A second order numerical method for two-parameter singularly perturbed time-delay parabolic problems

Document Type : Research Article

Authors

1 Department of Applied Mathematics, Adama Science and Technology University, Ethiopia

2 Department of Mathematics and Applied Mathematics, University of the Western Cape, South Africa

Abstract

In this article, a time delay parabolic convection-reaction-diffusion singularly perturbed  problem with two small parameters is considered. We investigate the layer behavior of the  solution for both smooth and non-smooth data. A numerical method to solve the problems described is developed using the Crank-Nicolson scheme to discretize the time-variable on a  uniform mesh while a hybrid finite difference is applied for the space-variable. The hybrid  scheme is a combination of the central, upwind and mid-point differencing on a piecewise uniform mesh of Shishkin type. The convergence analysis shows that the proposed method is  uniformly convergent of second order in both  space and   time. Numerical experiments  conducted on some test examples confirm the theoretical results.

Keywords

Main Subjects

References

[1] A. Ansari, S. Bakr, G. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205 (2007) 552–566.
[2] E.B. Bashier, K.C. Patidar, A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation, Appl. Math. Comput. 217 (2011) 4728–4739.
[3] T.A. Bullo, G.F. Duressa, G.A. Degla, Robust finite difference method for singularly perturbed two-parameter parabolic convection-diffusion problems, Int. J. Comput. Methods 18 (2021) 2050034.
[4] Z. Cen, A second-order finite difference scheme for a class of singularly perturbed delay differential equations, Int. J. Comput. Math. 87 (2010) 173–185.
[5] M. Chandru, P. Das, H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci. 41 (2018) 5359–5387.
[6] M. Chandru, T. Prabha, V. Shanthi, A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data, J. Comput. Appl. Math. 309 (2017) 11–27.
[7] P. Das, Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems, J. Comput. Appl. Math. 290 (2015) 16–25.
[8] P. Das, S. Natesan, Higher-order parameter uniform convergent schemes for Robin type reaction-diffusion problems using adaptively generated grid, Int. J. Comput. Methods 9 (2012) 1250052.
[9] P. Das, S. Natesan, Numerical solution of a system of singularly perturbed convection diffusion boundary value problems using mesh equidistribution technique, Aust. J. Math. Anal. Appl. 10 (2013) 1–17.
[10] P. Das, S. Natesan, Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh, CMES Comput. Model. Eng. Sci. 90 (2013) 463–485.
[11] P. Das, S. Natesan, Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations, Int. J. Comput. Math. 92 (2015) 562–578.
[12] M. Dehghan, Numerical solution of the three-dimensional advection–diffusion equation, Appl. Math. Comput. 150 (2004) 5–19.
[13] M. Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simul. 71 (2006) 16–30.
[14] E.P. Doolan, J.J. Miller, W.H. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, 1980.
[15] G.F. Duressa, T.B. Mekonnen, An exponentially fitted method for two parameter singularly perturbed parabolic boundary value problems, Commun. Korean Math. Soc. 38 (2023) 299–318.
[16] F. Erdogan, Z. Cen, A uniformly almost second order convergent numerical method for singularly perturbed delay differential equations, J. Comput. Appl. Math. 333 (2018) 382–394.
[17] K. Friedrichs, W. Wasow, Singular perturbations of non-linear oscillations, Duke Math. J. 13 (1946) 367–381.
[18] L. Govindarao, S.R. Sahu, J. Mohapatra, Uniformly convergent numerical method for singularly perturbed time delay parabolic problem with two small parameters, Iran. J. Sci. Technol. Trans. A: Sci. 43 (2019) 2373–2383.
[19] J. Gracia, E. O'Riordan, M. Pickett, A parameter robust second order numerical method for a singularly perturbed two-parameter problem, Appl. Numer. Math. 56 (2006) 962–980.
[20] B. Gunes, H. Duru, A computational method for the singularly perturbed delay pseudo-parabolic differential equations on adaptive mesh, Int. J. Comput. Math. (2023) 1–16.
