The approximate solution of a class of Fredholm integral equations with a weakly singular kernel (Q609202)

The authors consider weakly singular integral equations and singular integral equations as \[ a(x) \varphi(x) + b(x)\int\limits^1_{-1} \frac{\varphi(y)}{(y-x)^\alpha}dy = f(x), \, |x| < 1, \, 0< \alpha \leq 1, \tag{1} \] where \(\varphi(x)\) is the unknown function. A method for finding the numerical solution of equation (1) is presented. The Taylor series is used to remove the singularity and Legendre polynomials are used as a basis. The Legendre function of the second kind is used to remove the singularity in Cauchy type integral equations. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.