Using orthonormal wavelet basis in Petrov-Galerkin method for solving Fredholm integral equations of the second kind

Purpose - The purpose of this paper is to explain the choice of Alpert multi-wavelet as basis functions to discrete Fredholm integral equation of the second kind by using Petrov-Galerkin method. Design/methodology/approach - In this process, two kinds of matrices are obtained from inner product between basis of test space and trial space; some of them are diagonal with positive elements and some others are invertible. These matrices depend on type of selection of test and trial space basis. Findings - In this process, solution of Fredholm integral equation of the second kind is found by solving the generated system of linear equations. Originality/value - In previous work, convergence of Petrov-Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree. In point of computation, because of appearance of diagonal and invertible matrices, a small dimension system with a more accurate solution is obtained. The numerical examples illustrate these facts.