Convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method for solving Fredholm integro-differential equations of high orders

This work was intended as an attempt to motivate readers for a comparison study of constructions of Legendre multiwavelet and Chebyshev multiwavelet. It is also shown how to use them in Petrov-Galerkin approach for solving Fredholm integro-differential equation of high orders of the second kind. In fact, a numerical technique for the discretization method of Fredholm integro-differential equations is presented that yields linear system. The important point to note here is the convergence of presented methods. For the first time, two conditions are proved for convergence of Legendre and Chebyshev multiwavelets in Petrov-Galerkin method. The proof of these conditions with using linear algebra and matrix theory ensures that Petrov-Galerkin methods has a unique approximation. Finally, some relevent numerical examples, for which the exact solution is known, will indicate accuracy and applicability of the proposed method.