A Parameter Uniform Numerical Scheme for Singularly Perturbed Differential-difference Equations with Mixed Shifts

Document Type: Research Paper

Authors

Department of Mathematics, NIT Rourkela, Odisha, 769008, India

Abstract

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propose a hybrid finite difference scheme, in which we use the cubic spline difference method in the fine mesh regions and a midpoint upwind scheme in the coarse mesh regions. We establish a theoretical parameter uniform bound in the discrete maximum norm. To check the efficiency of the proposed methods, we consider test problems with delay, advance and the mixed parameters and the results are in agreement with our theoretical findings.

Keywords

Main Subjects

[1] Kokotovic, P.V., Khalil, H.K., O’Reilly, J., Singular perturbation methods in control: analysis and design, Academic Press, New York, 1986.

[2] Stein, R.B., A theoretical analysis of neuronal variability, Biophysical Journal, 5(2), 1965, 173-194.

[3] Stein, R.B., Some models of neuronal variability, Biophysical Journal, 7(1), 1967, 37-68.

[4] Tuckwell, H.C., Introduction to Theoretical Neurobiology, Cambridge University Press, Cam-bridge, UK, 1988.

[5] Wilbur, W.J., Rinzel, J., An analysis of Stein’s model for stochastic neuronal excitation, Biological Cybernetics, 45(2), 1982, 107–114.

[6] Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I., Robust Computational Techniques for Boundary Layers, CHAPMAN & HALL/CRC, Boca Ratan, 2000.

[7] Miller, J.J.H., O’Riordan, E., Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Co, Singapore, 2012.

[8] Mohapatra, J., Natesan, S., Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid, Numerical Mathematics: Theory, Methods and Applications, 3(1), 2010, 1-22.

[9] Swamy, D.K., Phaneendra, K., Babu, A.B., Reddy, Y.N., Computational method for singularly perturbed delay differential equations with twin or oscillatory behaviour, Ain Shams Engineering Journal, 6(1),2015, 391-398.

[10] Lange, C.G., Miura, R.M., Singular perturbation analysis of boundary value problems for differential-difference equation, SIAM Journal on Applied Mathematics, 42(3), 1982, 502-531.

[11] Lange, C.G., Miura, R.M., Singular perturbation analysis of boundary-value problems for differential-difference equations, III. Turning point problems, SIAM Journal on Applied Mathematics, 45(5), 1985, 708-734.

[12] Lange, C.G., Miura, R.M., Singular perturbation analysis of boundary-value problems for differential-difference equations, V. Small shifts with layer behavior, SIAM Journal on Applied Mathematics, 54(1), 1994, 249-272.

[13] Kadalbajoo, M.K., Sharma, K.K., Numerical analysis of boundary-value problem for singularly perturbed differential-difference equations: small shifts of mixed type, Journal of Optimization Theory & Application, 115(1), 2002, 145-163.

[14] Kadalbajoo, M.K., Sharma, K.K., ε-uniform fitted mesh method for singularly perturbed differential-difference equations: mixed type of shifts with layer behavior, International Journal of computer Mathematics, 81(1), 2004, 49-62.

[15] Kadalbajoo, M.K., Sharma, K.K., Numerical treatment of a mathematical model arising from a model of neuronal variability, Journal of Mathematical Analysis and Applications, 307(2), 2005, 606-627.

[16] Rao, R.N., Chakravarthy, P.P., A finite difference method for singularly perturbed differential-difference equations arising from a model of neuronal variability, Journal of Taibah University for Science, 7(3), 2013, 128-136.

[17] Rao, R.N., Chakravarthy, P.P., A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior, Applied Mathematical Modelling, 37(8), 2013, 5743-5755.

[18] Rao, R.N., Chakravarthy, P.P., An exponentially fitted tridiagonal finite difference method for singularly perturbed differential-difference equations with small shifts, Ain Shams Engineering Journal, 5(4),2014, 1351-1360.

[19] Mohapatra, J., Natesan, S., Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution, International Journal for Numerical Methods in Biomedical Engineering, 27(9), 2011, 1427-1445.

[20] Duressa, G., Reddy, Y.N., Domain decomposition method for singularly perturbed differential difference equations with layer behavior, International Journal of Engineering & Applied Sciences, 7(1), 2015, 86-102.

[21] Sirisha, L., Phaneendra, K., Reddy, Y.N., Mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, Ain Shams Engineering Journal, 9(4),2018, 647-654.

[22] Natesan, S., Bawa, R.K., Second-order numerical scheme for singularly perturbed reaction-diffusion Robin problems, Journal of Numerical Analysis, Industrial and Applied Mathematics, 2(3-4), 2007, 177-192.

[23] Priyadharshini, R.M., Ramanujam, N., Valanarasu, T., Hybrid difference schemes for singularly perturbed problem of mixed type with discontinuous source term, Journal of Applied Mathematics & Informatics, 28(5), 2010, 1035-1054.

[24] Mukherjee, K., Parameter-uniform improved hybrid numerical scheme for singularly perturbed problems with interior layers, Mathematical Modelling and Analysis, 23(2), 2018, 167-189.

[25] Li, G., Qiu, J., Hybrid weighted essentially non-oscillatory schemes with different indicators. Journal of Computational Physics, 229(21), 2010, 8105-8129.

[26] Li, G., Xing, Y., Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields. Computers & Mathematics with Applications, 75(6), 2018, 2071-2085.

[27] Dona, W.S., Lib, P., Wongc, K.Y., Gaod, Z., Symmetry-Preserving Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws. Journal of Computational Physics, 2017.

[28] Zhu, H., Gao, Z., An h-adaptive RKDG method with troubled-cell indicator for one-dimensional detonation wave simulations. Advances in Computational Mathematics, 42(5), 2016, 1081-1102.

[29] Kellogg, R.B., Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points. Mathematics of Computation, 32(144), 1978, 1025-1039.

[30] Stynes, M., Roos, H.G., The midpoint upwind scheme, Applied Numerical Mathematics, 23(3), 1997, 361-374.

[31] Ramesh, V.P., Kadalbajoo, M.K., Upwind and midpoint upwind difference methods for time-dependent differential difference equations with layer behaviour, Applied Mathematics and Computation, 202(2), 2008, 453-471.