Bernoulli Polynomials Method for Solving Integral Equations with Singular Kernel (Baghdad Science Journal, 2024)
There is always an interest in an effective technique to generate a numerical solution of integral equations with singular or weakly singular kernels more precisely because numerical methods have limitations. In this study, integral equations with singular or weakly singular kernels are solved using the Bernoulli polynomial approach. The primary goal of this study is to provide an approximate solution to such problems in the form of a polynomial in a series of straightforward steps. Also, assuming that the denominator of the kernel will never be zero or have an imaginary value due to the selected nodes of the unique two kernel variables. With the 4th and 8th-degree Bernoulli polynomials as an example, the current approach provides a solution very close to the exact solution in the test examples. While. The very modest magnitude of the errors in the test examples proves the effectiveness of the current strategy. Also, the ease with which a computer program can be implemented makes this technique very efficient.  Another objective is to determine the efficiency of the proposed method by comparing it with various approaches. The approximated solution for integral equations with singular or weakly singular kernels is demonstrated to significantly converge to the precise ones by using the Bernoulli polynomial and is superior to those found by other stated approaches. This guarantees the originality and high accuracy of the suggested method. Also, the convergent of the proposed method is discussed. The programs are implemented using the MATLAB program R2018a.