In this article, we will show the complex inversion formula for the inversion of the L2-transform and also some applications of the L2, and Post Widder transforms for solving singular integral equation with trigonometric kernel. Finally, we obtained analytic solution for a partial differential equation with non-constant coefficients.

The purpose of the present paper is to establish some new retarded weakly singular integral inequalities of Gronwall-Bellman type for discontinuous functions, which generalize some known weakly singular and impulsive integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of singular differential equations and singular impu...

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the pr...

Abstract: A one dimensional heat equation in a semi-infinite medium controlled through a heat source depending on the delayed heat flux at the extremum is studied. By reducing the problem to a delayed Volterra integral equation with a weakly singular kernel, we find conditions on the initial datum and on the source term of the equation to control the asymptotic behaviour of the mean temperatures.

We consider the h-p Galerkin boundary element method applied to a weakly singular integral equation of the rst kind on an open curved surface. We focus on the eeective computation of the Galerkin matrix, i.e. we introduce a numerical qudrature rule which is shown (for a model case) to converge exponentially fast with the number of kernel evaluations. Numerical experiments show that the applicat...

An integro-differential equation involving a convolution integral with a weakly singular kernel is considered. The kernel can be that of a fractional integral. The integro-differential equation is discretized using the discontinuous Galerkin method with piecewise constant basis functions. Sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of...

We consider a Fredholm integral equation of the second kind in L1([a, b],C), with a weakly singular kernel. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0([a, b],C) to apply it in L1([a, b],C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refi...

In this article, we consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. In the first part, we deal with two-dimensional integralalgebraic equations. Next, we analyze Volterra integral equations of the first kind with a degenerate matrix-kernel on the diagonal. Finally, the third part of the work is devoted to the analysis of degenerate integro...