[21] V. Gupta, M.K. Kadalbajoo, R.K. Dubey, A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters, Int. J. Comput. Math. 96 (2019) 474–499.
[22] M.K. Kadalbajoo, A. Jha, Exponentially fitted cubic spline for two-parameter singularly perturbed boundary value problems, Int. J. Comput. Math. 89 (2012) 836–850.
[23] A. Kaushik, Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation, Nonlinear Anal. Real World Appl. 9 (2008) 2106–2127.
[24] A. Kaushik, K. Sharma, M. Sharma, A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument, Appl. Math. Model. 34 (2010) 4232–4242.
[25] R.B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32 (1978) 1025–1039.
[26] N. Kopteva, Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, Computing 66 (2001) 179–197.
[27] D. Kumar, K. Deswal, Wavelet-based approximation for two-parameter singularly perturbed problems with Robin boundary conditions, J. Appl. Math. Comput. (2022) 1–25.
[28] M. Kumar, S. Kumar, A second order uniformly convergent numerical scheme for parameterized singularly perturbed delay differential problems, Numer. Algorithms 76 (2017) 349–360.
[29] S. Kumar, M. Kumar, High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay, Comput. Math. Appl. 68 (2014) 1355–1367.
[30] T. Linss, A posteriori error estimation for a singularly perturbed problem with two small parameters, Int. J. Numer. Anal. Model. 7 (2010) 3.
[31] J.J. Miller, E. O'Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in one and two Dimensions, World Scientific, 1996.
[32] J. Miller, E. O'Riordan, G. Shishkin, L. Shishkina, Fitted Mesh Methods for Problems with Parabolic Boundary Layers, Mathematical Proceedings of the Royal Irish Academy, JSTOR, (1998) 173–190.
[33] J.B. Munyakazi, A robust finite difference method for two-parameter parabolic convection-diffusion problems, Appl. Math. Inf. Sci. 9 (2015) 2877.
[34] J.D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002.
[35] N.T. Negero, A parameter-uniform efficient numerical scheme for singularly perturbed time-delay parabolic problems with two small parameters, Partial Differ. Equ. Appl. Math. 7 (2023) 100518.
[36] R. O'Malley, Two-parameter singular perturbation problems for second-order equations, J. Math. Mech. 16 (1967) 1143–1164.
[37] R. O'Malley, Boundary value problems for linear systems of ordinary differential equations involving many small parameters, J. Math. Mech. 18 (1969) 835–855.
[38] E. O'Riordan, M.L. Pickett, G.I. Shishkin, Singularly perturbed problems modeling reaction-convection-diffusion processes, Comput. Methods Appl. Math. 3 (2003) 424–442.
[39] E. O'Riordan, M. Pickett, G. Shishkin, Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems, Math. Comput. 75 (2006) 1135–1154.
[40] K.C. Patidar, A robust fitted operator finite difference method for a two-parameter singular perturbation problem, J. Differ. Equ. Appl. 14 (2008) 1197–1214.
[41] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandl. 3rd Int. Math. Kongr. Heidelberg (1904), Leipzig (1905).
[42] H.G. Roos, Z. Uzelac, The SDFEM for a convection-diffusion problem with two small parameters, Comput. Methods Appl. Math. 3 (2003) 443–458.
[43] V. Shanthi, N. Ramanujam, S. Natesan, Fitted mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers, J. Appl. Math. Comput. 22 (2006) 49–65.
[44] J. Singh, S. Kumar, M. Kumar, A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay, Numer. Methods Partial Differ. Equ. 34 (2018) 1849–1866.
[45] S. Sumit, S. Kumar, Kuldeep, M. Kumar, A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem, Comput. Appl. Math. 39 (2020) 1–25.
[46] R. Vulanovic, A higher-order scheme for quasilinear boundary value problems with two small parameters, Computing 67 (2001) 287–303.
[47] S. Yuzbasi, N. Sahin, Numerical solutions of singularly perturbed one-dimensional parabolic convection–diffusion problems by the Bessel collocation method, Appl. Math. Comput. 220 (2013) 305–315.
[48] W.K. Zahra, A.M. Mhlawy, Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, J. King Saud Univ. Sci. 25 (2013) 201–208.
Journal of Mathematical Modeling
Volume 11, Issue 4
December 2023
Pages 745-766

Files

Share

How to cite

Statistics