High performance parallel numerical methods for Volterra equations with weakly singular kernels

Abstract

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations

A parallel algorithm for large systems of Volterra integral equations of Abel type

“….. with respect to the convergence rate, for time-step NSWR Richardson methods, i.e., b = 1, the following theorem holds [3]:…”

“…• the use of a fully parallel and fast convergent WR method: a discrete non-stationary WR (NSWR) method of Richardson type based on a fractional linear method of order 4 [3,4]; • the evaluation of the numerical lag term by the FFT lag-block technique [12,13] implemented in parallel.…”

“…One way to face this problem is to use parallel architectures. With this aim, during last years parallel methods for VIEs have been proposed: some of them realize a parallelism "across the method" (see for example [8] and references quoted in [6]), while other ones, the iterative waveform relaxation (WR) methods [1,[3][4][5], realize a parallelism "across the system", so realize a massive parallelism, especially useful in solving large systems of VIEs.…”

Jacobi-spectral method for integro-delay differential equations with weakly singular kernels

“…The numerical solution of (1.1) with smooth kernels was investigated by spectral methods, which leads to a higher (exponential) convergence order [3], where it was shown that the results on the exponential order of convergence of the spectral method for the pantograph DDE [1,2] remain valid for pantograph-type integro-differential equation (1.1) with smooth kernels. Several numerical methods have been proposed for (1.1) without a delay (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][24][25][26][27][28]), where they show that its numerical treatment is not simple, due to the fact that the solution of (1.1) usually has a * Correspondence: ishtiaqali@comsats.edu.pk 2010 AMS Mathematics Subject Classification: 45J05, 65R20, 34K06, 34K28.…”

An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method

“…These equations are applied in many areas [1] such as reaction-diffusion problems in small cells [2], theory of elasticity, heat conductions, hydrodynamics, stereology [3], the radiation of heat from semi-infinite solids [4], and other applications. Such equations have been studied by several authors [5][6][7][8][9][10][11][12][13][14].…”