On the convergence of the homotopy analysis method to solve the system of partial differential equations

Subject Areas : Partial differential equations

A. Fallahzadeh ¹ , M. A. Fariborzi Araghi ² , V. Fallahzadeh ³

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran
3 - Department of Mathematics, Islamic Azad University, Arac Branch, Iran

Received: 2015-06-12 Accepted : 2015-10-20 Published : 2015-05-01

Keywords: Homotopy Analysis Method, System of partial differential equations, H-surface,

Abstract :

One of the efficient and powerful schemes to solve linear and nonlinear equationsis homotopy analysis method (HAM). In this work, we obtain the approximate solution ofa system of partial differential equations (PDEs) by means of HAM. For this purpose, wedevelop the concept of HAM for a system of PDEs as a matrix form. Then, we prove theconvergence theorem and apply the proposed method to defined the approximate solution ofsome systems of PDEs. Also, we show the region of convergence by plotting the H-surface.

References:

[1] S. Abbasbandy, Homotopy analysis method for the Kawahara equation, Nonlinear Analysis: Real World Applications 11 (2010) 307-312.

[2] S. Abbasbandy, Solitary wave solutions to the modified form of CamassaHolm equation by means of the homotopy analysis method, Chaos, Solitons and Fractals 39 (2009) 428-435.

[3] J. Biazar, M. Eslami, A new homotopy perturbation method for solving system of partial differential equations, Computers and Mathematics with Applications 62 (2011) 225-234.

[4] J. Biazar, M. Eslami, H. Ghazvini, Homotopy perturbation method for system of partial differential equations, International Journal of Nonlinear Sciences and Numerical simulations 8 (3) (2007) 411-416.

[5] M.A. Fariborzi Araghi, A. Fallahzadeh, On the convergence of the Homotopy Analysis method for solving the Schrodinger Equation, Journal of Basic and Applied Scientific Research 2(6) (2012) 6076-6083.

[6] M.A. Fariborzi Araghi, A. Fallahzadeh, Explicit series solution of Boussinesq equation by homotopy analysis method, Journal of American Science, 8(11) (2012).

[7] T. Hayat, M. Khan, Homotopy solutions for a generalized second-grade uid past a porous plate. Nonlinear Dyn 42 (2005) 395-405.

[8] S.J. Liao, Beyond pertubation: Introduction to the homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, (2003).

[9] S.J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Communication in Nonlinear Science and Numnerical Simulation, 14 (2009) 983-997.

[10] P. Roul, P. Meyer, Numerical solution of system of nonlinear integro-differential equation by Homotopy perturbation method, Applied Mathematical Modelling 35 (2011) 4234-4242.

[11] A. Sami Bataineh, M.S.M. Noorani, I.Hashim, Approximation analytical solution of system of PDEs by homotopy analysis method, Computers and Mathematics with Applications 55 (2008) 2913-2923.

[12] F. Wang, Y. An, Nonnegative doubly periodic solution for nonlinear teleghraph system, J.math.Anal.Appl. 338 (2008) 91-100.

[13] A.M. Wazwaz, The variational iteration method for solving linear and nonlinear system of PDEs, Comput, Math, Appl 54 (2007) 895-902.

[14] W. Wu, Ch. Liou, Out put regulation of two-time-scale hyperbolic PDE systems, Journal of Process control 11 (2001) 637-647.

[15] W. Wu, S. Liao, Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals, 26 (2005) 177-185.

[16] E. Yusufoglu, An improvment to homotopy perturbation method for solving system of linear equations, Computers and Mathematic with Applications 58 (2009) 2231-2235.