Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations

Transcription

1 Int. J. Adv. Appl. Math. and Mech. 3( ( (ISSN: IJAAMM Journal homepage: International Journal of Advances in Applied Mathematics and Mechanics Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations Research Article Hardik S. Patel, Ramakanta Meher Applied Mathematics and Humanities Department, S.V.National Institute of Technology, Surat, Gujarat, India Received 6 September 05; accepted (in revised version 07 November 05 Abstract: MSC: Laplace Adomian Decomposition Method (LADM is a combination of Adomian decomposition method (ADM and Laplace Transform. It is an approimate analytical method, which can be adapted to solve system of nonlinear partial differential equations. In this paper, Boussinesq-Burger equation has been solved by using Laplace Adomian Decomposition Method which gives an approimate analytical solution that converges faster to the eact solution by using only few iterates of the recursive relation 44A0 35R70 Keywords: Laplace Adomian Decomposition Method (LADM Boussinesq-Burger equation 05 The Author(s. This is an open access article under the CC BY-NC-ND license ( Introduction The soliton equation is one of the most prominent subjects in the field of nonlinear science. In the past several decades, a great number of efforts have been made to study various nonlinear soliton equations. The traditional methods of solving nonlinear wave equations include inverse scattering theory [,, Backlund transformation [3 5, Darbou transformation [6 and Painlev epansion method [7, etc. With the rapid development of nonlinear science, some new powerful solving methods have been developed, such as homogeneous balance method [8, Jacobi elliptic function method [9, method of bifurcation [0, F-epansion method [, and some approimate method such as HPM and ADM method [3 and method of auiliary equation [4, etc. The Laplace Adomian Decomposition Method (LADM is an approimate analytical method, which can be adapted to solve nonlinear ordinary and partial differential equations. Khuri [5, 6 used this method for the approimate solution of a class of nonlinear ordinary differential equations. Elgazery [7 eploits this method to solve Falkner Skan equation. This analytical technique basically illustrates how the Laplace transform be used to approimate the solutions of the nonlinear differential equations with the linearization of non-linear terms by using adomian polynomials. Jafari [ discussed numerical solutions of telegraph and laplace equations on cantor sets using local fractional laplace decomposition method. Kumar [3 used RDT method for the Solutions of the coupled system of Burgers equations and coupled Klein Gordon equation. Pirzada [4 discussed Solution of fuzzy heat equations using Adomian Decomposition method. Consider the nonlinear homogeneous Boussinesq Burger equation: u t v + uu = 0 ( v t u + (uv = 0, 0 ( Normally it arises in the study of fluid flow and describe the propagation of shallow water waves. Here and t represents the normalized space and time respectively where u(, t is the horizontal velocity field and v(, t represent the height of water surface above a horizontal level at the bottom. Corresponding author. address: hardy.nit@gmail.com (Hardik S. Patel, meher_ramakanta@yahoo.com (Ramakanta Meher

2 Hardik S. Patel, Ramakanta Meher / Int. J. Adv. Appl. Math. and Mech. 3( ( Basic idea of Laplace Adomian Decomposition Method (LADM Consider the system of partial differential equations in operator form D t u + R (u, v + N (u, v = g D t v + R (u, v + N (u, v = g (3 with initial conditions u(,0 = f ( v(,0 = f ( (4 Where D t is considered here as the first order partial differential operators, R and R are linear operator, N and N are nonlinear operators, g andg are inhomogenogenous terms. By applying Laplace transform to both side of Eq. (3 and using initial conditions (4, it obtain L [D t u + L [R (u, v + L [N (u, v = L [g L [D t v + L [R (u, v + L [N (u, v = L [g (5 Using the differentiation property of Laplace transform, it gives L [u = f ( + s s L [g s L [R (u, v s L [N (u, v L [v = f ( + s s L [g s L [R (u, v s L [N (u, v (6 The Laplace Adomian decomposition method decomposes the unknown functions u(, t and v(, t by an infinite series of components as u(, t = v(, t = u n (, t v n (, t (7 and the nonlinear operators N (u, v and N (u, v can be represented by an infinite series so called Adomian polynomials N (u, v = N (u, v = A n B n The Adomian polynomials can be generated for all forms of nonlinearity. They are determined by the following relations [ [ A n = d n ( n! dλ n N λ i u i i=0 λ=0 [ [ B n = d n ( n! dλ n N λ i v i (9 i=0 λ=0 Substituting Eqs. (7 and (8 into Eq. (6, it gives [ L u n (, t = f ( + s s L [g ( ([ [ s L R u n, [ L v n (, t = f ( + s s L [g ( ([ [ s L R u n, v n ( s L A n, v n ( s L B n, (0 Applying the linearity of Laplace transform in Eq. (0, it obtains the following recursively formula L [u n (, t = f ( + s s L [g s L [v n (, t = f ( + s s L [g s (R (u n, v n L L (A n, s (R (u n, v n L L (B n, ( s (8

