Computational Approach for Solving Fredholm Integral Equations

The Fredholm integral equation, a pivotal aspect of integral equations, arises in various applications across physics, engineering, and applied mathematics. This survey examines a computational approach for solving Fredholm integral equations, focusing on both the first and second kinds. For this purpose, the categories of general methods for solving integral equations are mentioned in summary. Also, the general classification of basis functions are reviewed. The properties and characteristics of entire domain and subdomain basis functions are mentioned. Finally, we survey the formulation of the computational approach in detail for solving Fredholm integral equations. The approach uses a set of entire domain functions in order to formulate an efficient numerical scheme. Employing entire domain functions in the implementation of the numerical scheme offers significant benefits and major advantages, including global representation of the solution, reduction of computational effort, fast convergence, simplicity in implementation, compactness of representation, avoidance of discontinuity issues, and numerical flexibility. Various examples are evaluated in order to show the effectiveness and accuracy of the approach. We also examine the convergence rate of the approach numerically. The results of implementation of the approach using a computational software are given for a better illustration. The results confirm the efficiency and accuracy of the approach to solve Fredholm integral equations.