Discussions on Solutions of Fredholm Integral Equations

Abstract

This thesis investigates the analytical and numerical solutions of Fredholm integral equa- tions of the second kind, focusing on both homogeneous and non-homogeneous cases. In the analytical framework, we explore solutions with degenerate kernels and methods to ap- proximate complex kernels by degenerate forms, providing efficient pathways for problem simplification. The study covers both theoretical and applied aspects, emphasizing cases where the kernel exhibits specific structures that facilitate closed-form solutions. Numer- ical methods are also developed and analyzed for solving these integral equations in both homogeneous and non-homogeneous settings. The numerical approaches are evaluated for their accuracy and computational efficiency, providing insights into practical implemen- tations and approximations for real-world applications. Through comprehensive analysis and comparison of methods, this research contributes to the effective solution of Fredholm integral equations, offering new perspectives for both theoretical exploration and applied problem-solving.