A Numerical Method for Solving Fredholm Integral Equations of the First Kind with Logarithmic Kernels and Singular Unknown Functions

Abstract

In this paper, we present a numerical method for solving singular Fredholm integral equations of the first kind. The method is based on the application of the shifted Chebyshev polynomials of the second kind using two techniques. By using the first technique, we solve singular Fredholm integral equations of the first kind with singular logarithmic kernels and singular unknown functions, while the second technique determines the associated data function for a specified exact solution. The unknown function is factorized into two functions; the first is badly-behaved and the second an unknown regular function. By expanding these two functions into shifted Chebyshev polynomials of the second kind through monomial basis functions, the unknown function’s singularity is completely removed. We develop an algebraic formula for evaluating the Chebyshev coefficients instead of using the Chebyshev integral formula. This minimizes the steps of the solution’s procedure and reduces the roundoff errors. The singularity of the kernel is treated by numerical integration. Two boundary integral equations related to the radar, electromagnetism, and scattering engineering problems are solved. The obtained results ensure the efficiency and the high accuracy of the presented method compared with other methods.