Application of the Homotopy Analysis Method for Solving the Systems of Linear and Nonlinear Integral Equations

Abstract

In this paper we indicate some applications of homotopy analysis method for solving the systems of linear and nonlinear integral equations. The method is based on the concept of creating function series. If the series converges, its sum is the solution of this system of equations. The paper presents conditions to ensure the convergence of this series and estimation of the error of approximate solution obtained when the partial sum of the series is used. Application of the method will be illustrated by examples.Keywords: homotopy analysis method, convergence, error estimation, system of Fredholm integral equations, system of Volterra integral equations.

A series solution of the boundary value problem arising in the application of fluid mechanics

“…As there are some limitations on the use of perturbation method such as the existence of the small parameter, the use of this method is restricted to many nonlinear problems. To overcome these pitfalls, some efficient analytical methods such as homotopy perturbation method (HPM) (He, 1999;He, 2005;Jalilpour et al, 2016), differential transformation method (Rashidi et al, 2015), laplace decomposition method (Khan, 2014), homotopy perturbation transform method (Madani et al, 2011;Morales-Delgado et al, 2016;G omez-Aguilar et al, 2017), variational iteration method (Słota, 2009;Faraz and Khan, 2010), Feng's first integral method (Yepez-Martineza et al, 2016) and homotopy analysis method (Brociek et al, 2016) are developed and several researchers have applied these methods for various nonlinear problems arising in different disciplines of science, engineering and technology (Dehghan and Salehi, 2011;Hetmaniok et al, 2015;Khan et al, 2012). HPM is an efficient method for solving nonlinear problems, which does not require the parameter to be small and this method was well tested and used by many researchers (Hetmaniok et al, 2012;Khan and Latifizadeh, 2013;Fathizadeh and Rashidi, 2009;Turkyilmazoglu, 2011).…”