Symmetry Analysis of 2+1 dimensional Burgers equation with variable damping

Symmetry Analysis of 2+1 dimensional Burgers equation with v ariable damping D. P a ndiara ja 1 and B. Ma yil V aganan 2 1 Departmen t o f Mathematics, Thiagara jar College, Madurai-6 2 5009, India 2 Departmen t of Applied Mathematics a nd Statistics, Madurai Kamar a j Univ ersit y , Madurai-62502 1 , India Abstract The symmetry classification of the t w o dimensional Burg ers equation with v ariable co- efficien t is considered. Symmetry algebra is found and a classification of its subalgebras, up to conjugacy , is obtained. Similarity reductions are p erfor med fo r eac h class. Keyw ords Lie symmetries, Symmetry analysis, Killing form. AMS Classification Num b ers 2 2 E60, 27E70, 34A05, 35G20. 1.In tro duction Gandarias [5] has studied Ty p e-I I hidden symmetries of the tw o dimensional Burg ers equation u t + uu x − u xx − u y y = 0 . (1.1) In [4], applications of (1.1) hav e b een disc ussed. In this pap er we pro vide a detailed symmetry analysis of the t w o dimensional Burgers equation with v ariable damping, viz., u t + uu x + α ( t ) u − u xx − u y y = 0 . (1.2) One o f the significant application of Lie symme try groups to differen tial equations is to ac hiev e a complete classification of its symmetry reductions. The symmetry prop erties and reductions of ce rtain differen tia l equations ha ve b een recen tly in v estigated (See references 1 0-12 in [1 ]). As described in [8], the classification of group in v ariant solutions requires a classific ation of subalgebras of the symmetry algebra in to conjuga cy classes under the adjoint action o f the symmetry group. This pap er is organized a s follows. In section 2, w e p erform a symmetry classification of (1.2) and a classification of one-dimensional, t w o-dimensional, three-dimensional subalgebras of the symmetry algebra. In section 3 , we ta bulate the reductions of (1 .2) under one-dimensional, t w o-dimensional, three-dimensional subalgebras. W e summarize the results in section 4. 2. The symmetry alb egra and classification of subalgebras Equation (1.2) is assumed to b e inv ariant under Lie group of infinitesimal tra nsforma t ions (Olv er [7], Blumen and Kumei [3]) x ∗ i = x i + ǫξ i ( x, y , t, u ) + O ( ǫ 2 ) , i = 1 , 2 , 3 , 4 , (2.1) 1 where ξ 1 = ξ , ξ 2 = η , ξ 3 = τ , ξ 4 = φ . Then the fo urth prolongation pr (4) V of V = τ ( x, y , t ; u ) ∂ t + ξ ( x, y , t ; u ) ∂ x + η ( x, y , t ; u ) ∂ y + φ ( x, y , t ; u ) ∂ u , (2.2) m ust satisfy pr (4) V Ω( x, y , t ; u ) | Ω( x,y ,t ; u )=0 = 0 , (2.3) where Ω is the RHS of (1.2). The followin g system of 12 detemining equations are obtained from (2.3) (See [6]). ( ξ 1 ) u = 0 , ( ξ 2 ) u = 0 , ( ξ 3 ) u = 0 , ( φ 1 ) u,u = 0 , ( ξ 3 ) y = 0 , ( ξ 3 ) x = 0 , − ( ξ 2 ) t − u ( ξ 2 ) x + ( ξ 2 ) x,x + ( ξ 2 ) y , y − 2( φ 1 ) y , u = 0 , φ 1 − ( ξ 1 ) t − u ( ξ 1 ) x + 2 u ( ξ 2 ) y + ( ξ 1 ) x,x + ( ξ 1 ) y , y − 2( φ 1 ) x,u = 0 , uα t ξ 3 + αφ 1 + 2 uα ( ξ 2 ) y + ( φ 1 ) t − uα ( φ 1 ) u + u ( φ 1 ) x − ( φ 1 ) x,x − ( φ 1 ) y , y = 0 , ( ξ 1 ) y + ( ξ 2 ) x = 0 , − ( ξ 1 ) x + ( ξ 2 ) y = 0 , 2( ξ 2 ) y − ( ξ 3 ) t = 0 . Solving these determining equations w e