Abstract
This article deals with 2D singularly perturbed parabolic delay differential equations. First, we apply implicit fractional Euler method for discretizing the derivative with respect to time and then we apply upwind finite difference method with bilinear interpolation to the locally one-dimensional problems with space shift. It is proved that the present finite difference method is almost first-order convergence in time and spatial directions. Numerical examples are given to illustrate the theoretical results.
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References
Amiraliyev GM, Cimen E (2010) Numerical method for a singularly perturbed convection–diffusion problem with delay. Appl Math Comput 216:2351–2359
Amiraliyeva IG, Erdogan F, Amiraliyev GM (2010) A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Appl Math Lett 23(10):1221–1225
Ansari AR, Bakr SA, Shishkin GI (2007) A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J Comput Appl Math 205(1):552–566
Briani M, Chioma C, Natalini R (2004) Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Numer Math 98:607–646
Clavero C, Jorge JC (2017) A fractional step method for 2D parabolic convection–diffusion singularly perturbation problems: uniform convergence and order reduction. Numer Algorithms 75(3):809–826
Clavero C, Jorge JC, Lisbona F, Shishkin GI (1998) A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection–diffusion problems. Appl Numer Math 27:211–231
Clavero C, Jorge JC, Lisbona F, Shishkin GI (2000) An alternating direction scheme on a nonuniform mesh for reaction–diffusion parabolic problems. IMA J Numer Anal 20:263–280
Das A, Natesan S (2018) Fractional step method for singularly perturbed 2d delay parabolic convection diffusion problems on Shishkin mesh. Int J Appl Comput Math 4(65):1–23
Das A, Natesan S (2019) Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Shishkin mesh. J Appl Math Comput 59:207–225
Farrell PA, Hegarty AF, Miller JJH, O’Riordan E, Shishkin GI (2004) Singularly perturbed convection diffusion problems with boundary and weak interior layers. J Comput Appl Math 166:133–151
Kadalbajoo MK, Sharma KK (2005) Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations. Comput Appl Math 24(2):151–172
Kadalbajoo MK, Sharma KK (2006) Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electron Trans Numer Anal 23:180–201
Kaushik M, Natesan S (2019) Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection-diffusion problems. J Appl Math Comput 60:51–86
Lange CG, Miura RM (1982) Singularly perturbation analysis of boundary-value problems for differential-difference equations. SIAM J Appl Math 42(3):502–530
Lange CG, Miura RM (1985) Singular perturbation analysis of boundary-value problems for differential-difference equations II. Rapid oscillations and resonances. SIAM J Appl Math 45(5):687–707
Miller JJH, O’Riordan E, Shishkin GI (2012) 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. Pte. Ltd., Singapore
Poole G, Boullion T (1974) A survey on M-matrices. SIAM Rev 16(4):419–427
Subburayan V, Ramanujam N (2012) Asymptotic initial value technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers. Appl Math Lett 25:2272–2278
Subburayan V, Ramanujam N (2013) An initial value technique for singularly perturbed convection-diffusion problems with a negative shift. J Optim Theory Appl 158:234–250
Subburayan V, Ramanujam N (2013) An initial value technique for singularly perturbed reaction–diffusion problems with a negative shift. Novi Sad J Math 43(2):67–80
Subburayan V, Ramanujam N (2021) Uniformly convergent finite difference schemes for singularly perturbed convection diffusion type delay differential equations. Differ Equ Dynam Syst 29(1):139–155
Süli E, Mayers DF (2003) An introduction to numerical analysis. Cambridge University Press, Cambridge
Zadorin AI (2022) Approaches to constructing two-dimensional interpolation formulas in the presence of boundary layers. J Phys Conf Ser 2182(012036):2–9
Acknowledgements
The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation. Further, the authors sincerely thank the DST-SERB for providing the fund under the scheme TARE, File No. TAR/2021/000053.
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Communicated by Zhaosheng Feng.
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Veerasamy, S., Srinivasan, N. Robust numerical method for space shift 2D singularly perturbed parabolic convection diffusion differential equations. Comp. Appl. Math. 42, 153 (2023). https://doi.org/10.1007/s40314-023-02289-8
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DOI: https://doi.org/10.1007/s40314-023-02289-8
Keywords
- Delay differential equations
- 2D parabolic equations
- Fractional step method
- Convection diffusion problems