Potential Symmetries and Differential Forms. for Wave Dissipation Equation

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 42, HIKARI Ltd, Potential Symmetries and Differential Forms for Wave Dissipation Equation V. G. Gupta and Kapil Pal Department of Mathematics, University of Rajasthan, Jaipur, India Copyright 2013 V. G. Gupta and Kapil Pal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The differential form method is applied to study the wave equation with dissipation and extended to determine potential symmetries. The symmetry group are given and group invariant solutions associated to the symmetries are obtained. Keywords: Differential-form, Symmetry-generators 1. Introduction The method of writing system of differential equation in terms of differential forms and finding their symmetries was introduced by Harrison and Estabrook [11]. Papachristou and Harrison [17-19], generalized the method to vector valued or Lie algebra valued differential forms and used in two-dimensional Dirac equation and the Yang Mills free field equations in Minkowski space-time. Edeled [3-6], developed the theory of differential forms and explore the use of differential form in physics. Gupta and Sharma [8] had worked on nonlinear diffusion equation with convection term. Haager et. al. [9] are analyzes the symmetries of a class of nonlinear telegraph equations are examined. Pakdemirli et. al. [15, 16] considered boundary layer equations for non-newtonian fluids. Davison and Kara [2] treated Burgers equation to obtain potential and approximate symmetries using differential form method. A generalized nonlinear Shröndinger equation with attention to both symmetries and Bäcklund

2 2062 V. G. Gupta and Kapil Pal transformations were considered by Harnad and Winternitz [10]. Web et. al. [20, 21] are analyzes a nonlinear magnetic potential equation with conservation laws, the Liouville equation and nonlinear Shröndinger equations for a type of MHD waves using differential form method. Kara et. al. [12], was studied the nonlinear wave equation with variable long wave velocity and the Gordon-type equations (in particular, the φ4-model equation). Ozer and Suhubi [14] considered nonvacuum Maxwell equations with nonlinear constitutive relations. In the present study, we obtained the determined equations for system of linear differential equation for wave equation with dissipation. And constricted a range of symmetry generators, translation and scaling etc. are related to conservation the laws, with the aid of these symmetries explicit new solution are derived. Finally, the group invariant solutions are obtained for all the cases for system of differential equation. 2. Determined Equations of system of linear differential equation 2.1: Consider the linear wave dissipation equation u tt + u t = u xx (2.1.1) For consideration of differential form of equation (2.1.1), we consider the following auxiliary system: v = u x, w = u t, w + w t = v x (2.1.2) we introduce the following 2-forms: = du dt v dx dt = u x dx dt v dx dt; = du dx w dt dx = u t dt dx w dt dx = w dx dt + dw dx + dv dt = w dx dt + w t dt dx + v x d x dt = w dx dt - w t dx dt + v x d x dt which gives the system (2.1.2) when annulled. Here we drop the wedge product to save writing. Consider the symmetry of equation (2.1.1) in the form X = t + x + + u v + w Lie derivatives of, and as = X d + d ( X ) = X ( dv dx dt ) + d ( dt du v dt + v dx ) = ( x + v x + v t + ) dt dx + ( u + v u t ) dt du + ( v + v v ) dt dv + ( w + v w ) dt dw + ( x v u ) dx du v v dx dv v w dx dw + v du dv + w du dw (2.1.3) The Lie derivatives of as = X d + d ( X ) = X ( dw dt dx ) + d ( dx du + w dt w dx ) = ( t w x w t ) dt dx + ( w u t ) dt du + ( w v ) dt dv + ( w w ) dt dw + ( u x w u ) dx du + ( v + w v ) dx dv + ( w + w w ) dx dw + v du dv + w du dw (2.1.4)

