Integration of a product in a two-dimensional Fredholm integral equation with a weakly singular kernel

This study highlights the feasibility of solving the Fredholm integral equation in cases where its kernel exhibits weak singularity, a challenging scenario that often arises in mathematical and physical applications. The primary objective of our research is to obtain a highly accurate solution, with a particular focus on two-dimensional cases. Addressing such equations requires careful consideration of the requisite mathematical conditions and the development of robust methods capable of handling the inherent complexities. In our previous work, we successfully demonstrated the effectiveness of the product integration method for solving the Fredholm equation in one-dimensional cases. This success served as the foundation and motivation to extend our approach to higher dimensions, where the complexity of the problem increases significantly. By adapting and refining the product integration method, we aim to achieve similar levels of accuracy and reliability in two-dimensional scenarios. Furthermore, the study introduces a practical application to showcase the utility of our approach. Before presenting this application, we rigorously prove the convergence of the approximate solution, ensuring that the methodology provides consistent and reliable results. This combination of theoretical analysis and practical demonstration underscores the robustness of our approach and its potential to address a wide range of problems involving weakly singular Fredholm integral equations in higher dimensions.