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IJATCA solicits original research papers for the December – 2022 Edition.
Last date of manuscript submission is December 30, 2022.

                                                   

Richardson extrapolation technique for singularly perturbed delay parabolic partial differential equation


Volume: 1 Issue: 1
Year of Publication: 2019
Authors: L. Govindarao, J. Mohapatra



Abstract

This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the Crank-Nicolson scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.

References

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  2. Bansal, K., Rai, P., & Sharma, K. K. (2017). 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.

  3. Das, A., & Natesan, S. (2018). Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations. International Journal of Computer Mathematics, 95(3), 490-510.

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  5. Govindarao, L., & Mohapatra, J. (2019). A second order numerical method for singularly perturbed delay parabolic partial differential equation. Engineering Computations, 36(2), 420-444.

  6. Kumar, M., & Rao, S. C. S. (2010). High order parameter-robust numerical method for time dependent singularly perturbed reaction-diffusion problems. Computing, 90(1-2), 15-38.

  7. Mohapatra, J., & Natesan, S. (2008). Uniformly convergent second-order numerical method for singularly perturbed delay differential equations. Neural, Parallel and Scientific Computations, 16(3), 353.

  8. Miller, J. J., O\"Riordan, E., & Shishkin, G. I. (2012). Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific.

  9. Reddy, N. R., & Mohapatra, J. (2015). An efficient numerical method for singularly perturbed two point boundary value problems exhibiting boundary layers. National Academy Science Letters, 38(4), 355-359.

  10. Salama, A. A., & Al-Amery, D. G. (2017). A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations. International Journal of Computer Mathematics, 94(12), 2520-2546.

  11. Shishkin, G. I., & Shishkina, L. P. (2010). A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation. Computational mathematics and mathematical physics, 50(12), 2003-2022.

Keywords

Singular perturbation, Delay parabolic problems, Crank-Nicolson scheme, Richardson extrapolation.




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