Abstract
The singularly perturbed parabolic convection-diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The Crank-Nicolson finite difference scheme for the temporal direction and the non-standard finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the spatial variable. The stability and uniform convergence analysis of the proposed method are studied. The method is shown to be uniformly convergent with a quadratic order of convergence in both temporal and spatial directions. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates, also provide more accurate results and a higher order of convergence than methods available in the literature.
Similar content being viewed by others
Data Availibility
No data was used for the study described in the article.
Code Availibility
Not applicable.
References
Gurney, W., Blythe, S., Nisbet, R.: Nicholson’s blowflies revisited. Nature 287(5777), 17–21 (1980)
Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197(4300), 287–289 (1977)
Marcus, C., Westervelt, R.: Stability of analog neural networks with delay. Phys. Rev. A 39(1), 347 (1989)
Franz, A.L., Roy, R., Shaw, L.B., Schwartz, I.B.: Effect of multiple time delays on intensity fluctuation dynamics in fiber ring lasers. Phys. Rev. E 78(1), 016208 (2008)
Marconi, M., Javaloyes, J., Barland, S., Balle, S., Giudici, M.: Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays. Nat. Photonics 9(7), 450 (2015)
Erneux, T.: Applied delay differential equations. Springer Science and Business Media, Berlin (2009)
Fiedler, B., Flunkert, V., Georgi, M., Hövel, P., Schöll, E.: Beyond the odd-number limitation of time-delayed feedback control. Handbook of chaos control. p. 73–84 (2008)
Farrell, P., Hegarty, A., Miller, J.M., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layers. Chapman and hall/CRC, Boca Raton (2000)
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)
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)
Bansal, K., Rai, P., Sharma, K.K.: Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. Differ. Equ. Dynam. Syst. 25(2), 327–346 (2017)
Woldaregay, M.M., Duressa, G.F.: Higher-order uniformly convergent numerical scheme for singularly perturbed differential difference equations with mixed small shifts. Int. J. Differ. Equ. 2020, 1–5 (2020)
Hailu, W.S., Duressa, G.F.: Uniformly convergent numerical method for singularly perturbed parabolic differential equations with non-smooth data and large negative shift. Res. Math. 9(1), 2119677 (2022)
Chandru, M., Prabha, T., Das, P., Shanthi, V.: A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms. Differ. Equ. Dynam. Syst. 27, 91–112 (2019)
Chandru, M., Das, P., Ramos, H.: Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Math. Methods Appl. Sci. 41(14), 5359–5387 (2018)
Das, P., Mehrmann, V.: Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. BIT Numer. Math. 56, 51–76 (2016)
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)
Kumar, K., Podila, P.C., Das, P., Ramos, H.: A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems. Math. Methods Appl. Sci. 44(16), 12332–12350 (2021)
Bansal, K., Sharma, K.K.: Parameter-robust numerical scheme for time-dependent singularly perturbed reaction-diffusion problem with large delay. Numer. Funct. Anal. Optim. 39(2), 127–154 (2018)
Kumar, D., Kumari, P.: Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay. Appl. Numer. Math. 153, 412–429 (2020)
Selvi, P.A., Ramanujam, N.: A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition. Appl. Math. Comput. 296, 101–115 (2017)
Gelu, F.W., Duressa, G.F.: A uniformly convergent collocation method for singularly perturbed delay parabolic reaction-diffusion problem. In: Abstract and applied analysis. vol. 2021. Hindawi (2021)
Cahlon, B., Kulkarni, D.M., Shi, P.: Stepwise stability for the heat equation with a nonlocal constraint. SIAM J. Numer. Anal. 32(2), 571–593 (1995)
Choi, Y., Chan, K.Y.: A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal. Theory Methods Appl. 18(4), 317–331 (1992)
Ewing, R.E., Lin, T.: A class of parameter estimation techniques for fluid flow in porous media. Adv. Water Resour. 14(2), 89–97 (1991)
Formaggia, L., Nobile, F., Quarteroni, A., Veneziani, A.: Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Vis. Sci. 2(2), 75–83 (1999)
Hu, M., Wang, L.: Triple positive solutions for an impulsive dynamic equation with integral boundary condition on time scales. Int. J. Appl. Math. Stat. 31, 43–66 (2013)
