Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations

Fractional V ariational Iteration Method for Fractional Nonlinear Diff erential Equations Guo-cheng W u * Interdisciplinary Institute for Nonlinear Science, National Engineering Laboratory of Modern Silk, Soochow University , Suzhou, 215021 Abstract Recently , fractional dif ferential equations have been investigated via the famous variational iteration method. However , all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional varia tional iteration m ethod is proposed. The Lagrange multipliers can be identified ex p licitly based on fractio nal variational theory . Keywords: Modified Riemann-Liouville deri vative; Frac tional co rrection functional; Fractional V ariational Iteration Method; 1 Introduction The variational iteration met hod [1 – 4] has been extens ively worked out for many years by numerous authors. S tarting from the pioneer ideas of the Inokuti–Sekine–Mura method, Ji -Huan He [3] developed th e variational iteration method (VIM) in (1999). In th is method, the equations are initially approxim ated with possible unknowns. A correction functional is established by the general Lagrange multiplier which can be identified o ptimally via the variatio nal theory . Besides, the VIM has no restrictions or unrea listic a ssumptions such as linearization or small parameters that are used in th e nonlinear operators [5 - 1 1]. In the last three decades, scientists and applied m athematicians have found fractional diff erential equations useful in va rious fields: rheology , quantitative biology , *Correspond ing author , Email addresses: wuguocheng2002@yahoo.com.cn. electrochemistry , scattering theory , dif fusion, transport theory , probability potential theory and elasticity [12]. Finding accurate and ef ficient methods for solving FDEs has been an active research undertaking. A que stion may naturally arise: Can we have a fractional variational met hod to derive approximate so lutions of FDEs? Although a number of useful attempts have been made to solve fractional equations v ia the VIM, the problem has not yet been completely re solved, i.e., m ost of the previous works avoid the term of fractional de rivative, handle them as re stricted variation and they cannot identify the fractional Langrange mu ltipliers explicitly in the correction function. Assume the following fractional dif ferential equ ation (, ) 0 , Du fu t Dt α α + = . at b ≤ ≤ ( 1 ) where () t D t Du α α is the famous Caputo’ s frac tional derivative defined as (1 ) () 1 (1 ) () , () t a m m t d m Dt Du u t α α α τ α τ τ + − = Γ+ − − ∫ 1. mm < α≤ + ( 2 ) Here α is a real constant called the orde r of the fractional derivative, and a is the initial point and Γ denotes the gamma function. I n t h e c a s e 01 , α < ≤ we re-write Eq. (1) in the form (, ) 0 du D u du fu t dt Dt dt α α +− + = , ( 3 ) Handling the term (, ) Du d u f ut Dt dt α α −+ as restricted variation, we can find the following variational it eration algorithms [2] 1 0 10 0 11 10 1 0 () () ( ) d () () ( ) d () () ( ) ( ) d . t n nn n t nn nn t nn n n nn n Du utu t f s Dt Du d u utu t f s Dt dt Du d u Du d u utu t f f s Dt dt Dt dt α α α α αα αα + + −− + − ⎧ =− + ⎪ ⎪ ⎪ ⎪ =− − + ⎨ ⎪ ⎪ ⎧⎫ =− − + − − + ⎪ ⎨⎬ ⎪ ⎩⎭ ⎩ ∫ ∫ ∫ ( 4 ) In the case 12 α <≤ , the above iteration formulas ar e also valid. W e can present Eq. (1) in the form 22 22 (, ) 0 , du Du du fu t dt Dt dt α α +− + = ( 5 ) and the following iteration formulae are suggested [2] 1 0 2 10 2 0 22 11 1 1 22 0 () () ( ) ( ) d () () ( ) ( ) d () () ( ) ( ) ( ) d . t n nn n t nn nn t nn n n nn n n Du utu t s t f s Dt Du du utu t s t f s Dt dt Du du D u du utu t s t f f s Dt dt Dt dt α α α α αα αα + + −− + − ⎧ =+ − + ⎪ ⎪ ⎪ ⎪ =+ − − + ⎨ ⎪ ⎪ ⎧⎫ =+ − − + − − + ⎪ ⎨⎬ ⎪ ⎩⎭ ⎩ ∫ ∫ ∫ (6) When α is close to 1, Eq. (4) is better while Eq. (6) is recommended for α approaching 2. In the above derivation, the Lag range multiplie r in the correction functional is identified approximately using the classical variational principle, due to the lack of fractional variational theory . Now the things are changing. Recently , Jumarie proposed a modified Riemann-Liouville de rivative [13]. W ith this kind of the fractional derivative, a generalized variat ional derivative is defined [13, 14], and variational approach for fractional partial dif ferential equations is proposed [15]. Based on the fractional variational theory , we can extend the VIM to a f ractional one and solve fractional dif ferential equ ations with a generalized Lagrange multiplier . 2 Properties of Modified Riemann-Liouville Derivative Comparing with the classical Caputo de rivative, the definition of modified Reimann-Liouville derivative is not required to satisfy higher integer -order derivative than α . Secondly , th α derivative of a constant is zero. Now we introduce some properties of the fracti onal derivative. Assume : f , R R → () x fx → denote a continuous (but not necessarily dif ferentiable) function in the inte rval [0, 1]. Through the fractional famous Riem ann Liouville integral 1 0 0 1 () = ( ) ( ) , > 0 , () x x If x x f d αα ξξ ξ α α − − Γ ∫ ( 7 ) The modified Riemann-Liouville de rivative is defined as [14] 0 0 1 () = ( ) ( ( ) ( 0 ) ) , (1 ) x x d Df x x f f d dx αα ξ ξξ α − −− Γ− ∫ ( 8 ) where [0, 1] x ∈ , 01 α << . In the next sections, we will use the following pr operties. (a). Fractional Leibniz product law [13] () () () 0 () = . x Du v u v u v α αα + ( 9 ) (b). Fractional integr ation by parts [13] () () =( ) / b ab a ab I uv u v I u v α αα α − ( 1 0 ) (c). Integration with respect to () dx α ( Lemma 2.1 of [14]) W e use the following equality for the integral w . r . t () dx α 1 0 00 11 () = ( ) ( ) ( ) ( ) , 0 1 . () ( 1 ) xx x If x x f d f d αα α ξξ ξ ξ ξ α αα − −= < ≤ ΓΓ + ∫∫ ( 1 1 ) In order to propose a FVIM for fractiona l nonlinear equations, firstly , we should consider fractional variationa l theory wh ich is em ployed to c onstruct a fractional correction functional, th en we need to identify the fractional Lagrange multiplie rs. 3 Fractional V ariational Theory Several versions of fractiona l variational approaches ha ve been proposed [15 - 17]. However , all of them can not applie d to establish a fractional va riational function al for fractional dif ferential equations. W ith Ju marie’ s fractional deri vative [13 - 15], we can readily establish a genera lized fraction al functional. W e now generally revisit the derivation of the fractional variational deriva tive [13, 14]. S tart from the functional 1 [ ] = ( , , ) ( ) (1 ) b ax a Jy Fxy D y d x α α α Γ+ ∫ ( 1 2 ) and find the necessary conditions for extrema. Let * () y x be the desired function and let R ε ∈ . Let * () = () () y xy x x εη + ( 1 3 ) be a family of curves