3 Potential symmetries and differential forms 2063 The Lie derivatives of as = X d + d ( X ) = X (dw dx dt)+ d (w dt w dx + dx dw + dt dv) = ( w x w t x + t ) dt dx + ( w u u ) dt du + ( w v v t ) dt dv + ( w w w t ) dt dw + (w u u ) dx du + (w v v x ) dx dv + (w w w x ) dx dw u du dv u du dw + ( w v ) dv dw (2.1.5) From equation (2.1.3), (2.1.4) and (2.1.5), we have the following system of determination equations table dt dx : x + v x + v t + = 0 (1) dt du : u + v u t = 0 (2) dt dv : v + v v = 0 (3) dt dw : w + v w = 0 (4) dx du : x v u = 0 (5) dx dv : v v = 0 (6) dx dw : v w = 0 (7) du dv : v = 0 (8) du dw : w = 0 (9) dt dx : t w x w t = 0 (10) dt du : w u t = 0 (11) dt dv : w v = 0 (12) dt dw : w w = 0 (13) dx du : u x w u = 0 (14) dx dv : v + w v = 0 (15) dx dw : w + w w = 0 (16) du dv : v = 0 (17) du dw : w = 0 (18) dt dx : w x w t x + t = 0 (19) dt du : w u u = 0 (20) dt dv : w v v t = 0 (21) dt dw : w w w t = 0 (22) dx du : w u u = 0 (23) dx dv : w v v x = 0 (24) dx dw : w w w x = 0 (25) du dv : u = 0 (26) du dw : u = 0 (27) dv dw : w v = 0 (28)

4 2064 V. G. Gupta and Kapil Pal From the determinate equation we obtain: x = 0, u = 0, v = 0, w = 0, t = 0, u = 0, v = 0, w = 0, v = 0, w = 0, u = 0, w = 0, u = 0, v = 0 and from the equations (2), (14), (21) and (25) u = v = w = x = t (2.1.6) From equation (2.1.6) we have ut = vt = wt = xt = tt = 0, ux = vx = wx = xx = tx = 0, uu = vu = wu = xu = tu = 0, uv = vv = wv = xv = tv = 0, uw = vw = ww = xw = tw = 0, (2.1.7) Adding the equations (2) and (21) ( u + v ) 2 t = 0 which gives from equations (14) and (25) t = ½ ( u + v ) (2.1.8) ( u + w ) 2 x = 0 which gives x = ½ ( u + w ) (2.1.9) from equation (2), (14), (21) and (25) (2 u + v + w ) 2 t 2 x = 0 which gives u = ½ ( v + w ) t x (2.1.10) This implies = [ ½ ( v + w ) t x ] u + A(t, x) (2.1.11) from equation (2.1.11) x = A x (2.1.12) and t = A t (2.1.13) using equation (2.1.12) in equation (1) = A x v ( x + t ) (2.1.14) using equation (2.1.13) in equation (10) = A t w ( x + t ) (2.1.15) from equation (19) t x + t = 0, we have A t + A tt = A xx (2.1.16) where A(t, x ) is an arbitrary function of x and t which is the solution of given equation (1) of wave propagation with dissipation, again from equation (2), (14), (21) and (25) we have u = v = w = c 3 (say) (2.1.17) which gives = c 3 t + c 1 (2.1.18) = c 3 x + c 2 (2.1.19) from (2.1.14) using (2.1.17) and (2.1.18) = A x + 2 c 3 v (2.1.20) from (2.1.15) using (2.1.18) = A t + 2 c 3 w (2.1.21) from (2.1.11) = c 3 u + A(t, x) (2.1.22) hence the require vector fields x 1 = t, x 2 = x, x 3 = t t x x + u u + 2 v v + 2 w w, x = A(t, x) u + A x v + A t w when v = 0 and w = 0 the commutation relation table given as follows x 1 x 2 x 3 x x x 1 x A` x x 2 x A`` x 3 x 1 x 2 0 x t x A` x x A`` x x A` x A`` ( x t x A` x x A``) 0

