An analytic technique, the homotopy analysis method (HAM), is applied to obtain the approximate analytical solutions of fourth-order integro-differential equations. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact and approximate solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħ, ...

Solving Nonlinear Klein-Gordon Equation with a Quadratic Nonlinear term using Homotopy Analysis Method H. Jafari , M. Saeidy, M. Arab Firoozjaee Department of Mathematics and Computer Science, University of Mazandaran, P. O. Box 47416-1467, Babolsar, Iran. Abstract In this paper, nonlinear Klein-Gordon equation with quadratic term is solved by means of an analytic technique, namely the Homotopy...

Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Frie...

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the present paper, we demonstrate that such a view is only valid in very special cases, and in general the homotopy analysis method is far more robust. In particular...

The generalized Taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of h can’t be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao’s method. We solve the problem. Through a detailed study of a simple ...

‎In this paper‎, ‎an efficient algorithm for solving a system of linear‎ ‎equations based on the homotopy analysis method is presented‎. ‎The‎ ‎proposed method is compared with the classical Jacobi iterative‎ ‎method‎, ‎and the convergence analysis is discussed‎. ‎Finally‎, ‎two‎ ‎numerical examples are presented to show the effectiveness of the‎ ‎proposed method.‎