In this paper we establish Gauss-type quadrature formulas for weakly singular integrals. An application of the quadrature scheme is given to obtain numerical solutions of the weakly singular Fredholm integral equation of the second kind. We call this method a discrete product-integration method since the weights involved in the standard product-integration method are computed numerically.

we develop and apply the product integration method to a large class of linear weakly singular volterra systems. we show that under certain sufficient conditions this method converges. numerical implementation of the method is illustrated by a benchmark problem originated from heat conduction.

Based on a general, operational approach, two new integral identities for the fundamental solutions of the potential and elastostatic problems are established in this paper. Non-singular forms of the conventional boundary integral equations (BIEs) are derived by employing these two identities for the fundamental solutions and the twoterm subtraction technique. Both the strongly(Cauchy type) and...

Let C: L L be a projective deformation of the second order of two totally pn focal pseudocongruences L and L of (m-l)-planes in projective spaces and n, 2m-i < n < 3m-l, and let K be a collineation realizing such a C. The deformation C is said to be weakly singular, singular, or a-strongly singular, s 3,4,..., if the collineation K gives projective deformations of order i, 2 or of all correspon...

Compact finite difference scheme is applied for a partial integro-differential equation with a weakly singular kernel. The product trapezoidal method is applied for discretization of the integral term. The order of accuracy in space and time is , where . Stability and convergence in  norm are discussed through energy method. Numerical examples are provided to confirm the theoretical prediction ...

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial ...

Abstract. After Zagier proved that the traces of singular moduli j(z) are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group PSL2(Z) have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize t...