3 5 Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations Matching both sides of Eq. (, it yields the following iterative relation L [u 0 = f ( + s s L [g L [v 0 = f ( + s s L [g ( L [u = s L [R (u 0, v 0 s L [A 0 L [v = s L [R (u 0, v 0 s L [B 0 (3 for k, the recursive relations are given by L [u k+ = s L [R (u k, v k s L [A k L [v k+ = s L [R (u k, v k s L [B k (4 Finally, by applying the inverse Laplace transform, we can evaluate u k and v k. 3. Application of Laplace Adomian Decomposition Method to Boussinesq -Burger equation Consider the general Boussinesq -Burger equation [9 of the form u t v + uu = 0 (5 v t u + (uv = 0, 0 (6 with initial conditions u(,0 = ck + ck ( k lnb tanh ( v(,0 = k k + lnb 8 sech The eact solution of Eqs. (5 and (6 is given by [3 u(, t = ck + ck ( ck tanh t k lnb ( v(, t = k k ck t + lnb 8 sech Applying Laplace transform to both sides of Eqs. (5-(6 and using initial conditions (7 and (8, we have L [D t u = [ [ v L L u u L [D t v = L [ 3 u 3 L [ (uv Using the differentiation property of Laplace transform, we get + [ v L L Or sl [u = ck + ck ( k lnb tanh ( sl [v = k k + lnb 8 sech L [u = ( ck s + ck ( k lnb tanh L [v = ( k ( k + lnb s 8 sech + L [ 3 u s L L [ v s L [ 3 u 3 [ u u [ (uv s [ L u u s L [ (uv (7 (8 (9 (0 ( ( (3 (4 (5 (6

4 Hardik S. Patel, Ramakanta Meher / Int. J. Adv. Appl. Math. and Mech. 3( ( Here the nonlinear terms uu and (uv can be represented by an infinite series so called Adomian polynomials [8 as uu = (uv = A n B n Where A n s and B n s are the Adomian polynomials to be determined. On substituting Eqs. (7 and (7 into Eqs. (5 and (6 respectively. it obtains [ L [ L u n = s v n = s ( ck + ck ( k lnb tanh ( k ( k + lnb 8 sech + + s L [ ( s L [ 3 ( 3 v n [ s L A n u n [ s L B n (7 (8 (9 Table. The absolute error in the solution of Boussinesq-Burger equation using two terms approimation for LADM at various points with c =, k = and b =. (, t u E act u L ADM v E act v L ADM u E act u HP M v E act v HP M (0., E E E E-05 (0., E E E E-04 (0., E E E E-04 (0., E E E E-04 (0., E E E E-03 (0., E E E E-05 (0., E-04.50E E-04.50E-04 (0., E E E E-04 (0., E E E E-03 (0., E E E E-03 (0.3, E E E E-05 (0.3, E E E E-04 (0.3, E E E E-04 (0.3, E-04.94E E-04.94E-03 (0.3, E E E E-03 (0.4,0..055E E E E-05 (0.4, E E E E-04 (0.4, E E E E-04 (0.4, E E E E-03 (0.4, E-04.86E E-04.86E-03 (0.5, E E E E-05 (0.5, E E E E-04 (0.5, E E E E-04 (0.5, E-04.74E E-04.74E-03 (0.5,0.5.95E E-03.95E E-03

