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Differential Equations and Dynamical Systems

, Volume 27, Issue 1–3, pp 113–140 | Cite as

\(\varepsilon \)-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

  • Komal Bansal
  • Kapil K. SharmaEmail author
Original Research
  • 80 Downloads

Abstract

The motive of the present work is to develop a parameter robust numerical scheme for the class of problems involving singularly perturbed parabolic differential-difference equations with delay, which often arise in computational neuroscience. The numerical schemes developed prior to this work are restricted either to the case of small values of delay argument or linear convergence with restriction on the mesh generation. In practice, the delay argument can be of arbitrary size. Parameter \(\varepsilon \) may take small enough values e.g., viscosity coefficient in Navier–Stokes equation for fluids with high Reynolds number. It is required to construct a higher order parameter robust numerical scheme without any restriction on the mesh generation for singularly perturbed parabolic differential-difference equations with state dependent delay of arbitrary size. A new class of non-standard finite difference method based on interpolation, \(\theta \)-method and Micken’s techniques is constructed to approximate the solution of singularly perturbed parabolic differential-difference equations with arbitrary values of delay. It is shown that proposed numerical scheme is parameter uniform convergent. It is proved that this method is unconditionally stable and is convergent for \(\frac{1}{2} \le \theta \le 1,\) without having any restriction on the mesh. Some numerical experiments are provided to illustrate the performance of the method.

Keywords

Singular perturbation Differential-difference equations Convection diffusion parabolic problem Fitted operator \(\theta \)-method Non-Standard finite difference method 

Mathematics Subject Classification

65L11 65M12 35K20 

Notes

Acknowledgements

The research work of first author is supported by U.G.C. (Letter no. F.17-7(J)/08(SA-1) dated 01-Feb-2012) New Delhi, India. The authors express their sincere thanks to Prof. Relja Vulanović, Professor and Coordinator of Mathematics, Department of Mathematical Sciences, Kent State University at Stark, North Canton, Ohio 44720, U.S.A., for his valuable contribution in the paper.

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Copyright information

© Foundation for Scientific Research and Technological Innovation 2017

Authors and Affiliations

  1. 1.Department of Mathematics (Center for Advance Study in Mathematics)Panjab UniversityChandigarhIndia
  2. 2.South Asian UniversityNew DelhiIndia

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