriv ati v e. The analog y with the cases of in teger order would induce one to adopt t he Caputo derivative in the t reatmen t of diﬀeren tial equatio ns of fractio nal order for p h ysi c al applic ations . In fact, in ph ysical problems, t he initi al conditio ns are usually expressed in terms of a give n n um b er of b ounded v alues assumed b y the ﬁeld v ari able and its deriv atives of in teger order, no matter if the gov erning ev olution equation ma y b e a generic integro- diﬀeren tial equation and therefore, in particular, a fractional diﬀeren tial equation. The liveliness of the ﬁeld of fractional integral and diﬀeren t ial equati o ns, b oth in applications and in pure t heory , is underlined b y several pap ers and some b o oks that ha v e app eared recen tly . W e would l ik e to conclude the prese n t lectures with brief hin ts to recen t inv estiga tions, which hav e not b een quoted explicitly up to now since not strictly related t o our results, but ha v e at t racted our attention. Naturall y , this listing is far from exhaustive. The in terested reader can ﬁnd more on problems and asp ects in several pap ers recen tly published or in press i n some conference volumes and sp ecial ized magazines. W e ﬁrst like to quote the most recen t b o ok b y Rubin [14], who starting from one-dimensional fracti o nal calculus dev elops the theory of multidimensional w eakly singular in tegral equations of ﬁrst ki nd, making hea v y use of t he Marchau d approac h. W e th us recognize that a l l existing b o oks on fractional calculus v ary widely from each other in their c haracter concerning problems treated and metho ds applied. W e quote also the Ph.D. thesis b y Mich alski [47], who treat s l inear and nonlinear problems o f fractional calculus (i n one and in several dimensions) in a very elegan t w ay . The imp ortance of using fractional metho ds in ph ysics for describing slow de c ay pr o c esses and pr o c es s es interme di ate b etwe en r elaxation and osc il lation was stressed b y Ni gmatullin [48] in 1984. Nonnenmac her and asso ciates published a series of pap ers (of whic h w e quote [49-50]) discussing v arious ph ysical asp ects of fr ac tional r elaxation . F racti onal relax ation is o verall a p eculiarit y of a class of vis c o elastic b o dies whic h are extensiv ely treated b y Mainardi [24], to whic h w e refer for details and addi- tional bibliograph y . The fractional calculus ﬁnds imp ortan t applications in diﬀeren t areas of applied science including ele ctr o chemistry , see e.g. [51-54], ele c tr omagnetism , see e. g. [55-56], r adiati on physics , see e.g . [5 7-59], and c ontr ol the ory , see e.g. [6 0 -64]. Y et anot her ﬁeld of applications of fractional calculus i s that of f r actiona l p artial diﬀer enti al e quations ( F P D E ), including certain equati o ns o f f r actional diﬀusion , in tro duced to explain the phenomena of anomalous diﬀusion in complex or fractal systems. W e refer again to Mainardi [24] for a mathematical treatmen t of a rele- v an t F P D E , referred to as the time fr actional diﬀusion-wave e quation , with some applications a nd related references. R. Gor enﬂo and F. Maina r di 263 APPENDIX: THE MITT AG-LEFFLER TYPE FUNCTIONS In this App endix we shall consider the Mitt ag-Leﬄer function and some of the related functions whic h are relev an t for their connection with fractional cal culus. It is our purp ose to pro vide a rev i ew of the main prop erti es o f t hese functions i ncluding their Laplace transforms. A.1 The Mittag- L eﬄer functions E α ( z ) , E α,β ( z ) The Mit t ag-Leﬄer function E α ( z ) with α > 0 is so named from the great Sw edish mathematician who in tro duced it at the b eginning of this cen tury in a sequence of ﬁv e notes, see [65-69]. The function is deﬁned b y the following series represen tati on, v alid in t he whole complex plane, E α ( z ) := ∞ X n =0 z n Γ( αn + 1 ) , α > 0 , z ∈ C . ( A . 1) It t urns out that E α ( z ) is a n entir e function of order ρ = 1 /α and t yp e 1 . This prop ert y is still v alid but wit h ρ = 1 / Re { α } , if α ∈ C with p o s itive r e al p art , as formerly noted by Mittag-Leﬄer himself in [6 8]. In the limit for α → 0 + the analyticity in the whole complex plane is lost since E 0 ( z ) := ∞ X n =0 z n = 1 1 − z , | z | < 1 . ( A. 2) The Mittag-Leﬄer function pro vides a simple generalization of t he exp onen tial function b ecause of the substitution of n ! = Γ( n + 1) with ( α n )! = Γ( αn + 1) . P articular cases of (A.1 ), from whic h elemen tary functions a re recov ered, are E 2  + z 2  = cosh z , E 2  − z 2  = cos z , z ∈ C , ( A. 3) and E 1 / 2 ( ± z 1 / 2 ) = e z h 1 + erf ( ± z 1 / 2 ) i = e z erfc ( ∓ z 1 / 2 ) , z ∈ C , ( A. 4) where erf (erfc) denotes the (complemen tary) error function deﬁned as erf ( z ) := 2 √ π Z z 0 e − u 2 du , erfc ( z ) := 1 − erf ( z ) , z ∈ C . In (A.4) for z 1 / 2 w e mean the principal v alue o f the square ro ot of z in the complex plane cut along the the negativ e real ax is. With this c hoice ± z 1 / 2 turns out to b e p ositive/negativ e for z ∈ I R + . Since the iden tities i n (A.3) are trivi al, we presen t the pro of only for (A.4). Av oid- ing t he inessen tial p olidrom y wi th the substitution ± z 1 / 2 → z , w e write E 1 / 2 ( z ) = ∞ X m =0 z 2 m Γ( m + 1) + ∞ X m =0 z 2 m +1 Γ( m + 3 / 2) = u ( z ) + v ( z ) . ( A. 5) 264 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der Whereas the ev en part is easily recognized to b e u ( z ) = exp( z 2 ) , only after some manipulations the o dd part can b e pro v ed to b e v ( z ) = exp( z 2 ) erf ( z ) . T o this end w e need to recall the following series represe n tati on for the error function, see e. g . the handbo ok of the Bat eman Pro ject [70] or t hat by Abramo witz and Stegun [71 ] , erf ( z ) = 2 √ π e − z 2 ∞ X m =0 2 m (2 m + 1)! ! z 2 m + 1 , z ∈ C and note that (2 m + 1) ! ! := 1 · 3 · 5 . . . · (2 m + 1) = 2 m +1 Γ( m + 3 / 2) / √ π . An alternative pro of is obtai ned b y recognizing, after a term-wise diﬀeren tiation of the series represen tation in ( A .5), that v ( z ) satisﬁes the diﬀeren t ial equation in C , v ′ ( z ) = 2  1 √ π + z v ( z )  , v (0) = 0 , whose sol ution can immediat ely b e c hec k ed t o b e v ( z ) = 2 √ π e z 2 Z z 0 e − u 2 du = e z 2 erf ( z ) . A straig h tforw ard generalization of the Mittag-Leﬄer function, original l y due to Agarw al in 1953 based on a note by Hum b ert, see [7 2 -74], is obtained b y replacing the additiv e constan t 1 i n the argumen t o f t he Gamma function in (A.1) b y an arbitrary complex parameter β . Lat er, when we shall deal with Lapla ce transform pairs, the parameter β is required to b e p ositiv e as α . F or the new function w e agree to use the follo wing notat ion E α,β ( z ) := ∞ X n =0 z n Γ( αn + β ) , α > 0 , β ∈ C , z ∈ C . ( A. 6) P articular simple cases are E 1 , 2 ( z ) = e z − 1 z , E 2 , 2 ( z ) = sinh ( z 1 / 2 ) z 1 / 2 . ( A. 7) W e note t hat E α,β ( z ) is still an en tire function of order ρ = 1 /α and t yp e 1 . In these lectures we hav e preferred to use only the original Mittag-Leﬄer function (A.1) since our problems dep end on only a single parameter α , the order of fractional in tegration of diﬀeren ti ation. How ev er, for completeness, w e list hereafter the general functional relations for the generalized