International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 Lie s Symmetries of (+)dim PDE S. Padmasekaran and S. Rajeswari Department of Mathematics Periyar University Salem - 66 Tamil Nadu India. e mail: padmasekarans@yahoo. co.in e mail: rajeswarisa@gmail. com Astract: In this paper we apply Lie s Classical Method to a (+) dimensional PDE equation to identify all possile solution for which this equation admits an exact solution and we otain Lie algera of infinitesimal symmetries is spanned y the six vector fields. We conclude that there is an infinite group of point transformations are invariant using invariant form method.we otained local symmetry classifications. MSC : 5AC and 5R. Key Words : Lie Group Transformation Transform Methods Lie Classical Method Symmetries and Invariant.. Introduction In recent years there have een so many research to find symmetries for system of nonlinear (+) dimensional PDEs. In this paper we consider the (+) dimensional PDE wy v wx u wt + u + vy. () wt + wx + wyy. () The aove system () is equivalent to In the present work we first apply Lie classical method (Bluman and Cole Bluman and Kumei George M.Murphy Olver Bluman and Anco5 Olver P.J68 Iragimov7 Daniel Zwillinger9 Sachdev.P. L. and Tesdall M. and Hunter.K ) to the second order PDE() and show that this system is invariant under a group containing six parameters using invariant form method. ISSN: -57 Page 8
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 We consider a scalar second order PDE in three variales F (x x x u u u u u u u u u u ). Here u u u. x u x u u x u u x u u x u u x x u u x x () u u x u u x x The one-parameter Lie group of transformations x x x u x + ξ (x x x u ) + O( ) x + ξ (x x x u ) + O( ) x + ξ (x x x u ) + O( ) u + η(x x x u) + O( ) () leaves PDE () invariant if and only if its second extension also leaves () invariant. Let ξi (x x x u) i + η(x x x u) xi u (5) e the infinitesimal generator of (). Let () ξi (x x x u) i + + () ηi (x x x u u u u ) + + η(x x x u) xi u i ui () ηi i (x x x u u u u u u u u u u ) i () e the second extended infinitesimal generator of (5)where ηi ui i is given y () Ux + Uu ux ξx ux ξx ux ξx ux ξu ux ξu ux ux ξu ux ux () η ξu ux η () η Ux + Uu ux ξx ux ξx ux ξx ux ξu ux ux ξu ux ux Ux + Uu ux ξx ux ξx ux ξx ux ξu ux ux ξu ux ux ξu ux (j) () Ux x + Uux ux + Uuu ux + Uu ux x ξx x ux ξux ux ux ξx ux x ξuu ux ux ξu ux x ux ξu ux ux x ξx x uxx ξux ux ξx ux x ISSN: -57 (7) (8) and ηi i y η (6) Page 8 (9)
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 ξuu ux ξu ux x ux ξx x ux ξux ux ux ξx ux x ξuu ux ux ξu ux x ux ξu ux ux x () () η Ux x + Uux ux + Uux ux + Uuu ux ux + Uu ux x ξx x ux ξux ux ξx ux x ξuu ux ux ξ ux ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ux ξx ux x ξuu ux ux ξux ux ξu ux x ux ξx x ux ξux ux ux ξx ux xt ξuu ux ux ux ξux ux ux ξu ux x ux ξu ux ux x ξx ux x ξx ux x ξx ux x ξu ux ux x ξu ux ux x ξu ux ux x () () Ux x + Uux ux + Uux ux + Uuu ux ux + Uu ux x ξx x ux ξux ux ux ξx ux x ξuu ux ux ξ ux ux ξu ux x ux ξx x ux ξux ux ux ξx ux x ξuu ux ux ux ξux ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ux ξx ux xt ξuu ux ux ξux ux ξuu ux ux ξu ux ux x ξx ux x ξx ux x ξx ux x ξu ux ux x ξu ux ux x () () Ux x + Uux ux + Uuu ux + Uu ux x ξx x ux ξux ux ux ξx ux x η η ξuu ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ξx ux x ξuu ux ξu ux x ux ξx x ux ξux ux ux ξx ux x ξuu ux ux ξu ux x ux ξu ux ux x () η () η () Ux x + Uux ux + Uux ux + Uuu ux ux + Uu ux x ξx x ux ξux ux ξx ux x ξux ux ux ξuu ux ux ux ξ ux ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ux ξx ux x ξuu ux ux ξux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ξx ux x ξuu ux ux ξux ux ux ξu ux ux x ξu ux ux x ξx ux x ξx ux x ξx ux x ξu ux ux x () Ux x + Uux ux + Uuu ux + Uu ux x ξx x ux ξux ux ux ξx ux x ξuu ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ux ξx ux x ξux ux x ξuu ux ux ξu ux x ux ξu ux ux x ξx x ux ξux ux ξx ux x ξuu ux ξu ux x ux (5) We then have () F (x x x u u u u u u u u u u ) If u (x x x ) is an invariant solution of () corresponding to (5) admitted y PDE () then ISSN: -57 Page 8
