Boubaker Wavelet Functions for Solving Higher Order Integro-Differential Equations

Abstract

In the present paper, the properties of Boubaker orthonormal polynomials are used to construct new Boubaker wavelet orthonormal functions which are continuous on the interval [0, 1). Then, a Boubaker wavelet orthonormal operational matrix of the derivative is obtained with the new general procedure. The matrix elements can be expressed in a simple form that reduces the computational complexity. The collocation method of the Boubaker orthonormal wavelet functions together with the application of the derived operational matrix of the derivative are then utilized to transform the higher-order integro-differential equation into a solution of linear algebraic equations. As a result, the solution of the original problem reduces to the solution of a linear system of algebraic equations and can be sufficiently solved by an approximate technique. The main advantage of the suggested method is that the orthonormality property greatly simplifies the original problem and leads to easy calculation of the coefficients of expansion. Special attention is needed to perform the convergence analysis. The error is analyzed when a sufficiently smooth function is expanded in terms of the Boubaker orthonormal wavelet functions, then an estimation of the upper bound of the error is calculated. The results obtained by the technique in the current work are reported by solving some numerical examples and the accuracy is checked by comparing the results with the exact solution.

An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation

“…Volterra-Fredholm integro-differential equations (VFIDEs) have arisen in different areas of science and engineering. Population dynamics, oceanopraphy, fluid mechanics, financial mathematics, plasma physics, artificial neural networks, electromagnetic theory and biological processes are among these fields (see, e.g., [10,29]).…”

A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh

“…Volterra-Fredholm integro-differential equations (VFIDEs) have arisen in different areas of science and engineering. Their implementations can be found in electrostatics [14], biological models [37], atomic physics [20], astronomy [16], potential theory [17], fluid dynamics [2] and electromagnetic theory [32] (see, also references therein).…”

“…Tau method was given for one and twodimensional VFIDEs in [36,37]. Collocation method with Boubaker wavelet functions was proposed for high-order VFIDEs in [32]. Integral collocation approximation was employed for high-order VFIDEs in [1].…”