A stable finite difference scheme and error estimates for parabolic singularly perturbed PDEs with shift parameters

Abstract This article presents a stable finite difference approach for the numerical approximation of singularly perturbed differential-difference equations (SPDDEs). The proposed scheme is oscillation-free and much accurate than conventional methods on a uniform mesh. Error estimates show that the scheme is linear convergent in space and time variables. By using the Richardson extrapolation technique, the obtained results are extrapolated in order to get better approximations. Some numerical examples are taken from literature to validate the theory, showing good performance of the proposed method.

[1]  Wil H. A. Schilders,et al.  Uniform Numerical Methods for Problems with Initial and Boundary Layers , 1980 .

[2]  Abhishek Das,et al.  Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations , 2018, Int. J. Comput. Math..

[3]  Kapil K. Sharma,et al.  Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments , 2017, Numerical Algorithms.

[4]  R. Stein.  Some models of neuronal variability. , 1967, Biophysical journal.

[5]  Devendra Kumar.  An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience , 2018 .

[6]  M. K. Kadalbajoo,et al.  A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations☆ , 2011 .

[7]  P Lánský,et al.  Generalized Stein's model for anatomically complex neurons. , 1991, Bio Systems.

[8]  Shaaban A. Bakr,et al.  A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations , 2007 .

[9]  K. Sharma,et al.  Numerical Treatment for the Class of Time Dependent Singularly Perturbed Parabolic Problems with General Shift Arguments , 2017 .

[10]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[11]  R. Chakravarthy.  A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory , 2014 .

[12]  Sunil Kumar,et al.  Higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations , 2018 .

[13]  R. Stein.  A THEORETICAL ANALYSIS OF NEURONAL VARIABILITY. , 1965, Biophysical journal.

[14]  Y. Wong,et al.  Pollution-free finite difference schemes for non-homogeneous helmholtz equation , 2014 .

[15]  Srinivasan Natesan,et al.  ε-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations , 2017, Int. J. Comput. Math..

[16]  A. A. Salama,et al.  A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations , 2017, Int. J. Comput. Math..

[17]  P Lánský,et al.  On approximations of Stein's neuronal model. , 1984, Journal of theoretical biology.

[18]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[19]  V. P. Ramesh,et al.  Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior , 2008, Appl. Math. Comput..

[20]  Yau Shu Wong,et al.  Efficient and Accurate Numerical Solutions for Helmholtz Equation in Polar and Spherical Coordinates , 2015 .