Mittag-Leﬄer function (A.6) , whic h in v olve b oth the t w o parameters α , β , see [18] and [70] , E α,β ( z ) = 1 Γ( β ) + z E α,β + α ( z ) , ( A. 8) E α,β ( z ) = β E α,β +1 ( z ) + αz d dz E α,β +1 ( z ) , ( A. 9)  d dz  p  z β − 1 E α,β ( z α )  = z β − p − 1 E α,β − p ( z α ) , p ∈ I N . ( A. 10) R. Gor enﬂo and F. Maina r di 265 A2. The Mittag-L eﬄer functions of r ational or der Let us no w consider the Mittag-Leﬄer functions of rational order α = p/q with p , q ∈ I N relatively prime. The relev an t functional relations, that we quote from [18], [ 70], t urn out to b e  d dz  p E p ( z p ) = E p ( z p ) , ( A. 11) d p dz p E p/q  z p/q  = E p/q  z p/q  + q − 1 X k =1 z − k p/ q Γ(1 − k p/q ) , q = 2 , 3 , . . . , ( A. 12) E p/q ( z ) = 1 p p − 1 X h =0 E 1 /q  z 1 /p e i 2 π h/p  , ( A. 13) and E 1 /q  z 1 /q  = e z " 1 + q − 1 X k =1 γ (1 − k /q , z ) Γ(1 − k /q ) # , q = 2 , 3 , . . . , ( A. 14) where γ ( a, z ) := R z 0 e − u u a − 1 du denotes the inc omplete gamma function . Let us now sk etc h the pro of for the ab ov e functional relations. One easily recognizes that the rela tions (A. 11) and (A.12) are i mmediate conse- quences of the deﬁnition (A.1). In order to pro ve the relation (A.13) we need to recall the iden tity p − 1 X h =0 e i 2 π hk /p = ( p if k ≡ 0 (mo d p ) , 0 if k 6≡ 0 (mo d p ) . ( A. 15) In fact, using this iden tity and the deﬁnition (A.1 ), w e ha ve p − 1 X h =0 E α ( z e i 2 π h/p ) = p E αp ( z p ) , p ∈ I N . ( A. 16) Substituting in the a b o v e relat i on α/p instead of α and z 1 /p instead of z , we obtain E α ( z ) = 1 p p − 1 X h =0 E α/p  z 1 /p e i 2 π h/p  , p ∈ I N . ( A. 17) Setting ab o v e α = p/q , we ﬁnally obtain (A.13). T o prov e the relatio n (A.1 4 ) we consider (A.12) for p = 1 . Multiplying b oth sides b y e − z , we obtai n d dz h e − z E 1 /q  z 1 /q i = e − z q − 1 X k =1 z − k/q Γ(1 − k /q ) . ( A. 18) Then, upo n integration of t his and recalling the deﬁnition of the incomplete gamma function, we arrive at ( A.14). 266 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der The relation (A.14) sho ws ho w the Mittag-Leﬄer functions of ra t ional order can b e expressed in terms of exp onen t ials and incomplete gamma functions. In particular, taking in (A.14) q = 2 , w e no w can verify ag ain t he relation (A. 4). In fact, from (A.14) w e obtai n E 1 / 2 ( z 1 / 2 ) = e z  1 + 1 √ π γ (1 / 2 , z )  , ( A. 19) whic h is equiv alen t to (A.4) if w e use the relation erf ( z ) = γ (1 / 2 , z 2 ) / √ π , see e.g. [70-71]. A3. The Mittag-L eﬄer inte gr al r epr esentation and asymptotic exp ans ions Man y of the most imp ortan t properties of E α ( z ) follow from Mittag-Leﬄer’s inte gr al r epr esentation E α ( z ) = 1 2 π i Z H a ζ α − 1 e ζ ζ α − z dζ , α > 0 , z ∈ C , ( A. 20) where the path of in tegrati o n H a (the Hankel p ath ) is a lo op whic h starts and ends at −∞ and encircles the circular disk | ζ | ≤ | z | 1 /α in the p ositive sense: − π ≤ arg ζ ≤ π on H a . T o prov e (A.20), expand the i ntegrand i n p o w ers of ζ , i n tegrate term-b y-term, and use Hankel’s in tegral for the recipro cal of the Gamma function. The integrand in (A.20 ) has a branch -p oint at ζ = 0. The complex ζ -plane is cut along the negative real axis, and in the cut plane the integrand is single-v a l ued: the principal branc h of ζ α is take n in the cut plane. The in tegrand has p oles at the p oin ts ζ m = z 1 /α e 2 π i m/α , m in teger, but only those of the p oles lie in the cut plane for which − α π < arg z + 2 π m < α π . Th us, the n um b er of the p oles inside H a i s either [ α ] or [ α + 1], according to t he v alue of arg z . The in tegral represen tation of the generalized Mittag-Leﬄer function turns out to b e E α,β ( z ) = 1 2 π i Z H a ζ α − β e ζ ζ α − z dζ , α , β > 0 , z ∈ C . ( A. 21) The most intere sting prop erties of the Mittag -Leﬄer function are asso ciated with its asymptot ic deve lopmen ts as z → ∞ in v arious sectors of t he complex plane. These prop erties can b e summarized as follows. F or the case 0 < α < 2 w e hav e E α ( z ) ∼ 1 α exp( z 1 /α ) − ∞ X k =1 z − k Γ(1 − αk ) , | z | → ∞ , | arg z | < απ / 2 , ( A . 