Convergence analysis for numerical solution of Fredholm integral equation by Sinc approximation K. Maleknejad a,⇑ , R. Mollapourasl b , M. Alizadeh a a Department of Mathematics, Karaj Branch, Islamic Azad University, Faculty of Science, Karaj, Iran b Department of Mathematics, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Tehran 16788, Iran article info Article history: Received 5 December 2009 Received in revised form 25 September 2010 Accepted 29 September 2010 Available online 7 October 2010 Keywords: Integral equation Fredholm type Collocation method Sinc approximation abstract In this study one of the new techniques is used to solve numerical problems involving inte- gral equations known as Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding accurate solutions. So, in this article, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then by a theorem we show error in the approximation of the solution decays at an exponential rate. Finally, numerical examples are included to demonstrate the validity and applicability of the technique. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The purpose of this paper is to develop high order numerical method for Fredholm integral equations of the first kind defined by Z b a kðs; tÞf ðtÞdt ¼ gðsÞ; 1 < a 6 s 6 b < 1; ð1Þ where k(s, t) and g(s) are known functions and f(t) is the solution to be determined. This type of equations appear in many science and engineering fields, and in many cases, we can not solve this equation analytically to find an exact solution. So that, by using numerical methods we try to estimate a solution for this equation. Regarding to this fact that Fredholm integral equation of the first kind is one of the ill-posed problems, so employing every numerical method dose not work for that, also, without convergence analysis of numerical method we can not guar- antee the numerical scheme. So, choosing a powerful numerical scheme appears a little hard. Numerical and theoretical methods for solving integro-differential and integral equations have been studied by many authors so far [1–9]. Some of them usually use techniques based on an expansion in terms of some basis functions or use some quadrature formulas, and the convergence rate of these methods are usually of polynomial order with respect to N where N represents the number of terms of the expansion or the number of points of the quadrature formula. On the other hand, in [10] it is shown that if we use the Sinc method the convergence rate is O exp c ffiffiffiffi N p     with some c > 0. Although this convergence rate is much faster than that of polynomial order. 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.09.034 ⇑ Corresponding author. Tel.: +98 261 340 74 81; fax: +98 261 341 02 78. E-mail addresses: maleknejad@iust.ac.ir (K. Maleknejad), mollapour@iust.ac.ir (R. Mollapourasl), alizadeh@kiau.ac.ir (M. Alizadeh). URL: http://webpages.iust.ac.ir/maleknejad/ (K. Maleknejad). Commun Nonlinear Sci Numer Simulat 16 (2011) 2478–2485 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns