MSC:Keywords: Cauchy kernel Weakly singular Taylor series Galerkin method Legendre functions a b s t r a c t A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. 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.
In this paper, we transform ζ(s) to appropriate integral forms and for numerical computing of these integrals, we introduce a method based on Gauss-Hermite and Gauss-Laguerre quadratures. By using the zeta function, we compute the prime counting function π(x) numerically. Some relations are new and three examples are given to show the good accuracy of the method.
This paper is concerned with the numerical solution for a class of weakly singular Fredholm integral equations of the second kind. The Taylor series of the unknown function, is used to remove the singularity and the truncated Taylor series to second order of k(x, y) about the point (x , y ) is used. The integrals that appear in this method are computed exactly and some of these integrals are computed with the Cauchy principal value without using numerical quadratures. The solution in the Legendre polynomial form generates a system of linear algebraic equations, this system is solved numerically. Through numerical examples, performance of the present method is discussed concerning the accuracy of the method.
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