Fredholm integro-differential equation with weak singularities: a combined analytical and numerical study

This study is dedicated to the analytical and numerical investigation of a nonlinear Fredholm integro-differential equation, specifically one that is characterized by weakly singular kernels. The primary challenge addressed is the weak singularity in the kernel, which complicates the solution process. To tackle this, we employ a combination of two techniques: the Nyström method and the product integration method. The Nyström method is used for approximating the solution to the Fredholm integro-differential equation, while the product integration technique is applied to handle the singular behavior of the kernel effectively. In addition to developing these numerical methods, we rigorously prove the existence and uniqueness of the solution within the appropriate functional spaces. The theoretical results are complemented by practical demonstrations through numerical examples, which are presented to verify and illustrate the accuracy and efficiency of the proposed approach. The results show that the combined methods can efficiently address the challenges posed by weakly singular kernels in Fredholm integro-differential equations.