22) E α ( z ) ∼ − ∞ X k =1 z − k Γ(1 − αk ) , | z | → ∞ , απ / 2 < arg z < 2 π − απ / 2 . ( A. 23) R. Gor enﬂo and F. Maina r di 267 F or the case α ≥ 2 we ha ve E α ( z ) ∼ 1 α X m exp  z 1 /α e 2 π i m/α  − ∞ X k =1 z − k Γ(1 − αk ) , | z | → ∞ , ( A. 24) where m takes all in teger v al ues suc h that − απ / 2 < arg z + 2 π m < απ / 2 , and arg z can assume any v alue b etw een − π a nd + π inclusiv e. F rom the asymptotic properties (A.2 2 -24 ) and the deﬁnition of the order of an en tire function, w e infer that the Mittag-Leﬄer function is an e ntir e f unction of or der 1 /α for α > 0; in a certain sense eac h E α ( z ) is the simplest entire fun ction of its order, see Phragm ´ en [ 75]. The Mittag -Leﬄer function also furnishes examples and coun ter-examples for the growth and other prop erti es of entire functions of ﬁnite order, see Buhl [76] . A4. The L aplac e tr ansform p ai rs r elate d to the Mittag-L eﬄer functions The Mittag-Leﬄer functions are connected to the Laplace in tegral through the equation Z ∞ 0 e − u E α ( u α z ) d u = 1 1 − z = Z ∞ 0 e − u u β − 1 E α,β ( u α z ) du , α , β > 0 . ( A. 25) The i n tegral at t he L.H. S. was ev aluated by Mittag-Leﬄer who sho w ed that the region of its con vergen ce contains the unit circle and is b ounded b y the line Re z 1 /α = 1. The ab ov e integral is fundamen ta l in the ev aluation of the Laplace transform of E α ( − λ t α ) and E α,β ( − λ t α ) with α , β > 0 and λ ∈ C . Since these functions turn out to play a key role in problems of fractional calculus, w e shall in tro duce a sp ecial notation for them. Putting in (A.25) u = st and u α z = − λ t α with t ≥ 0 and λ ∈ C , a nd using the sign ÷ for the juxtap osit i on of a function dep ending on t with its Laplace transform depending on s , we get t he following Laplace transform pairs e α ( t ; λ ) := E α ( − λ t α ) ÷ s α − 1 s α + λ , R e s > | λ | 1 /α , ( A. 26) and e α,β ( t ; λ ) := t β − 1 E α,β ( − λ t α ) ÷ s α − β s α + λ , R e s > | λ | 1 /α . ( A. 27) W e note that the results (A.26-27) , but with a diﬀeren t notation, were used b y Hum b ert and Aga rw al [72-74] to obtain a n um b er of functional relati ons sat i sﬁed b y E α ( z ) and E α,β ( z ) . Of course the results (A.2 6 -27) can also be o btained formally b y Laplace transforming term by term the series (A.1) and (A.6) wi th z = − λ t α , respecti vely , and summing the resulting series. 268 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der W e ﬁnd w orth while to list the following relations for the functions e α,β easily obtained from (A. 8-9) : e α,β ( t ; λ ) = t β − 1 Γ( α ) − λ e α,β + α ( t ; λ ) , ( A. 28) and d dt e α,β +1 ( t ; λ ) = e α,β ( t ; λ ) . ( A . 