ACCURATE NUMERICAL SCHEME FOR SINGULARLY PERTURBED TIME DELAYED PARABOLIC DIFFERENTIAL EQUATIONS

ACCURATE NUMERICAL SCHEME FOR SINGULARLY PERTURBED TIME DELAYED PARABOLIC DIFFERENTIAL EQUATIONS

N. T. Negero, G. F. Duressa

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Abstract

For the numerical solution of the singularly perturbed parabolic convection- di usion equation with large time delays, a novel class of  tted operator  nite di erence method is constructed using the Mickens-type scheme. Since the perturbation param- eter is the source for the simultaneous occurrence of time-consuming and high-speed phenomena in physical systems that depend on present and past history, our study here is to capture the e ect of the parameter on the boundary layer. The time derivative is suitably replaced by a Crank-Nicolson-based scheme, followed by the spatial derivative, which is replaced by a non-standard  tted operator scheme. First-order error bounds in space and second-order error bounds in time are established to provide numerical results.

Keywords

singular perturbation; large time delay; parabolic convection-di usion prob- lem; denominator function; uniformly convergent.