Bahuguna, D., Abbas, S., Dabas, J.: Partial functional differential equation with an integral condition and applications to population dynamics. Nonlinear Anal. Theory Methods Appl. 69(8), 2623–2635 (2008)
Das, P., Rana, S.: Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis. Math. Methods Appl. Sci. 44(11), 9419–9440 (2021)
Das, P., Rana, S., Ramos, H.: On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. J. Comput. Appl. Math. 404, 113116 (2022)
Das, P., Rana, S., Ramos, H.: A perturbation-based approach for solving fractional-order Volterra-Fredholm integro differential equations and its convergence analysis. Int. J. Comput. Math. 97(10), 1994–2014 (2020)
Sekar, E., Tamilselvan, A.: Singularly perturbed delay differential equations of convection-diffusion type with integral boundary condition. J. Appl. Math. Comput. 59(1), 701–722 (2019)
Debela, H.G., Duressa, G.F.: Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition. J. Egyptian Math. Soc. 28(1), 1–16 (2020)
Sharma, A., Rai, P.: A hybrid numerical scheme for singular perturbation delay problems with integral boundary condition. J. Appl. Math. Comput. 68(5), 3445–3472 (2022)
Elango, S., Tamilselvan, A., Vadivel, R., Gunasekaran, N., Zhu, H., Cao, J., et al.: Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition. Adv. Differ. Equ. 2021(1), 1–20 (2021)
Hailu, W.S., Duressa, G.F.: Parameter-uniform cubic spline method for singularly perturbed parabolic differential equation with large negative shift and integral boundary condition. Res. Math. 9(1), 2151080 (2022)
Gobena, W.T., Duressa, G.F.: Parameter-uniform numerical scheme for singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition. Int. J. Differ. Equ. 2021, 1–16 (2021)
Gobena, W.T., Duressa, G.F.: Fitted operator average finite difference method for singularly perturbed delay parabolic reaction diffusion problems with non-local boundary conditions. Tamkang Journal of Mathematics. (2022)
Gobena, W.T., Duressa, G.F.: An optimal fitted numerical scheme for solving singularly perturbed parabolic problems with large negative shift and integral boundary condition. Results Control Optim. 9, 100172 (2022)
Sharma, N., Kaushik, A.: A uniformly convergent difference method for singularly perturbed parabolic partial differential equations with large delay and integral boundary condition. Journal of Applied Mathematics and Computing. 1–23 (2022)
Hailu, W.S., Duressa, G.F.: Accelerated parameter-uniform numerical method for singularly perturbed parabolic convection-diffusion problems with a large negative shift and integral boundary condition. Results Appl. Math. 18, 100364 (2023)
Clavero, C., Gracia, J., Jorge, J.: High-order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers. Numer. Methods Partial Differ. Equat. Int. J. 21(1), 149–169 (2005)
Mickens, R.E.: Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore (2005)
Bansal, K., Sharma, K.K.: Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Numer. Algorithms 75(1), 113–145 (2017)
Munyakazi, J.B.: A robust finite difference method for two-parameter parabolic convection-diffusion problems. Appl. Math. Inform. Sci. 9(6), 2877 (2015)
Das, P.: A higher order difference method for singularly perturbed parabolic partial differential equations. J. Differ. Equ. Appl. 24(3), 452–477 (2018)
Doolan, E.P., Miller, J.J., Schilders, W.H.: Uniform numerical methods for problems with initial and boundary layers. Boole Press (1980)
Das, P., Natesan, S.: Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations. Int. J. Comput. Math. 92(3), 562–578 (2015)
Shakti, D., Mohapatra, J., Das, P., Vigo-Aguiar, J.: A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction-diffusion problems with arbitrary small diffusion terms. J. Comput. Appl. Math. 404, 113167 (2022)
Acknowledgements
The authors would like to express their gratitude to the anonymous referees for their helpful suggestions that helped to improve the quality of this paper.
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Ethical approval
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hailu, W.S., Duressa, G.F. Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition. Differ Equ Dyn Syst (2023). https://doi.org/10.1007/s12591-023-00645-y
Accepted:
Published:
DOI: https://doi.org/10.1007/s12591-023-00645-y
Keywords
- Singularly perturbed problem
- Parabolic convection-diffusion equations
- Non-standard finite difference method
- Integral boundary condition
- Large negative shift