Physical Models for Sustainability using Fredholm Integro-Differential Equations: Applicability and Analysis of Chebyshev Polynomial Method

Abstract

Numerical analysis is concerned with the mathematical derivation, explanation and evaluation/analysis of algorithms, models and methods used to obtain numerical solutions for mathematical problems. This paper explores the reliability of the Chebyshev Polynomial Method (CPM) for solving a specific class of equations known as the second-order Fredholm Integro-Differential Equations (FIDEs). A series expansion of the Chebyshev polynomial is derived, used in solving these integral equations, and later on examined in terms of accuracy and convergence of solutions. The evaluation process involves a hybrid approach, combining manual methods and mathematical programs like MAPLE and MATLAB. In addition, three numerical examples were solved in which two truncation points are considered per each example. Furthermore, the performance of the CPM is reported in terms of accuracy, convergence, suitability, reliability and effectiveness in the context of the exact solution.