Solving Nonlinear Klein-Gordon Equation with a Quadratic Nonlinear term using Homotopy Analysis Method
Subject Areas : Operation Research
H.
جعفری
1
(Department of Mathematics and Computer Science, University of Mazandaran, P. O. Box 47416-1467, Babolsar, Iran.)
م.
سعیدی
2
(Department of Mathematics and Computer Science, University of Mazandaran, P. O. Box 47416-1467, Babolsar, Iran.)
م.
عرب فیروزجایی
3
(Department of Mathematics and Computer Science, University of Mazandaran, P. O. Box 47416-1467, Babolsar, Iran.)
Keywords: Homotopy Analysis Method, Adomian decomposition method, KLEIN-GORDON, Partial Differential Equation, Homotopy Perturbation method,
Abstract :
In this paper, nonlinear Klein-Gordon equation with quadratic term is solved by means of an analytic technique, namely the Homotopy analysis method (HAM).Comparisons are made between the Adomian decomposition method (ADM), the exact solution and homotopy analysis method. The results reveal that the proposed method is very effective and simple.
[1] Ayub M., Rasheed A., and Hayat T. Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int. J. Eng. Sci.,41, 2091-2103, 2003.
[2] Adomian G. Solving frontier problems of physics: the decomposition method, Dordrecht, Kluwer Academic Publishers,1994.
[3] Adomian G., and Rach R., On the solution of algebraic equations by the decomposition method, Math. Anal. Appl, 105,141-166, 1985.
[4] Hayat T., Khan M., and Ayub M. On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. Eng. Sci., 42, 123-135, 2004.
[5] Hayat T., Khan M., and Ayub M. Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J. Math. Anal. Appl., 298, 225-244, 2004.
[6] Hayat T., Khan M., and Asghar S. Homotopy analysis of MHD flows of an Oldroyd 8- constant fluid, Acta Mech., 168, 213-232, 2004.
[7] Hayat T., and Khan M., Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear Dyn., 42, 395-405, 2005.
[8] Hayat T., Khan M., and Ayub M., On non-linear flows with slip boundary condition, ZAMP, 56, 1012-1029, 2005.
[9] He J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech., 35(1), 37-43, 2000.
[10] He J. H., Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350(12), 87-88, 2006.
[11] He J.H., Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20(10), 1141-1199, 2006.
[12] Chandra Basak K., Chandra Ray P., and Kumar Bera R., Solution of non-linear Klein- Gordon equation with a quadratic non-linear term by Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 2007, 14( 3), 718-723, 2009.
[13] Liao S. J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[14] Liao SJ., An approximate solution technique which does not depend upon small parameters: a special example, Int. J. Nonlinear Mech., 30, 37180, 1995.
[15] Liao SJ., An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics, Int. J. Nonlinear Mech., 32(5), 815-822, 1997.
[16] Liao SJ., An explicit totally analytic approximation of Blasius viscous flow problems, Int. J. Non-Linear Mech., 34(4), 75-78, 1999.
[17] Liao SJ., Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Boca Raton: Chapman & Hall, 2003.
[18] Liao SJ., On the homotopy anaylsis method for nonlinear problems, Appl. Math. Comput. , 147, 499-513, 2004.
[19] Liao SJ., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput., 169, 1186-1194, 2005.
[20] Liao SJ., A new branch of solutions of boundary-layer ¤ows over an impermeable stretched plate, Int. J. Heat Mass Transfer, 48, 2529-2539, 2005.
[21] Liao SJ., and Pop I., Explicit analytic solution for similarity boundary layer equations, Int. J. Heat Mass Transfer, 47, 75-78, 2004.