In this paper, a computational method based on a hybrid of parabolic and block-pulse functions is proposed to solve a system of linear and special nonlinear Fredholm integral equations of the second kind. The convergence and error bound are analyzed. Numerical examples are given to illustrate the efficiency of the method. | Turk J Math (2017) 41: 787 – 796 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Convergence Analysis of Parabolic Basis Functions for Solving Systems of Linear and Nonlinear Fredholm Integral Equations Yousef JAFARZADEH∗, Bagher KERAMATI Department of Mathematics, Semnan University, Semnan, Iran Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, a computational method based on a hybrid of parabolic and block-pulse functions is proposed to solve a system of linear and special nonlinear Fredholm integral equations of the second kind. The convergence and error bound are analyzed. Numerical examples are given to illustrate the efficiency of the method. Key words: Hybrid, parabolic functions, block-pulse functions, Fredholm integral equations, fixed-point iteration method, error bound analysis 1. Introduction Second kind integral equations have recently attracted attention due to their wide applications in various areas of science and engineering; for example, many problems in plasma physics [6] or electrical engineering [7] result in solving some second kind integral equations. Usually the explicit solution of an integral equation system is difficult to derive. Hence it is necessary to seek efficient numerical solutions. There are many different basis functions such as the Adomian decomposition method [2, 21], Legendre collocation method [18], Tau method [9], method of Taylor’s expansion [10, 19], homotopy perturbation method [11], method of spline collocation [5, 13, 17], Runge–Kutta [15, 25], Sinccollocation method [8, 20], block-pulse functions [4, 12] and hat function [3] that have been used to get approximate solutions of integral equations. Recently, the idea of hybrid functions has been exploited to improve the convergence rate of the numerical solution of integral equations. For example, the combination of block-pulse .
