The Numerical Simulation of the Qualitative Behaviour of Volterra Integro-Differential Equations
Abstract
We consider the qualitative behaviour of exact and approximate solutions of integral and integro-differential equations with fading memory kernels. Over long time intervals the errors in numerical schemes may become so large that they mask some important properties of the solution. One frequently appeals to stability theory to address this weakness, but it turns out that, in some of the model equations we have considered, there remains a gap in the analysis. We consider a linear problem and we solve the equation using simple numerical schemes. We outline the known stability behaviour of the problem and derive the values of lambda at which the true solution bifurcates. We give the corresponding analysis for the discrete schemes and highlight that, for particular stepsizes, the methods give unexpected behaviour and we show that, as the step size of the numerical scheme decreases, the bifurcation points tend towards those of the continuous scheme. We illustrate our results with some numerical examples.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 2001
- Accession Number
- ADP013718
Entities
People
- Jason A. Roberts
- John T. Edwards
- Neville J. Ford