29) A remark able prop erty satisﬁed b y the functions e α ( t ; λ ) , e α,β ( t ; λ ) when λ i s p ositi ve and 0 < α ≤ 1 , 0 < α ≤ β ≤ 1 , resp ectiv ely , is to b e c ompletely monotone for t > 0 . W e recall that a function f ( t ) i s told t o b e c ompletely monotone for t > 0 if ( − 1) n f ( n ) ( t ) ≥ 0 for all n = 0 , 1 , 2 , . . . and for all t > 0 , and that a suﬃcien t condition for this i s the existence of a nonnegativ e lo cally in tegrable function K ( r ) , r > 0 , referred to a s t he sp e ctr al functi o n , with which f ( t ) = R ∞ 0 e − rt K ( r ) dr . F or more details see e.g. the b o ok b y Berg & F orst [7 7]. Excluding the trivial case α = β = 1 for whic h e 1 ( t ; λ ) = e 1 , 1 ( t ; λ ) = e − λ t , w e can pro v e the existence of the corresp onding sp ectral functions using the complex Brom wich form ula to inv ert t he Laplace transform i n (A.26-27) and b ending the Brom wich path into the H ank el path, as w e ha ve already shown in the sp ecial case e α ( t ) := e α ( t ; 1) in § 3. As an exercise in complex analysis (we kindly in vite the reader to carry it out) w e obtain the integral represen tations [A.30-33 ], e α ( t ; λ ) := Z ∞ 0 e − r t K α ( r ; λ ) dr , 0 < α < 1 , λ > 0 , ( A. 30) with s p e ctr al f unction K α ( r ; λ ) = 1 π λ r α − 1 sin ( απ ) r 2 α + 2 λ r α cos ( απ ) + λ 2 ≥ 0 , ( A. 31) and e α,β ( t ; λ ) := Z ∞ 0 e − r t K α,β ( r ; λ ) dr , 0 < α ≤ β < 1 , λ > 0 , ( A . 32) with s p e ctr al f unction K α,β ( r ; λ ) = 1 π λ sin [( β − α ) π ] + r α sin ( β π ) r 2 α + 2 λ r α cos ( απ ) + λ 2 r α − β ≥ 0 . ( A. 33) R. Gor enﬂo and F. Maina r di 269 Historically , t he complete monotonicit y of the Mittag-Leﬄer function in t he neg- ative real axis, i.e. of E α ( − x ) , for x ∈ I R + , when 0 < α < 1 , w as ﬁrst conjectured b y F eller using probabili stic metho ds a nd rigorously prov ed b y P ollard in 1948 [78 ]. Only recen tly , Sc hneider [79] has prov ed a t heorem for the complete monotonicity of the generalized Mittag-Leﬄer function in the negative real axis. He pro ved that E α,β ( − x ) , for x ∈ I R + , is completely mo not o ne i ﬀ 0 < α ≤ 1 and β ≥ α . Our conditions for e α,β ( t, λ ) to be completely monotone app ear more restricti ve than those b y Sc hneider for E α,β ( − x ) ; ho w eve r, we m ust note that in our case (A .27) the factor t β − 1 precedes the generalized Mittag-Leﬄer function. W e note that, up to our knowledge, in the handb o o ks con taining tables for the Laplace transforms, the Mit t ag-Leﬄer function is ignored so that the transform pairs (A.26-27) do not app ear if not in the sp ecial cases α = 1 / 2 and β = 1 , 1 / 2 , writ ten ho w ev er in terms of the error and complemen tary error fun ctions, see e.g. [71] . In fact, i n t hese cases we can use (A . 4) and (A.2 8 ) and reco ve r from (A. 2 6-27) the t w o Laplace transform pairs 1 s 1 / 2 ( s 1 / 2 ± λ ) ÷ e 1 / 2 ( t ; ± λ ) = e λ 2 t erfc ( ± λ √ t ) , λ ∈ C , ( A . 34) 1 s 1 / 2 ± λ ÷ e 1 / 2 , 1 / 2 ( t ; ± λ ) = 1 √ π t ∓ λ e 1 / 2 ( t ; ± λ ) , λ ∈ C . ( A. 35) W e also obtai n the related pairs 1 s 1 / 2 ( s 1 / 2 ± λ ) 2 ÷ 2 r t π ∓ 2 λ t e 1 / 2 ( t ; ± λ ) , λ ∈ C , ( A. 36) 1 ( s 1 / 2 ± λ ) 2 ÷ ∓ 2 λ r t π + (1 + 2 λ 2 t ) e 1 / 2 ( t ; ± λ ) , λ ∈ C , ( A. 37) In the pair (A.36) w e ha ve used the prop erties 1 s 1 / 2 ( s 1 / 2 ± λ ) 2 = − 2 d ds  1 s 1 / 2 ± λ  , d n ds n ˜ f ( s ) ÷ ( − t ) n f ( t ) . The pair (A . 37) is easily obtained by noting that 1 ( s 1 / 2 ± λ ) 2 = 1 s 1 / 2 ( s 1 / 2 ± λ ) ∓ λ s 1 / 2 ( s 1 / 2 ± λ ) 2 . 270 F r actional Calculus: Inte gr al and Diﬀer ential Equatio ns of F r actional Or der A.5 A dditional r efer enc es for the Mittag-L eﬄer typ e f unctions W e note that the Mittag-Leﬄer type functions are unkno w n to the ma jority of sci- en tists, b ecause t hey are ignored in the common b o oks on sp ecial functions. Thanks to our suggestion t he new 2000 Mathematics Subje ct Classi ﬁc ation has included these functions, see the item 33E12: Mittag-L eﬄer functions and gener alizations . A description of the most imp o rt an t prop erties of these functions wi th relev an t reference s can b e found in the