Muntz Function Collocation Method for 2D Fredholm Integral Equations of the Second Kind

This paper presents a novel numerical approach for solving two-dimensional Fredholm integral equations of the second kind (2DFIEs) using M¨untz functions and a collocation method. In the proposed approach, both the unknown function and the integral kernel are approximated by M¨untztype orthogonal functions, and the integral equation is transformed into a system of linear equations using two-dimensional collocation points; the approximate solution is then obtained by solving this linear system. The approximation of bivariate functions is analyzed in the Sobolev space. Several numerical examples are provided to demonstrate the accuracy and efficiency of the method. The numerical results confirm that the two-dimensional M¨untz collocation method offers a flexible and effective tool for solving complex 2DFIEs, with potential applications in scientific and engineering problems involving integral equations.