Numerical Solution of Weakly Singular Integral Equations by Using Taylor Series and Legendre Polynomials

In this paper, we use Taylor series and Legendre functions of the second kind to remove singularity of the weakly singular Fredholm integral equations of the second kind with the kernel $k(x,y)=frac{1}{(x-y)^alpha},; 0< alphaleq1$. Legendre polynomials are used as a basis and some integrals that appear in this method are computed with Cauchy principal value sense without using any numerical quadrature. Three examples are given to show the efficiency of the method.