that satisfy the boundary conditions, which we can set, for simplicity , as () = ( ) = 0 . ab η η As () a D y x α is a linear operator , we have ** () = () () () , aa a D y xD y xD y xx αα α εη + ( 1 4 ) so that by substituting Eqs. (13) and (14) in to Eq. (12), for each () x η , we have ** 1 =[ ] = ( , , ) ( ) . (1 ) b aa a JJ F x y D y D d x Γα α αα εε η ε η ++ + ∫ ( 1 5 ) Note that [] J ε is a function of ε only and it attains its extrem um at =0 ε . Diff erentiating Eq. (15) with respect to ε gives 1 =[ ] ( ) , (1 ) b ax a ax dJ F F Dd x d Γα yD y α α α ηη ε ∂∂ + +∂ ∂ ∫ ( 1 6 ) so that a necessary co ndition for () J ε t o h a v e a n e x t r e m u m i s f o r / dJ d ε to vanish for all admissible () x η which leads to the result 1 [ ( ) ( )]( ) = 0. (1 ) b ax a ax FF xD x d x Γα yD y αα α ηη ∂∂ + +∂ ∂ ∫ ( 1 7 ) The integral in Eq. (17) can be rewritte n, using the fractional Leibniz formula and integration by parts, in the f orm 1 () ( ) (1 ) ( ) 1 =( ) | ( ) ( ) ( ) () ( 1 ) () 1 =( ) ( ) ( ) , (1 ) ( ) b ax a ax b b aa x a ax ax b ax a ax F Dx d x Γα Dx FF x xD d x Dy x Γα Dy x F xD d x Γα Dy x αα α α α αα αα α η η ηη η ∂ +∂ ∂∂ − ∂+ ∂ ∂ − +∂ ∫ ∫ ∫ ( 1 8 ) so that we have 1 [ ( ) ]() ( ) = 0 . (1 ) ( ) b ax a ax FF Dx d x Γα yD y x αα α η ∂∂ − +∂ ∂ ∫ ( 1 9 ) As () x η is arbitrary , the Euler-L agrange equation for the fractional variational principle is () = 0 . ax ax FF D yD y α α ∂∂ − ∂∂ ( 2 0 ) Similarly , we can derive higher order fractional Euler -Lagrange equation (1 ) ( ) = 0 . kk ax k ax FF D yD y α α ∂∂ +− ∂∂ ( 2 1 ) When 1 α= , Eq. (21) can turn out to be the Euler-Lagrange equation in usual sense. 4 Fractional V ariational Iteration Method In Ref. [18], we proposed a fractional VI M for two kinds of fractional diff usion equation. In this section, we solve two frac tional nonlinear equations to illustrate the fractional iteration method’ s efficiency . Example.1. As the first example, we consid er a tim e-fractional dif fusion equation. Previously , Oldham and Spanier [19] solv ed a fractional dif fusion equation that contains first order derivative in sp ace and half order derivative in time. Recently , F . Mainardi [20, 21] investigat ed analytically the time-fractional dif fusion wave equations. In space fractional dif fusion process, G . Gorenflo and F . Mainardi [22] obtained this fractional mode l by replacing the second orde r space derivative with a suitable fractional der ivative operator . The analytical fractional dif fusion equa tion in time is governed by the equation [23] 2 2 ( ,) ( ,) ( ( ) ( ,) ) , ux t ux t F x ux t D tx x α α ∂∂ ∂ =− ∂∂ ∂ 01 , < α≤ ( 2 2 ) where t α α ∂ ∂ is the Caputo derivative, with initial condition (, 0 ) () ux f x = . W e replace the fractional Caputo derivativ e with the modified Riemann-Liouville derivative in Eq. (22), and assume 1 D = , () Fx x = − which leads to 2 2 (, ) (, ) ( (, ) ) , u xt u xt x u xt tx x α α ∂∂ ∂ =+ ∂∂ ∂ 01 , < α≤ ( 2 3 ) with initial condition 2 (, 0 ) ux x = . Then a corrected functional for Eq. (23) can be constructed as follows ~~ 2 1 2 0 (, ) 1 ( ,) ( ( ,) ) (, ) (, ) ( , ) { } ( ) . (1 ) t nn n nn ux ux t x ux t ux t u x t t d Γα xx λ α α + α ∂τ ∂∂ =+ τ − − τ +∂ τ ∂ ∂ ∫ (24) with the property from Eqs. (9 - 1 1 ), (, ) t λ τ must satisfy (, ) 0, t λ α α ∂τ = ∂τ and 1( , ) 0 . t t λ τ= + τ= ( 2 5 ) Therefore, (, ) t λ τ can be identified as (, ) 1 . t λ τ =− Substituting the initial value 2 00 (, ) (, 0 ) () ux t ux f xx = == into the iteration formulation as follows 2 1 2 0 ( , ) ( ,) ( ( ,) ) 1 (, ) (, ) { } ( ) . (1 ) t nn n nn ux ux t x ux t ux t u x t d Γα xx α α + α ∂τ ∂ ∂ =− − − τ +∂ τ ∂ ∂ ∫ W e can derive 2 2 00 0 1 2 0 2 2 (, ) (, ) ( (, ) ) 1 (, ) { } ( ) (1 ) (2 3 ) , (1 ) t α ux ux t x ux t ux t x d Γα xx xt x Γα α α α ∂τ ∂ ∂ =− − − τ +∂ τ ∂ ∂ + =+ + ∫ ( 2 6 ) 22 2 2 2 (2 3 ) ( 8 9 ) (, ) , (1 ) (1 2 ) αα x tx t ux t x Γα Γ α ++ =+ + ++ ( 2 7 ) 22 2 2 3 2 3 (2 3 ) ( 8 9 ) (2 6 2 7 ) (, ) , (1 ) (1 2 ) (1 3 ) αα α x tx t x t ux t x Γα Γ α Γ α ++ + =+ + + ++ + ( 2 8 ) ……. Finally , the exact solution is 0 (, ) l i m (, ) l i m ( ) , (1 ) ii n n nn i kt ux t u x t E k t i α α α →∞ →∞ = == = Γ+ α ∑ ( 2 9 ) where 22 (1 ) ( 3 1) ii kx x =+ + − . If we assume 1 2 α = in Eq. (29), we can derive 1 2 () E kt is the exact so lution of the fractional dif fusion equation [23]. Example.2. In order to illustrate the fr actional VIM for higher fractional-order equations, we only consider the simple initial value problem in [23] (2 2) 1, 0 < 1, 0 1 . yy x α =+ α ≤ ≤ ≤ ( 3 0 ) (0)=0 y and () (0)=1. y α Through this paper , () m y α is defined by ...... ax ax m DD y αα 1 4 4 2 44 3 . Construct the following functional ~ (2 ) 2 1 0 1 (, ) (, ) { ( ) 1 } ( ) (1 ) x nn n n yx t y x t y y d Γα λ α α + =+ − ξ − ξ + ∫ (2 ) 2 1 0 () () ( 2) 0 1 (( ) 1 ) ( ) (1 ) 1 = | ( ) ( ) | ( ) (1 ) t α nn n n x nn x n x n δ y δ y δλ yy d Γα δ y λδ y λτ δ y τλ δ yd Γα α + α αα α ξ= ξ= =+ − ξ − ξ + +− + ξ + ∫ ∫ Similarly , set the coefficients of () n δ u τ and () n δ y α to zero () 1| 0 x λ α ξ= −= , (2 ) 0 λ α = and |0 . x λ ξ= = Then, the generalized Lagrange multipliers for Eq. (31) can be identified as () . (1 ) x λ Γ α ξ− = + α ( 3 1 ) By the fractional variational theory a nd similar manipulation, we can derive more generalized Lagrange m ultipliers (1 ) () () (1 ) , (1 ( 1) ) m m x λ Γ m −α ξ− =− + −α for higher fractional nonlinear ordi nary dif ferential equations () () , 0 < 1 . m yN y α = α≤ From Eq. (31), we can check when 1 α = , λ x = ξ− is the multiplier for the Riccati equation () '' 2 1. yy = + ( 3 2 ) The iteration formulation for Eq. (13) can be rewritten as (2 ) 2 1 0 1( ) () () { 1 } ( ) (1 ) (1 ) x nn n n x y xy x y y d ΓΓ α α α + ξ− =+ − − ξ +α +α ∫ . W ith the fractional Jumarie-T aylor series [13] and the initial value () 0 (0) (0) , (1 ) (1 ) yx x yy ΓΓ αα α =+ = + α+ α W e can obtain 2 (2 ) 2 10 0 0 0 1( ) () () { 1 } ( ) , (1 ) (1 ) (1 ) (1 2 ) x xx x yx y x y y d ΓΓ Γ αα α αα ξ− =+ − − ξ = + +α Γ +α +α + α ∫ (2 ) 2 21 1 1 0 24 5 2 6 2 1( ) () () { 1 } ( ) (1 ) (1 ) (1 2 ) 2 (1 3 ) (1 ) (1 2 ) (1 ) (1 4 ) (1 ) (1 2 ) (1 5 ) (1 4 ) (1 ) (1 6 ) . x x yx y x y y d ΓΓ xx Γ x Γ x ΓΓ Γ Γ Γ Γ Γ Γ x ΓΓ α αα αα α α α ξ− =+ − − ξ +α +α +α + α =+ + +α + α +α + α +α + α + + +α +α +α + α ∫ 2 () y x is the second approximate form of Eq. (30). W e can compare the solutions with the exact solutio ns through Figure.1 when assuming 1. α → W e note that the same r esult can be obtained by the Fractional Decomposition method [25]. 5 Conclusion V ariational Iteration Method has proven as an efficien t tool to so lve nonlinear dif ferential equations of integ er order . In this paper , fractional variational iteration method is proposed. 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