Analytic methods to solve Fredholm integral equations

This manuscript presents an innovative semi-analytical approach for solving linear Fredholm integral equations (FIEs) of the first as well as second kinds. Utilizing the properties of Fourier and Mellin transformations, we derive analytical solutions that substantially improve the comprehension and resolution of these equations. A key innovation of our approach is the ability to effectively manage non-smooth kernels through the degeneration of kernel functions, facilitating their separation and simplification. Empirical examples illustrate the method’s effectiveness, demonstrating superior numerical stability and convergence rates compared to existing techniques. This work not only fills a critical gap in the literature but also provides a robust framework for future research in integral equations, paving the way for advancements in various scientific and engineering applications.