On Fully Discrete Collocation Methods for Solving Weakly Singular Integro‐differential Equations

Abstract

Abstract. In order to find approximate solutions of Volterra and Fredholm integrodifferential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non-smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

Product Integration for Weakly Singular Integro-Differential Equations

“…Also ∂ n−1 G(t, s)/∂t n−1 is continuous and bounded in the region ∆, but it has a discontinuity at t = s. Note that in [6] a different formula for J i is derived. The operators J i , i = 0, .…”

“…Since this is rarely possible in concrete applications, there arises the question how to approximate these integrals, and it is of interest to derive error estimates for the approximate solutions. In [6] this problem in the case of special quadrature formulas is studied.…”

A Problem With Parameter for the Integro-Differential Equations

“…The theory of control problems for a system of ordinary differential equations and for a system of integro-differential equations in partial derivatives, with parameters, is rapidly developing and used in various fields of applied mathematics, biophysics, biomedicine, chemistry, etc. Control problems, also called as boundary value problems with parameters and parameter identification problems for systems of ordinary differential and integro-differential equations with parameters, are intensively studied by many authors [3,4,8,9,17,18,19,20,24,25]. To find solutions to these problems, methods of the qualitative theory of differential equations, variational calculus and optimization theory, the method of upper and lower solutions, etc.…”

A new operational approach for solving weakly singular integro--differential equations

“…Weakly singular integro-differential equations have been solved using different techniques. For example, the methods including the spline collocation method [9,10], the discrete collocation method [11][12][13], the discrete Galerkin method [14,15], the Legendre multiwavelets method [16], the piecewise polynomial collocation method with graded meshes [17], Homotopy perturbation method [18] to determine the approximate solutions. Some problems of mathematical physics are described in terms of second order linear and nonlinear singular Volterra integro-differential equations of the following form [19][20][21][22]:…”