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Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD Thesis, Shanghai Jiao Tong University, Shanghai.

has been cited by the following article:

• TITLE: Application of HAM for Nonlinear Integro-Differential Equations of Order Two

  AUTHORS: Zainidin Eshkuvatov, Davron Khayrullaev, Muzaffar Nurillaev, Shalela Mohd Mahali, Anvar Narzullaev

  KEYWORDS: Integral-Differential Equations, Homotopy Analyses Method, Iterative System, Algebraic Equations, Gauss-Legendre Quadrature Formula

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.11 No.1, January 17, 2023

  ABSTRACT: In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.