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 u (x x x ) is an invariant surface of (5) if (u (x x x )) when u (x x x ) or ξi (x (x x x )) i η(x x x (x x x )) xi (6) And u (x x x ) solves () if F (x x x u u u u u u u u u u ). Equation (6) is known as the invariant surf ace condition for an invariant solutions corresponding to (5). We otain the invariant solutions using the invariant form method descried elow: Lie s Classical Method is applied to a single PDE () and we otain Lie algera of infinitesimal symmetries is spanned y the six vector fields.. Lie s Classical Method to single PDE We now seek the following group of infinitesimal transformations w t x y w + W (t x u) t + T (t x y u) x + (t x y u) y + Y (t x y u) (7) under which () is invariant. Then Wt + Ww wt Tt wt Tw wt t wx w wx wt Yt wy Yw wt wy h +wx Wx + Ww wx Tx wt Tw wt wx x wx w wx Yx wy Yw wx wy i +Wyy + Wwy wy + Www wy + Ww wyy Tyy wt Twy wt wy Ty wty Tww wy wt Tw wyy wt Tw wy wty yy wx wy wx wy y wxy wy wxy ww wy wx w wyy wx w wy wxy Yyy wy Ywy wy Yww wy Yw wy wyy Tw wy wty w wy wxy Yy wyy Yw wy wyy. (8) Now replacing the highest derivative term wyy using () equation (8) ecomes Wt + Ww wt Tt wt Tw wt t wx w wx wt Yt wy Yw wt wy h +wx Wx + Ww wx Tx wt Tw wt wx x wx w wx Yx wy Yw wx wy i +Wyy + Wwy wy + Www wy Tyy wt Twy wt wy Ty wty Tww wy wt ISSN: -57 Page 8
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 Tw wyy wt Tw wy wty yy wx wy wx wy y wxy wy wxy ww wy wx w wy wxy Yyy wy Ywy wy Yww wy Tw wy wty w wy wxy [Ww w wx Yw wy Yy Yw wy ] wt + wx. (9) Equating the coefficients of wt wx wt wt wy wty wxy to zero we otain Tw w Yw Tx Yx Ty y. Again equating the derivativess of wy wx wx wt wy to zero we otain the six linear partial differential equations Www Wx t W w x + Yy Wt + Wyy Yy Tt Wwy Yt Yyy () () (). () () (5) Integrating equation () we otain Ww k (x t y). (6) W k (x t y)w + k (x t y). (7) Again integrating equation (6) Integrating with respect to y equation () yields Y Tt y + h (t). (8) Putting equation (8) into equation(5) and integrating with respect to y we arrive at k y Ttt + h y + k (x t). (9) 5 ISSN: -57 Page 85
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 Sustituting (7) in (8) and integrating with respect to x we get Z Ttt y + Tt x + xyh + k (x t)dx + (t). 8 () Inserting equations () and (7) into equation () we otain wkx + kx t. But we find that (x t). Therefore kx t kx. () Using () and integrating with respect to x equation () ecomes xy Z x Tttt y + Ttt + h + k dt + x (t) + H (y t) k 8 () where H (y t) is a constant of integration. Sustituting (7) () and () in equation () we get wkt + kt + wkyy + kyy. () Now collecting the different powers of w we get two equations kt + kyy kt + kyy () (5) x y x x Ttttt y + Tttt + h + k (t) + x (tt) + Ht + Tttt + Hyy. 8 8 6 (6) Using () equation (5) reduces to Now equating the different powers of x to zero we get y 5 Ttttt y + h Tttt + 8 6 tt k + Ht + Hyy. (7) (8) (9) Solving equations (7) and (8) we otain that a t + t + c h t + t + h d t + d. T h () 6 ISSN: -57 Page 86
