Abstract
This paper deals with numerical treatment of singularly perturbed parabolic differential difference equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diffusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial differential equation is solved using implicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε >N−1and first order convergence for the case ε «N−1, whereNis number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme.
Highlights
Differential equations with delay terms play an important role in modeling many process in Computational Bio-science, Control Theory, Economics and Engineering [5]
To overcome the drawback associated with standard numerical methods, we developed a numerical scheme using backward Euler in time and exponentially fitted operator FDM in space, which treat the problem very well
Using the theory applied in asymptotic method we develop exponentially fitted numerical scheme to solve the singularly perturbed BVPs in (20)-(21)
Summary
Differential equations with delay terms play an important role in modeling many process in Computational Bio-science, Control Theory, Economics and Engineering [5]. The feasibility of recording single neuron movement induces the development of accurate mathematical models of neuronal variability. In modeling of spiking movement of neuron to any level of exactness, one has to consider special features of each kind of neuron and its input processes [3]. In 1965, Stein [16] developed a mathematical model for the stochastic movement of neuron. The author [17] generalized his model to handle a distribution of past synaptic potential amplitudes.
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