5 54 Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations Table. L and L error norm for Boussinesq-Burger equation use two terms approimation for LADM and HPM at various points. Laplace Adomain Decomposition Method (LADM Homotopy Perturbation Method (HPM Error in case of two terms approimation for u(, t Error in case of two terms approimation for v(, t Error in case of two terms approimation for u(, t Error in case of two terms approimation for v(, t L L L L L L L L E-4 9.E-4.485E-3.485E E-4 9.E-4.485E-3.485E E E-4 9.4E-4.639E E E-4 9.4E E E E-4.766E E E E-4.766E E E-4.039E-3.86E-3.340E E-4.039E-3.86E E-4.9E-4.074E-3.93E-3.85E-4.9E-4.074E-3.93E-3 Table 3. The absolute error in the solution of Boussinesq Burger equation using two terms approimation for LADM at various points with c =, k = and b =. (, t u E act u L ADM v E act v L ADM u E act u HP M v E act v HP M (0., E-07.95E E-07.95E-06 (0., E E E E-06 (0., E E E E-05 (0., E E E E-05 (0.,0.5.00E E-04.00E E-04 (0.,0..046E E E E-06 (0., E E E E-06 (0., E E E E-05 (0., E E E E-05 (0.,0.5.34E-04.4E-04.34E-04.4E-04 (0.3,0..7E E-07.7E E-07 (0.3, E E E E-06 (0.3, E-05.88E E-05.88E-05 (0.3, E E E E-05 (0.3, E E E E-05 (0.4,0..0E E-07.0E E-07 (0.4, E E E E-06 (0.4, E E E E-05 (0.4, E E E E-05 (0.4, E E E E-05 (0.5,0..600E E E E-07 (0.5,0..0E E-06.0E E-06 (0.5, E E E E-05 (0.5, E E E E-05 (0.5, E E E E-05

6 Hardik S. Patel, Ramakanta Meher / Int. J. Adv. Appl. Math. and Mech. 3( ( Table 4. L and L error norm for Boussinesq Burger equation use three terms approimation for LADM and HPM at various points. Laplace Adomain Decomposition Method (LADM Homotopy Perturbation Method (HPM Error in case of three terms approimation for u(, t Error in case of three terms approimation for v(, t Error in case of three terms approimation for u(, t Error in case of three terms approimation for v(, t L L L L L L L L E-5.0E E-5.409E E-5.0E E-5.409E E-5.34E E-5.4E E-5.34E E-5.4E E-5.453E-4 5.9E E E-5.453E-4 5.9E E E-5.539E E E E-5.539E E E E-5.596E-4.03E E E-5.596E-4.03E E-5 Fig.. Comparison of two terms LADM solution (Lines and eact solution (Circle of u(, t for Boussinesq-Burger equation when c =, k = and b =. Fig.. One soliton approimate solution of u(, t for Boussinesq-Burger equation when c =, k = and b =. where the Adomian polynomials are given by A 0 = u 0 u 0, A = u u 0 + u 0 A = u u 0 + u u 0 A 3 = u 3 + u. u, u + u u 0, (30 u + u u + u u 3 0, and B 0 = (u 0v 0, B = (u 0v + u v 0, B = (u 0v + u v + u v 0, (3 B 3 = (u 0v 3 + u v + u v + u 3 v 0,.

7 56 Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations Fig. 3. Comparison of two terms LADM solution (Lines and eact solution (Circle of v(, t for Boussinesq-Burger equation when c =, k = and b =. Fig. 4. One soliton approimate solution of v(, t for Boussinesq-Burger equation when c =, k = and b =. Now from Eq. (8, we obtain the following recursively formula as L [u 0 = ( ck s + ck ( k lnb tanh L [v 0 = ( k ( k + lnb s 8 sech (3 L [u = L [v = s L s L [ v0 [ 3 u 0 3 s L [A 0 For k, the recursive relations are given by L [u k+ = L [v k+ = s L s L [ vk [ 3 u k 3 s L [B 0 (33 s L [A k s L [B k (34 By applying the inverse Laplace transform in Eqs. (3 and (33, it obtains u 0 = 4 4 tanh ( ln( v 0 = 8 sech ( ln( (35 u = 6 t cosh ( ln( v = 0.065t sinh( cosh ( (36

8 Hardik S. Patel, Ramakanta Meher / Int. J. Adv. Appl. Math. and Mech. 3( ( cosh 6 ( cosh 4 ( t cosh 5 ( ( sinh cosh 3 ( ( sinh cosh ( cosh ( ( sinh u = ( ( cosh ( sinh cosh ( ( sinh cosh 5 ( t sinh ( cosh 4 ( ( cosh cosh 3 ( ( sinh cosh 7 ( cosh 6 ( ( sinh v = ( ( cosh ( sinh 9. (37 Hence, the approimate solutions of Boussinesq -Burger equation is given by u(, t =u 0 (, t + u (, t + u (, t +... = 4 ( 4 tanh ln( t 6 cosh ( ln( cosh 6 ( cosh 4 ( t cosh 5 ( ( sinh cosh 3 ( ( sinh cosh ( cosh ( ( sinh ( ( cosh ( sinh 8 + (38 v(, t =v 0 (, t + v (, t + v (, t +... = 8 sech ( ln( t sinh( cosh ( cosh ( ( sinh cosh 5 ( t sinh ( cosh 4 ( ( cosh cosh 3 ( ( sinh cosh 7 ( cosh 6 ( ( sinh ( ( cosh ( sinh 9 + Eqs. (38 and (39 represents the approimate solution of Eqs. (5 and (6 having it s numerical values have been represented in Table and Table 3. (39 4. Conclusion Here Laplace Adomain Decomposition Method (LADM has been successfully developed and applied for solution of Boussinesq-Burger equation. By keen observation we concluded that the Laplace Adomain Decomposition Method (LADM is a powerful and efficient technique that can be used for finding approimate analytical solution of a system of nonlinear partial differential equation. Acknowledgment The authors are thankful to the Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, for providing research facilities.

