Numerical Solution of Second-Order Fredholm Integro-Differential Equations Using the Chebyshev Collocation Method

This paper explores the reliability of the Chebyshev collocation Method (CCM) as a numerical approach used 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. This method transforms FIDEs and the conditions into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. The selected FIDEs encompass diverse practical scenarios hailing from various scientific disciplines. These equations undergo a rigorous numerical analysis to evaluate both the convergence and accuracy of the obtained numerical solutions. The evaluation process involves a hybrid approach, combining manual methods and mathematical programs like MAPLE and MATLAB. In addition, to ensure the analysis is thorough, three numerical examples were solved at N=3 and N=5 for each examples. Furthermore, the performance of the Chebyshev collocation method is reported in terms of accuracy, convergence, suitability, reliability and effectiveness in the context of the exact solution. Moreover, the numerical results obtained show that the Chebyshev collocation method yields high-accuracy solutions, with good convergence and requires little computational effort, thereby confirming its reliability as a good approach for the numerical solution of second-order FIDEs. Finally, it is concluded that the Chebyshev collocation method is very effective and simple.