A class of finite difference schemes for singularly perturbed delay differential equations of second order

In this paper, we proposed a new class of finite difference schemes for solving singularly perturbed delaydifferential equation of second order. The proposed schemes are oscillation-free and more accurate than conventionalschemes on a uniform mesh. These schemes are easily adaptable on special meshes like Shishkin mesh or Bakhvalovmesh and are uniformly convergent with respect to the perturbation parameter. The error analysis has been carried outand numerical examples are presented to show the accuracy and efficiency of the proposed schemes.

PDF

___

[1] Amiraliyev GM, Cimen E. Numerical method for a singularly perturbed convection-diffusion problem with delay. Applied Mathematics and Computation 2010; 216: 2351-2359. doi:10.1016/j.amc.2010.03.080
[2] Bakhvalov NS. On the optimization of the methods for solving boundary value problems in the presence of boundary layers. USSR Computational Mathematics and Mathematical Physics 1969; 9: 841-859. doi:10.1016/0041- 5553(69)90038-X
[3] Doolan ER, Miller JJH, Schilders WHA. Uniform Numerical Methods for Problems With Initial and Boundary Layers. Dublin, Ireland: Boole Press, 1980.
[4] Eloe PW, Raffoul YN, Tisdell CC. Existence, uniqueness and constructive results for delay differential equations. Electronic Journal of Differential Equations 2005; 121: 1-11.
[5] Farrell PA, Hegatry AF, Miller JJH, O’Riordan E, Shishikin GI. Singularly perturbed convection diffusion problems with boundary and weak interior layers. Journal of Computational and Applied Mathematics 2004; 166: 133-151. doi:10.1016/j.cam.2003.09.033
[6] Gartland EC. Graded-mesh difference schemes for singularly perturbed two-point boundary value problems. Mathematics of Computation 1988; 51(184): 631-657. doi:10.1090/S0025-5718-1988-0935072-1
[7] Jain MK. Numerical Solution of Differential Equations. 2nd ed. New York, NY, USA: A Halsted Press Book. John Wiley & Sons, Inc., 1984.
[8] Kadalbajoo MK, Sharma KK. Numerical analysis of singularly perturbed delay differential equations with layer behavior. Journal of Computational and Applied Mathematics 2004; 157: 11-28. doi:10.1016/j.amc.2003.06.012
[9] Kadalbajoo MK, Sharma KK. Parameter-Uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electronic Transactions on Numerical Analysis. 2006; 23: 180-201.
[10] Kadalbajoo MK, Sharma KK. A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. Applied Mathematics and Computation 2008; 197: 692-707. doi:10.1016/j.amc.2007.08.089
[11] Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential-difference equations. SIAM Journal on Applied Mathematics 1982; 42: 502-531. doi:10.1137/0142036
[12] Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential-difference equations II. Rapid oscillations and resonances. SIAM Journal on Applied Mathematics 1985; 45: 687-707. doi:10.1137/0145041
[13] Lange CG, Miura RM. Singular perturbation analysis of boundary-value problems for differential-difference equations V. Small shifts with layer behavior. SIAM Journal on Applied Mathematics 1994; 54: 249-272. doi:10.1137/S0036139992228120
[14] Miller JJH, Riordan EO, Shishkin IG. Fitted Numerical Methods for Singular Perturbation Problems. Singapore: Word Scientific Publishing, 2012.
[15] Roos HG, Stynes M, Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations. Berlin, Germany: Springer-Verlag, 2008.
[16] Subburayan V, Ramanujam N. An initial value technique for singularly perturbed convection-diffusion problems with a negative shift. Journal of Optimization Theory and Applications 2013; 158: 234-250. doi:10.1007/s10957- 012-0200-9
[17] Subburaya, V, Ramanujam N. An initial value technique for singularly perturbed reaction-diffusion problems with a negative shift. Novi Sad Journal of Mathematics 2013; 43: 67-80.
[18] Subburayan V. A parameter uniform numerical method for singularly perturbed delay problems with discontinuous convection coefficient. Arab Journal of Mathematical Sciences 2016; 22: 191-206. doi:10.1016/j.ajmsc.2015.07.001
[19] Sun ZZ. Numerical Methods of Partial Differential Equations. 2nd ed. Beijing, China: Science Press, 2012. (in Chinese)
[20] Valanarasu T, Ramanujam N. An asymptotic initial-value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term. Journal of Applied Mathematics and Computing 2007; 23: 141-152. doi:10.1007/BF02831964
[21] Valanarasu T, Ramanujam N. Asymptotic initial-value method for second order singular perturbation problems of reaction-diffusion type with a discontinuous source term. Journal of Optimization Theory and Applications 2007; 133: 371-383. doi:10.1007/s10957-007-9167-3
[22] Wang K, Wong YS. Pollution-free finite difference schemes for non-homogeneous Helmholtz equation. International Journal of Numerical Analysis and Modeling 2014; 11: 787-815.
[23] Wang K, Wong YS, Deng J. Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates. Communications in Computational Physics 2015; 17: 779-807. doi:10.4208/cicp.110214.101014a
[24] Xuefei H, Wang K. New finite difference methods for singularly perturbed convection-diffusion equations. Taiwanese Journal of Mathematics 2018; 22: 949-978. doi:10.11650/tjm/171002