PARAMETER UNIFORM NUMERICAL METHOD FOR SINGULARLY PERTURBED PARABOLIC DIFFERENTIAL DIFFERENCE EQUATIONS

M. M. WOLDAREGAY, G. F. DURESSA

Abstract


In this paper, a numerical study is made for solving singularly perturbed differential difference equations with small advance and delay parameters. To approximate the advance and delay terms a Taylor series expansion has been used. The resulting singularly perturbed parabolic PDE is solved by using non-standard finite difference method on uniform mesh in $ x $-direction and implicit Runge-Kutta method is used for the resulting system of IVPs in $ t $-direction. The method is shown to be accurate of order one.  A convergence analysis has been carried out to show $\varepsilon-$ uniform convergence of the proposed scheme. Two numerical examples are considered to investigate parameter uniform convergence of the proposed method.

Full Text:

PDF

References


R.K. Alexander, Stability of Runge--Kutta methods for stiff ordinary differential equations, SIAM journal on numerical analysis,

(4) 1147-1168, 1994.

C.TH. Baker and G.A. Bocharov and F.A. Rihan,

A report on the use of delay differential equations in numerical modelling in the biosciences Citeseer, 1999.

[3] K. Bansal and K.K. Sharma, $varepsilon-$ Uniform Numerical Technique for the Class of Time Dependent Singularly Perturbed Parabolic Problems With State Dependent Retarded Argument Arising from Generalised Stein's Model of Neuronal Variability,

Differential Equations and Dynamical Systems, 1-28, 2016.

K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments,

Numerical Algorithms, 75 (1) 113-145, 2017.

K. Bansal, P. Rai and K.K. Sharma, Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments, Differential Equations and Dynamical Systems, 25 (2) 327-346, 2017.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 Journal of the Nigerian Mathematical Society

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.