Waveform relaxation methods for Volterra integro-differential equations

Parsons, Wade William (1999) Waveform relaxation methods for Volterra integro-differential equations. Doctoral (PhD) thesis, Memorial University of Newfoundland.

[English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (18MB) [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

Abstract

This thesis is concerned with the solution of systems Volterra integro-differential equations by the application of waveform relaxation methods. This is a timely topic since such methods can often be implemented efficiently on parallel architectures. It derives convergence results for both the regular kernel and the weakly singular kernel cases, and although our primary concern is with numerical methods, we consider both analytic and numerical solutions. -- In Chapter 1 we study the history of waveform relaxation methods and try to bring the reader up to date with what is presently known about these methods. We emphasize their application to the solution of systems of ordinary differential equations and Volterra integral equations. In each case, we consider both the continuous- time and discrete-time methods and give convergence results for each. Therefore, this chapter will set the stage for the application of waveform relaxation techniques to the solution of Volterra integro-differential equations in Chapter 2. -- Chapter 2 is the main chapter of the thesis and contains all of the original results from my research. It begins by giving the standard resolvent representation of the analytic solution of Volterra integro-differential equations, with both regular kernels and weakly singular kernels. It then considers continuous-time iteration waveform relaxation methods, in which we assume that the resulting equations can be solved exactly. We prove that these methods converge uniformly on all bounded intervals. -- However, the main body of results in this chapter, concern the collocation solution of the iterates that result when waveform relaxation methods are applied to Volterra integro-differential equations. We will consider convergence, both as the steplength tends to zero and as the number of iterations tend to infinity. We study the effect various iterative methods used to solve the resulting implicit nonlinear algebraic equations have on the convergence and complete the discretization by taking into account the use of quadrature to solve the integrals in the method. Throughout the chapter we include numerical examples which illustrate the various theorems, with tables of results and discussion placed in a section at the end of this chapter. -- In Chapter 3, we point out that a major source of applications of Volterra integro- differential equations are the Volterra partial integro-differential equations. We also mention topics not considered in the main body of the thesis. These include numerical stability of Volterra integro-differential equations, and the use of graded meshes for the solution of Volterra integro-differential equations, with weakly singular kernels.

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