5 Potential symmetries and differential forms 2065 where x A` = A t u, x A`` = A x u Symmetry groups are G 1 : ( t +, x, u), G 2 : ( t, x +, u), G 3 : ( t e, x e, u e ), G : ( t, x, u + A), Solution for symmetry groups u (1) = f (t, x ), u (2) = f (t, x ), u (3) = e f (t e, x e ), u = f (t, x ) + A(t, x) Conclusion The proposed method has been successfully applied to analyzing the wave equation. Potential and Lie point symmetry have been obtained for the wave equation. Further, using Lie point symmetry groups, the solution of the problem have been obtained. One can simply write the differential equations as a set of second order equations and then the differential forms can be written by inspection. The method is also easy to apply for symbolic computation for Lie point symmetry, cf. Edelen [7]. Carminati et al.[1] introduces a useful computer program liesymm in MAPLE is use the proposed method. Thus, the proposed method has been extended to solve a large class of problems in nonlinear differential equations. REFERENCES [1] Carminati J., Devitt J. S. and Fee G. J., Isogroups of differential equations using algebraic computing, J. Symbolic Comp., 14 (1992) [2] Davison, A. H., and Kara, A. H., Symmetries and Differential Forms, Journal of Nonlinear Mathematical Physics, 15(1) (2008) [3] Edelen, D. G. B., Isovector fields for problems in the mechanics of solids and fluids, Internat, J. Engrg. Sci., 20 (1982) [4] Edelen, D. G. B., On solving problems in the mechanics of solids and fluids by a generalized method of characteristics, Internat, J. Engrg. Sci., 26 (1988) [5] Edelen, D. G. B., Order-independent mehod of charactristics, Internat, J. Theoret. Phys., 28 (1989) [6] Edelen, D. G. B., Implicit similarities and inverse isovector methods, Arch. Rat. Mech. And Anal., 82 (1983) [7] Edelen, D. G. B., Programs for calculation of isovector fields in the reduce.2 environment, center for the application of mathematics, Lehigh University, (1981). [8] Gupta V. G., and Sharma P., Symmetries and the differenatial form for a nonlinear diffusion equation with convection term, Acta Universitatis Apulensis, No. 28, (2011),

6 2066 V. G. Gupta and Kapil Pal [9] Haager G., Baumann G. and Nonnenmacher T. F., Symmetries of nonlinear telegraph equation in strong fields, Mathematical and Computational Applications, Vol 1, No. 2, (1996) [10] Harnad J. and Winternitz P., Pseudopotentials and Lie symmetries for the generalized nonlinear schrdinger equation, J. Math. Phys. 23 (1982), [11] Harrison B. K. and Estabrook F. B., Geometric approach to invariance groups and solution of partial differential systems, J. Maths.Phys., 12 (1971) [12] Kara A. H., Bokhari A. H. and Zaman F. D., On the Exact Solutions of the Nonlinear Wave and φ4-model Equations, Journal of Nonlinear Mathematical Physics, Vol. 15, Supplement 1 (2008), , [13] Ovsyannikov, L. V., The group analysis of differential equations, Nauka, Moscow, (1978). [14] Ozer S. and Suhubi E. S., Equivalence transformations for first order balance equations, Internat. J. Engrg. Sci. 42 (2004) [15] Pakdemirli M., Yurusoy M. and Kucukbursa A., Symmetry group of boundary layer equations of a class of non-newtonian fluids, Internat. J. Non- Linear Mech., 31 (1996) [16] Pakdemirli M., Yurusoy M., Equivlence transformations applied to exterior calculus approach for finding symmetries: an example of non- Newtonian fluid flow, Internat. J. Non-Linear Mech., 37 (1999) [17] Papachristou C. J. and Harrison B. K., Isogroups of differential idealsof vector-valued differential forms, application to partial differential equations, Acta Appl. Math., 11 (1988) [18] Papachristou C. J. and Harrison B. K., Symmetry groups of partial differential equations associated with vector valued differential forms, Proceedings of the XV International Colloquium in Group Theoretical Methods in Physics, Editor R. Gilmore, Singapore, World Scientific. (1987), [19] Papachristou C. J. and Harrison B. K., Some aspects of isogroup of the selfdual Yang Mills system, J. Math. Phys., 28 (1987) [20] Webb G. M., Similarity considerations and conservation laws for magnetostatic atmospheres, Solar Phys., 106 (1986) [21] Webb G. M., Brio M. and Zank G. P., Symmetries of the triple degenerate DNLS equations for weakly nonlinear dispersive MHD waves, J. Plasma Phys., 54 (1995) Received: June 1, 2013