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Bansal, Komal; Rai, Pratima; Sharma, Kapil K.

Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. (English) Zbl 1372.65227

Differ. Equ. Dyn. Syst. 25, No. 2, 327-346 (2017).
This paper proposes two numerical schemes for numerical approximation of solutions to time-dependent singularly perturbed parabolic convection-diffusion problems with general shift arguments in the reaction term. Linear convergence of the numerical algorithm in space and time for both schemes is obtained. Some numerical experiments are presented to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)

Cited in 36 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B25 Singular perturbations in context of PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

singular perturbation; differential-difference equations; interpolation; finite difference scheme; parabolic convection-diffusion problems; linear convergence; algorithm; numerical experiment
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References:

[1] Ansari, A., Bakr, S., Shishkin, G.: A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J. Comput. Appl. Math. 205(1), 552-566 (2007) · Zbl 1121.65090 · doi:10.1016/j.cam.2006.05.032
[2] Bashier, E., Patidar, K.: A second-order fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. J. Differ. Equ. Appl. 17(05), 779-794 (2011) · Zbl 1227.65070 · doi:10.1080/10236190903305450
[3] Bashier, E., Patidar, K.C.: An almost second order fitted mesh numerical method for a singularly perturbed delay parabolic partial differential equation. Neural Parallel Sci. Comput. 18(2), 137-154 (2010) · Zbl 1214.65043
[4] Bashier, E.B., Patidar, K.C.: A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl. Math. Comput. 217(9), 4728-4739 (2011) · Zbl 1221.65213
[5] Clavero, C., Jorge, J., Lisbona, F.: A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems. J. Comput. Appl. Math. 154(2), 415-429 (2003) · Zbl 1023.65097 · doi:10.1016/S0377-0427(02)00861-0
[6] Farrell, P., Hegarty, A.: On the determination of the order of uniform convergence. In: Proceedings of the 13th IMACS World Congress on Computational and Applied Mathematics (IMACS, 1991), pp. 501-502 (1991) · Zbl 1219.65110
[7] Kaushik, A., Sharma, K., Sharma, M.: A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument. Appl. Math. Modell. 34(12), 4232-4242 (2010) · Zbl 1201.65131 · doi:10.1016/j.apm.2010.04.020
[8] Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32(144), 1025-1039 (1978) · Zbl 0418.65040 · doi:10.1090/S0025-5718-1978-0483484-9
[9] Kumar, D., Kadalbajoo, M.K.: A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations. Appl. Math. Model. 35(6), 2805-2819 (2011) · Zbl 1219.65110 · doi:10.1016/j.apm.2010.11.074
[10] Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic type, vol. 23. American Mathematical Society, Providence (1988)
[11] Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific Publishing Co. Inc., River Edge (1996) · Zbl 0915.65097 · doi:10.1142/2933
[12] Musila, M., Lánskỳ, P.: Generalized stein’s model for anatomically complex neurons. Biosystems 25(3), 179-191 (1991) · doi:10.1016/0303-2647(91)90004-5
[13] Ramesh, V., Kadalbajoo, M.K.: Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behavior. Appl. Math. Comput. 202(2), 453-471 (2008) · Zbl 1151.65072
[14] Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, vol. 24. Springer, Berlin (2008) · Zbl 1155.65087
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