On the Computation of Blow‐Up Solutions for Nonlinear Volterra Integrodifferential Equations

P G Dlamini,M Khumalo

doi:10.1155/2012/878497

Abstract

We make use of an adaptive numerical method to compute blow‐up solutions for nonlinear ordinary Volterra integrodifferential equations (VIDEs). The method is based on the implicit midpoint method and the implicit Euler method and is named the implicit midpoint‐implicit Euler (IMIE) method and was used to compute blow‐up solutions in semilinear ODEs and parabolic PDEs in our earlier work. We demonstrate that the method produces superior results to the adaptive PECE‐implicit Euler (PECE‐IE) method and the MATLAB solver of comparable order just as it did in our previous contribution. We use quadrature rules to approximate the integral in the VIDE and demonstrate that the choice of quadrature rule has a significant effect on the blow‐up time computed. In cases where the problem contains a convolution kernel with a singularity we use convolution quadrature.

Highlights

We demonstrate that the method produces superior results to the adaptive PECE-implicit Euler PECE-Implicit Euler Method (PECE-IE) method and the MATLAB solver of comparable order just as it did in our previous contribution
Many solutions of differential equations modeling physical problems blow-up in finite time. the phenomenon of blow-up is said to have occurred at Tb < ∞ if the solution of the differential equation becomes unbounded at t Tb
Budd et al 2 proposed the use of moving mesh partial differential equation methods MMPDE for solving 1.1

Summary

Introduction

Many solutions of differential equations modeling physical problems blow-up in finite time. the phenomenon of blow-up is said to have occurred at Tb < ∞ if the solution of the differential equation becomes unbounded at t Tb. Many solutions of differential equations modeling physical problems blow-up in finite time. The reaction-diffusion equation is a semilinear parabolic equation of the form ut t, x − Δu t, x f t, x, u , t > 0, x ∈ Ω ⊂ Rd, u 0, x u0 x ≥ 0, x ∈ Ω, 1.1 u t, x 0, t > 0, x ∈ ∂Ω, Mathematical Problems in Engineering where Δu t, x is referred to as the diffusion term and f t, x, u as the reaction term It is well known from blow-up theories that for sufficiently large initial function u0 x the solution of. Ma et al 11 implement the moving mesh methods to solve reaction-diffusion equations with nonlocal nonlinear terms such as 1.2 which blows-up in finite time

Description of Methods Used

Numerical Computation

Using Block-by-Block Method

Using Convolution Quadrature

Numerical Examples

Discussion