Existence of solution and solving the integro-differential equations system by the multi-wavelet Petrov-Galerkin method

In this paper, we discuss about existence of solution for integro-differential system and then we solve it by using the petrov-galerkin method. in the petrov-galerkin method choosing the trial and test space is important, so we use alpert multi-wavelet as basis functions for these spaces. orthonormality is one of the properties of alpert multi-wavelet which helps us to reduce computations in the process of discretizing and we drive a system of algebraic equations with small dimension which it leads to approximate solution with high accuracy. we compare the results with similar works and it guarantees the validity and applicability of this method.