Integrating Bernstein and Improved Block-Pulse Functions for Solving Linear Fredholm Integro-Differential Equations

Abstract

Mathematical modeling of real-life problems usually results in some form of functional equations, e.g. algebraic equations, differential equations, integral equations and others. The occurrence of differential equations and integral equations is common in many areas of the sciences and engineering. In particular, the conversion of boundary value problems in differential equations to integro-differential equations, with limits of integration, considered as constant, is termed Fredholm integro-differential equations In this study, issues involving linear Fredholm integro-differential equations are numerically solved using a hybrid of orthogonal functions. To solve these problems, a hybrid method combining improved block-pulse functions and Bernstein is proposed. To convert the solution of integro-differential equations to the solution of algebraic equations, the operational matrices of derivative for this function, together with the hybrid functions, are presented. To demonstrate the practicality and accuracy of the proposed approach in this study, we provide some test problems Examples are given to highlight the accuracy and effectiveness of the proposed method.