A wavelet Petrov-Galerkin method for solving integro-differential equations

In this paper, we solve integro-differential equation by using the Alpert multiwavelets as basis functions. We also use the orthogonality of the basis of the trial and test spaces in the Petrov-Galerkin method. The computations are reduced because of orthogonality. Thus the final system that we get from discretizing the integro-differential equation has a very small dimension and enough accuracy. We compare the results with [M. Lakestani, M. Razzaghi, and M. Dehghan, Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math. Probl. Eng. 2006 (2006), pp. 1-12, Article ID 96184] and [A. Ayad, Spline approximation for first-order Fredholm integro-differential equation, Stud. Univ. Babes-Bolyai. Math., 41(3), (1996), pp. 1-8] which used a much larger dimension system and got less accurate results. In [Z. Chen and Y. Xu, The Petrov-Galerkin and iterated Petrov-Galerkin methods for second kind integral equations, SIAM J. Numer. Anal. 35(1) (1998), pp. 406-434], 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.