9 58 Application of Laplace Adomian Decomposition Method for the soliton solutions of Boussinesq-Burger equations References [ M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Vol. 49, Cambridge university press, 99. [ M. Wadati, Stochastic korteweg-de vries equation, Journal of the Physical Society of Japan 5 (8 ( [3 G. L. Lamb Jr, Elements of soliton theory, New York, Wiley-Interscience, 300 p. 9 95: ( [4 M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method, backlund transformation and an infinite number of conservation laws, Progress of theoretical physics 53 (( [5 K. Konno, M. Wadati, Simple derivation of backlund transformation from riccati form of inverse method, Progress of Theoretical Physics 53 (6 ( [6 V. B. Matveev, M. A. Salle, Darbou transformations and solitons, Springer, 99. [7 F. Cariello, M. Tabor, Painleve epansions for nonintegrable evolution equations, Physica D: Nonlinear Phenomena 39 ( ( [8 M. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to eact solutions of nonlinear equations in mathematical physics, Physics Letters A 6 ( ( [9 S. Liu, Z. Fu, S. Liu, Q. Zhao, Jacobi elliptic function epansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A 89 ( (00 69-âĂŞ74. [0 J. Li, L. Zhang, Bifurcations of traveling wave solutions in generalized pochhammer chree equation, Chaos, Solitons & Fractals 4 (4 ( [ Y. Zhou, M. Wang, Y. Wang, Periodic wave solutions to a coupled kdv equations with variable coefficients, Physics Letters A 308 ( ( [ J.-L. Zhang, M.-L. Wang, Y.-M. Wang, Z.-D. Fang, The improved f-epansion method and its applications, Physics Letters A 350 ( ( [3 S. M. El-Sayed, D. Kaya, An application of the adm to seven-order sawada kotara equations, Applied mathematics and computation 57 ( ( âĂŞ0. [4 Sirendaoreji, A new auiliary equation and eact travelling wave solutions of nonlinear equations, Physics Letters A 356 ( ( [5 S. A. Khuri, A laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics (4 ( [6 S. Khuri, A new approach to bratus problem, Applied Mathematics and Computation 47 ( ( [7 N. S. Elgazery, Numerical solution for the falkner-skan equation, Chaos, Solitons & Fractals 35 (4 ( âĂŞ [8 G. Adomian, Solving frontier problems of physics: The decomposition method kluwer academic publisher, Boston, MA. [9 J. Zhang, Y. Wu, X. Li, Quasi-periodic solution of the (+ -dimensional boussinesq-burgers soliton equation, Physica A: Statistical Mechanics and its Applications 39 ( [0 Z. Liang, Z. Li-Feng, L. Chong-Yin, Some new eact solutions of jacobian elliptic function about the generalized boussinesq equation and boussinesq? burgers equation, Chinese Physics B 7 ( ( [ A. A. Rady, M. Khalfallah, On soliton solutions for boussinesq-burgers equations, Communications in Nonlinear Science and Numerical Simulation 5 (4 ( [ H. Jafari, & H. K. Jassim, Numerical solutions of telegraph and laplace equations on cantor sets using local fractional laplace decomposition method, International Journal of Advances in Applied Mathematics and Mechanics (3 ( [3 A. Kumar, & R. Arora, Solutions of the coupled system of Burgers equations and coupled Klein Gordon equation by RDT Method. International Journal of Advances in Applied Mathematics and Mechanics ( ( [4 U. M. Pirzada, & D. C. Vakaskar, Solution of fuzzy heat equations using Adomian Decomposition method, International Journal of Advances in Applied Mathematics and Mechanics 3( ( Submit your manuscript to IJAAMM and benefit from: Regorous peer review Immediate publication on acceptance Open access: Articles freely available online High visibility within the field Retaining the copyright to your article Submit your net manuscript at editor.ijaamm@gmail.com