Galley Proof Analysis in Theory and Applications Anal. Theory Appl., Vol. 29, No. 3 (2013), pp. 197-207 DOI: 10.4208/ata.2013.v29.n3.1 1 A Method for Solving Fredholm Integral Equation 2 of the First Kind Based on Chebyshev Wavelets 3 M. Bahmanpour 1 and M. A. Fariborzi Araghi 2, ∗ 4 1 Department of Mathematics, Sama Technical and Vocational Training College, 5 Islamic Azad University, Khorasgan Branch, Isfahan, Iran 6 2 Department of Mathematics, Central Tehran Branch, Islamic Azad University, 7 P.O.Box 13185.768, Tehran, Iran 8 Received 7 October 2011; Accepted (in revised version) xx xxxx 9 Available online xxx 10 11 12 Abstract. In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other. Key Words: First kind Fredholm integral equation, Galerkin and Modified Galerkin method, 13 Legendre wavelets, Chebyshev wavelets. 14 AMS Subject Classifications: 65R20, 65T60 15 16 1 Introduction 17 Many inverse problems in sciences and engineering lead to the solution of the following 18 integral equation of the first kind, 19  b a k( x, t)y(t)dt = g( x), a ≤ x ≤ b, (1.1) where g( x) and k( x, t) are known functions and y( x) is the unknown function to be de- 20 termined. In general, this kind of integral equation is ill-posed for a given kernel k and 21 driving term g [6]. For this reason, the special method should be introduced to solve it. 22 In recent years, several numerical methods for approximating the solution of Eq. (1.1) 23 are known. Among these methods, the methods based on the wavelet are more attrac- 24 tive and considerable. The wavelets technique allows the creation of very fast algorithms 25 ∗ Corresponding author. Email addresses: m fariborzi@iauctb.ac.ir, fariborzi.araghi@gmail.com (M. A. Fariborzi Araghi) http://www.global-sci.org/ata/ 197 c 2013 Global-Science Press