get ξ = q 2 x + c 2 + c 1 Z e − R αdt dt, (2.4) η = q 2 y + p, (2.5) τ = q t + r, (2.6) φ = − q 2 u + c 1 e − R αdt , (2.7) under the condition that ( q t + r ) α t = 0 (2.8) Th us w e hav e the fo llowing theorems: Theorem 1. If α ( t ) = α 0 , a real constan t then symmetry alg ebra L c of (1.2) can b e written as L c = V 1 + V 2 + V 3 + V 4 where V 1 = − e − α 0 t α 0 ∂ x + e − α 0 t ∂ u , V 2 = ∂ t , V 3 = ∂ y and V 4 = ∂ x pro vided q = 0 , r 6 = 0 T able 1- Comm utato r table for the Lie algebra L c : 2 V 1 V 2 V 3 V 4 V 1 0 α 0 V 1 0 0 V 2 − α 0 V 1 0 0 0 V 3 0 0 0 0 V 4 0 0 0 0 Theorem 2 . If α ( t ) 6 = 0 is an a rbitrary function of time t, then the general elemen t of the symmetry algebra L of (1.2) can b e written as L = V 1 + V 2 + V 3 where V 1 = R e R − αdt dt∂ x + e R − αdt ∂ u , V 2 = ∂ x and V 3 = ∂ y . It should b e noted here that the generators V 1 , V 2 , V 3 comm ute. The problem of en umeration of all subalgebras of a giv en finite-dimensional Lie algebra L is essen tial for the group analysis of differen tial equations. F or this group Int L of the inner automor- phisms can b e found. Under t he op era t io n of an automorphism, ev ery algebra transforms into a subalgebra of the same dimensionalit y . During the construction of optimal systems for a giv en Lie alg ebra L a sp ecial role is play ed b y the cen ter Z. The p oint is that ev ery vec tor z ∈ Z is inv erian t relative to a n automorphism A ∈ I ntL . Th us t he central elemen ts g enerate subalgebras (ideals), whic h cannot c hanged by any automorphisms. They can b e included as a direct term in an y subalgebra of the Lie algebra L. Th us if the optimal syste ms are kno wn fo r t he factor algebra L/Z, then they can b e considered a s kno wn for the entire Lie algebra L. It is sfficien t to describ e metho ds for constructing opt ima l systems only for Lie algebras with n ull cen ters [8]. Case 1 . α is a constan t. W e assume that α = α 0 . The comm utator ta ble gives the follo wing information ab o ut the structure o f the Lie a lg ebra L c . The Lie algebra L c can b e written a s L c = Z ⊕ L 2 where Z = { V 3 , V 4 } is the cen ter of the Lie algebra and L 2 = { V 1 , V 2 } W e shall pro ceed to construct the optimal system no w. First the mapping ad(V) is calculated b y t he equation ad ( V ) < x > = [ x, V ] , where V is the general v ector, V = V 1 V 1 + V 2 V 2 and x ∈ L 2 and it turns out to b e ad(V)=  α 0 V 2 − α 0 V 1 0 0  The genaral automorphism of the gro up Int L 2 can find b y construting one-para meter g roups of Automorphisms A i ( t ) for ev ery basis v ector V 1 , V 2 T o find the group A 1 ( t ) corresp ondingto V 1 , it is necces sary to solve the equation ∂ t x ′ = [ x ′ , V ], solving w e get the matrix o f auto morphism 3 A 1 ( t )=  1 − α 0 t 0 1  T o find the group A 2 ( t ) corresp ondingto V 2 , it is necces sary to solve the equation ∂ t x ′ = [ x ′ , V ], solving w e get the matrix o f auto morphism A 2 ( t )=  e α 0 t 0 0 1  Here it is con v enien t to assume t = 1 α 0 a in A 1 ( t ) and t = 1 α 0 lnb in A 2 ( t ) A = A 1 ( 1 α 0 a ) oA 2 ( 1 α 0 ) l nb (2.9) The genaral automorphism A ∈ I ntL 2 is obtained and it dep ends on t w o parameters, a, b : A =  b − a 0 1  No w, it is p ossible to find the comp onen ts of the v ector x ′ = A ( x ) in the bais { V 1 , V 2 } : x ′ 1 = bx 1 − ax 2 , x ′ 2 = x 2 (2.10) The decomp osition of the space L 2 , on the classes o f similar vec tors, can b e found using the in v ariance prop ert y of the killing fo r m. The killing form is 1 α 2 0 K ( x, x ) = ( x 2 ) 2 , (2.11) and is inv ariant of the group I ntL 2 . Since K ( x, y ) 6 = 0, it is not degenerate and the algebra L 2 is semi simple. The vec tors are separated in to tw o disjoin t classes: 1) K ( x, x ) > 0, 2) K ( x, x ) = 0 A represen tative of class (1) is V 2 and a r epresen tative o f class (2) is V 1 . Hence Θ 1 = { V 1 , V 2 } . Th us the sub algebras of the symmetry alg ebra L c is giv en in the following table: T able 3 Classification of subalgebras of symmetry algebra L c Dimension Subalgebra 4 1-dimensional subalgebra L 1 = ( aV 3 + bV 4 + V 1 ) L 2 = ( aV 3 + bV 4 + V 2 ) L 3 = ( aV 3 + bV 4 ) 2-dimensional subalgebra L 4 = ( aV 3 + bV 4 , V 1 ) L 5 = ( aV 3 + bV 4 , V 2 ) L 6 = ( V 3 , V 4 ) 3-dimensional subalgebra L 7 = ( aV 3 + bV 4 , V 1 , V 2 ) Case 2 . α is arbitrary The commu tator table giv es the follow ing informat io n ab out the structure of the Lie algebra L . The Lie a lg ebra L itself a cen ter hence a maximal ideal. Th us the subalgebras of the symmetry algebra can b e classified a s in t able 4. T able 4 Classification of subalgebras of symmetry algebra L Dimension Subalgebra 1-dimensional subalgebra L 1 = ( aV 1 + bV 2 + cV 3 ) 2-dimensional subalgebra L 2 = ( V 1 , V 2 ) L 3 = ( V 1 , V 3 ) L 4 = ( V 2 , V 3 ) 3-dimensional subalgebra L 5 = ( V 1 , V 2 , V 3 ) 3. Symmetry reduction for (1.2) F or brevit y , we presen t only t w o represen tative reductions that to o f o r the case when α is arbitrary b elow. The complete details o f reduced equations for eac h subalgebra when α is either a constan t or an y function of t are presen ted in ta ble forms. When α is arbit r a ry , the reductions to PDEs and t hen to ODEs a r e g iv en resp ectiv ely in App endix-A and App endix-B. Similarly , when α is a constan t, the resp ectiv e reductions are given in App endix-C and App endix-D. 3.1 Reduction under one dimensional subalgebra V 1 + V 2 + V 3 5 The c haracteristic equation for aV 1 + bV 2 + cV 3 is dx 1 + R e R − α ( t ) dt = dy 1 = dt 0 = du e R − α ( t ) dt (3.1) In t egrating the c haracteristic equation w e get three similarity v aria bles ξ = x − y (1 + Z e − R α ( t ) dt ) , η = t, u = y e − R α ( t ) dt + F ( ξ , η ) (3.2) In terms of the similarit y v a riables, (1.2) is reduced to (1 + (1 + Z e − R α ( η ) dη ) 2 ) F ξ ξ − F η − F F ξ − α ( η ) F = 0 (3.3) W e giv e a complete table of reductions t o PDEs for all one dimensional subalgebras in App endix-A. 3.1 Reduction under tw o dimensional sub algebra [ V 2 , V 3 ] Consider the algebra giv en b y V 2 and V 3 . S ince [ V 2 , V 3 ] = 0, we b egin with V 2 = ∂ ∂ x . The similarit y v ariables for this generators are g iv en b y ξ = y , η = t, u = F ( ξ , η ). Using these v ariables (1.2) reduces to a PDE F ξ ξ − F η + α ( η ) F = 0 . (3.4) In order to p erform second reduction of t he ab ov e equation, w e firstly write V 3 in terms of new v ariables ξ , η and F ( ξ , η ) as V 3 = Conclusions The symmetry classification of the t w o dimensional Burgers equations with constant and v ariable co efficien ts, viz., u t + uu x + α 0 u = u xx − u y y = 0 , (3.5) u t + uu x + α ( t ) u = u xx − u y y = 0 , (3.6) ha v e b een carried out. F or, the symmetry a lgebras and their subalgebras, up to conjugacy , are en umerated. Successiv e reductions to PDEs with tw o indep enden t v ariables a nd then to ODEs of b oth second and first orders are p erformed for each subalgebra. As most of the reduced equations are of the standard for ms or can b e c hanged to canonical forms through simple transformatio ns w e disp ense with the problem of writing solutio ns to these equations. References 1. Azad H and Mustafa M T 2007 J. Math. A na l. Appl . 333 1 180-118 8 6 2. Barbara Abraham Shrauner and Keshlan S Go vinder 2006 J. Nonlin. M ath. Phys.) 13 612-622 3. Bluman G W and Kumei S Symmetries and Diff e r ential Equations , Spring er- V erlag, New Y ork, 1989. 4. Edw ards M P and Broadbridge P 1995 Z. Angew. Math. Phys. 46 595 - 622 5. Gandaria s M L 200 8 J. Math. Anal. Appl. 348 752-759 6. Gerd Baumann Symm etry A nalysis of Differ ential Equations with Mathematic a , Springer- V erlag, New Y ork, 2000. 7. Olv er P J Applic ations of Lie Gr oups to Di ff er ential Equations , Springer- V erlag, New Y ork, 1986. 8. Ovsiannik o v L V Gr oup analysis o f Differ en tial Equations , Academic Press, New Y ork, 1982 App endix-A S ubal g ebra Reduced eq uation V 1 F ξ ξ − F η − α ( η ) F = 0 V 2 F ξ ξ − F η − α ( η ) F = 0 V 3 F ξ ξ − F η − F F ξ − α ( η ) F = 0 V 1 + V 2 F ξ ξ − F η − α ( η ) F = 0 V 2 + V 3 2 F ξ ξ − F η − F F ξ − α ( η ) F = 0 V 1 + V 3 (1 + ( R e − R α ( η ) dη ) 2 ) F ξ ξ − F η − F F ξ − α ( η ) F = 0 V 1 + V 2 + V 3 (1 + (1 + R e − R α ( η ) dη ) 2 ) F ξ ξ − F η − F F ξ − α ( η ) F = 0 App endix B S ubal g ebra Reduced eq u ation [ V 1 , V 2 ] W r + α ( r ) W = 0 [ V 1 , V 3 ] W r + α ( r ) W = 0 [ V 2 , V 3 ] W r + α ( r ) W = 0 [ V 1 , V 2 , V 3 ] W r + α ( r ) W = 0 App endix C 7 S ubal g ebra C ond ition Reduced eq u ation aV 3 + bV 4 + V 1 a 6 = 0 , b 6 = 0 ( a 2 + ( b + e − α 0 η ) 2 ) F ξ ξ − F F ξ − F η − α 0 F = 0 a = 0 , b 6 = 0 F ξ ξ − F η − α 0 F = 0 a 6 = 0 , b = 0 ( a 2 + e − 2 α 0 η α 2 0 ) F ξ ξ − F F ξ − F η − α 0 F = 0 a = 0 , b = 0 F ξ ξ − F η = 0 aV 3 + bV 4 + V 2 a 6 = 0 , b 6 = 0 F ξ ξ + F ηη − aF F ξ + bF ξ + aF η − α 0 F = 0 a = 0 , b 6 = 0 F ξ ξ + F ηη − F F ξ + bF ξ − α 0 F = 0 a 6 = 0 , b = 0 F ξ ξ + F ηη − F F ξ + aF η − α 0 F = 0 a = 0 , b = 0 F ξ ξ + F ηη − F F ξ − α 0 F = 0 aV 3 + bV 4 a 6 = 0 , b 6 = 0 ( a 2 + b 2 ) F ξ ξ − aF F ξ − F η − α 0 F = 0 a = 0 , b 6 = 0 F ξ ξ − F F ξ − F η − α 0 F = 0 a 6 = 0 , b = 0 F ξ ξ − F η − α 0 F = 0 App endix D S ubal g ebra C ond ition Reduced eq u ation [ aV 3 + bV 4 , V 1 ] a 6 = 0 , b 6 = 0 W r + α 0 W = 0 a = 0 , b 6 = 0 W r + α 0 W = 0 a 6 = 0 , b = 0 W r + α 0 W = 0 [ aV 3 + bV 4 , V 2 ] a 6 = 0 , b 6 = 0 ( a 2 + b 2 ) W r r − aW W r − α 0 W = 0 a = 0 , b 6 = 0 W r r − W W r − α 0 W = 0 a 6 = 0 , b = 0 W r r − α 0 W = 0 [ V 3 , V 4 ] W r + α 0 W = 0 [ aV 3 + bV 4 , V 1 , V 2 ] a 6 = 0 , b 6 = 0 W r + α 0 W = 0 8