Product Integration for Weakly Singular Integro-Differential Equations

Abstract

On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

“…So a variety works focused its interest to tackle the equations of this form which are usually numerical methods, including an operational method [7], Bernstein series [8], Block boundary value method [9], Partition of the interval and introduction of additional parameters [10], Smoothing transformation and spline collocation [11], The asymptotic estimations of the solution [12], Product integration [13], collocations methods as Spline, Piecewise Polynomial, and Spectral respectively [14][15][16], but it is well known that the results of numerical methods have an error rate.…”

A Novel Projection Method for Cauchy-Type Systems of Singular Integro-Differential Equations

“…An essential class of these problems is singular integro-differential equations with boundary conditions (cf. [10][11][12][13][14][15][16]).…”

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.…”