Performance Analysis of Two-Point Block Iterative Method for Solving Fredholm Integral Equations of the First Kind

Abstract

This study develops a numerical approach for solving first-kind Fredholm integral equations involving semi-smooth kernel. The integral equation is first discretised using the composite Simpson's $\frac{3}{8}$ quadrature scheme, resulting in a system of algebraic equations. A Two-Point Block Arithmetic Mean (2-BLAM) iterative method is then applied to obtain the numerical solution of the resulting system. The computational performance of the proposed approach is investigated through two numerical test problems and benchmarked against the conventional Gauss–Seidel (GS) and Arithmetic Mean (AM) iterative methods. Performance is assessed in terms of number of iterations, computational time and maximum absolute error. The numerical results demonstrate that the 2-BLAM method requires considerably fewer iterations and shorter execution times than the GS and AM methods, while maintaining comparable solution accuracy. These results indicate that combining composite Simpson's $\frac{3}{8}$ discretisation with the 2-BLAM iteration provides a computationally efficient approach for treating first-kind Fredholm integral equations with semi-smooth kernels.