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 Sustituting equations (9) and () in equation () we get a h y + kt +. () Clearly we conclude that h a k t + a. () Putting () () into equations () and () we find that a a y + at + x + xy + t + a x + d t + d + a 8 a a k x + t + a t + a + d x + H (y t). 8 () () Sustituting ()-() in equation () we conclude that a must e zero. Inserting a and equations () ()into equations () () (8) and (7) finally we get T t + c y + t + h Y a x + xy + d t + d + x + a a) W y+ w + (a t + a ) + d x + H (y t) with a (5) (6) (7) (8) + Ht + Hyy. Special Case : For H k from (5)-(8) we otain T t + c Y y + t + h x + xy + d t + d W yw + d x. (9) (5) (5) (5) 7 ISSN: -57 Page 87
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 Consequently +T +Y +W x t y w x + xy + d t + d + [t + c] x t + y + t + h + yw + d x y w y xy yw x +t + + +t + x t y x y w +d t +x + d + c + h. x w x t y (5) Thus the Lie algera of infinitesimal symmetries of () is spanned y the six vector fields. x y +t + x t y xy yw +t + x y w t + x x w x 5 t 6 y (5) The commutation relations etween these vector fields are given y the following tale. The entry in row i and column j representing [i j ]: Tale 5 6 5 6 5 5 6 6 The corresponding local one parameter groups are the following : G : (u x y t) (σs ) x y exp s t exp s exp s 8 ISSN: -57 Page 88
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 G : (u x y t) (σs ) G G G5 G6 (u x y t) (u x y t) (u x y t) (u x y t) : : : : xy yw exp s t exp s exp s (t exp s x exp s) (w x + s y t) (w x y t + s) (w x y + s t). (σs ) (σs ) (σs5 ) (σs6 ) (55) where σsj exp(sj ) j 6. Since each local Lie group Gj is a symmetry group exponentiation shows that if w w(x y t) is a solution of () then so are w (x y t) w (x y t) w (x y t) w (x y t) w5 (x y t) w6 (x y t) x y w exp( s) t exp( s) exp( s) xy yw w exp( s) t exp( s) exp( s) w (t exp( s) x exp( s)) w (x s y t) w (x y t s) w (x y s t). (56). CONCLUSION : In this paper we found new symmeries for the equation() and the infinite group of point transformations are invariant. And also we found the corresponding local one parameter groups (55). The Lie algera of infinitesimal symmetries of () is spanned y the six vector fields eqn (5). The corresponding local one parameter groups are otained eqn (55). And also each local Lie group Gj is a symmetry group exponentiation shows that if w w(x y t) is a solution of () then so are eqn(56). Acknowledgements: This research work was supported y the University Grants Commission - Special Assistance Programme (UGC-SAP) New Delhi India through the letter N o.f.5/7/drs /6 (SAP ) dated Sept. 6. References [] G. W. Bluman and J. D. Cole The general similarity solution of the heat equation J. Math. Mech. 8 (969) 5-7. [] G. W. Bluman and S. Kumei Symmetries and Differential Equations Springer-Verlag New York 989. 9 ISSN: -57 Page 89
International Journal of Mathematics Trends and Technology (IJMTT) - Volume 5 Numer 6 Novemer 7 [] George M. Murphy Ordinary Differential Equations and their Solutions D. Van Nostrand Company Inc New Jersey 96. [] P. J. Olver Equivalance Invariants and Symmetry (Berlin Springer-Verlag 986). [5] G. W. Bluman and S. C. Anco Symmetries and Integration methods for Differential Equations (New York Springer-Verlag ). [6] Olver P. J. Applications of Lie Groups to Differential Equations Springer-Verlag New York 986. [7] Iragimov N. H. CRC Hand Book of Analysis of Differential Equations Exact Solutions and Conservation Laws CRC Press New York 99. [8] Olver P. J. Equivalence Invariants and Symmetry Camridge University Press Camridge 996. [9] Daniel Zwillinger Handook of Differential Equations Academic Press New York 99. [] Sachdev P. L. A Compendium on Nonlinear Ordinary Differential Equations John-Wiley & Sons Inc New York 997. [] Sachdev P. L. and Mayil Vaganan B. Exact free surface flows for shallow water equations Stud. Appl. Math. 9: 57-76 (995). [] Tesdall M. and Hunter K. Self-similar solutions for weak shock reflection Siam J. Appl. Math. 6: (). ISSN: -57 Page 9