third volume of the Bateman Pro ject [70], in the c hap- ter X V I I I dev oted to misc el lane ous functions . The sp ecialized t reat ises where great atten tion is dev oted to the Mittag-Leﬄer type functions are those b y Dzherbash yan [18], [ 2 2]. F or t he in terested readers w e also recommend t he classical treati se on com- plex functions b y Sansone & Gerretsen [8 0 ], where a suﬃcien tly detailed treatmen t of the original Mittag-Leﬄer function is give n. Since the t i mes of Mittag-Leﬄer seve ral scien tists ha ve recognized the imp ortance of the Mitt ag-Leﬄer type functions, pro- viding intere sting results and applications, which unfortun ately are not muc h k no wn. As pioneering works of mathematical nature in the ﬁeld of fractional in tegral and diﬀeren tial equations, w e li k e to quote those b y Hille & T amarkin and b y Barret. In 1930 Hil le & T amarkin [81] ha ve provided the solution of the Ab el integral equation of the second k i nd in t erms of a Mittag-Leﬄer function, whereas in 1956 B arret [82] has expressed the general solution of the linear fractional diﬀeren tial equation wit h constan t co eﬃcien ts in terms of Mittag-Leﬄer functions. As former applications in ph ysics w e li ke to quote the con tributions by K.S. Cole (1933), quoted b y H. T. Da vis [15, p. 287] in connection with nerv e conduction, and b y F.M. de Oliv eira Castro (1939) [83] and B. Gross (1947) [84] in connection with dielectrical and mec hanical relaxation, resp ectively . Subsequen tly , in 19 71, Caputo & Mainardi [28 ] hav e prov ed that the Mittag-Leﬄer function is presen t whenev er deriv atives of fractional order are i n tro duced in the constitutiv e equati ons of a l i near visco elastic b o dy . Since then, sev eral o ther authors ha ve p oin ted out the rel ev ance of the Mittag -Leﬄer function for fractional v isco elastic mo dels, see Mai nardi [24 ]. In recen t times the attention of mathematicians tow ards the Mit tag-Leﬄer typ e functions has i ncreased from b oth the analyti cal and nu merical p oint of v iew, ov erall b ecause of their relat i on with the fractional calculus. In addition to the b o oks a nd pap ers already quoted in t he text, here w e would like to dra w the reader’s attention to the most recen t pap ers on the Mitta g-Leﬄer t yp e functions, e.g. Al Saqabi & T uan [85], Kilbas & Saigo [86], Gorenﬂo, Luc hk o & Rogozin [87] and Mainardi & Gorenﬂo [88] . Since the fractional calculus has actually recalled a wide interes t for its applications in diﬀeren t areas o f ph ysics and engineering, we exp ect that so o n t he Mittag-Leﬄer function will exit from its isolated life as Cinder el la (using the term coined by F.G. T ricomi in the 19 5 0s for the inc o mplete Gamma function). R. Gor enﬂo and F. Maina r di 271 REFERENCES 1. Ross, B. (Editor): F r actional Calculus and i ts Applic ations , Lecture Notes in Mathematics # 457 , Springer V erlag, Berlin 1975. [ Pro c. In t. Conf. held at Univ. of N ew Hav en, USA, 1974] 2. Oldham, K.B. and J. Spanier: The F r actional Calculus , Academic Press, New Y ork 1974. 3. McBride, A.C.: F r actional Calculus and Inte g r al T r ans forms of Gener aliz e d F unc- tions , Pitman Researc h Not es in Mathematics # 31, P itman, London 1979. 4. McBride, A. C. and G.F. Roac h (Editors): F r a c tional Calculus , Pitman Researc h Notes in Mathematics # 138, Pit ma n, London 198 5 . [ Pro c. In t. W o rk shop. held at Univ. of Strathcly de, UK, 1984] 5. Samko S.G., Kilbas, A . A. and O.I. Marichev: F r actional Inte gr als and Derivatives, The ory and Applic ati ons , Gordon and Breach, Amsterdam 1993. [Engl. 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