Numerical solution of a boundary value problem including both delay and boundary layer
Abstract
Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.
Keyword : singular perturbation, boundary-value problem, finite difference method, delay differential equation, Shishkin mesh
How to Cite
Cimen, E. (2018). Numerical solution of a boundary value problem including both delay and boundary layer. Mathematical Modelling and Analysis, 23(4), 568-581. https://doi.org/10.3846/mma.2018.034
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
[1] G.M. Amiraliyev and E. Cimen. Numerical method for a singularly perturbed convection–diffusion problem with delay.Applied Mathematics and Computation ,216(8):2351–2359, 2010. https://doi.org/10.1016/j.amc.2010.03.080
[2] V.B. Andreev. The green function and a priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes.Differential Equations, 37(7):923–933, 2001. https://doi.org/10.1023/A:1011949419389
[3] M. Cakir.A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. Mathematical Modelling and Analysis,21(5):644–658,2016. https://doi.org/10.3846/13926292.2016.1201702
[4] E. Cimen. A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term.Journal of Mathematical Analysis, 8(1):202–211, 2017.
[5] E. Cimen and G.M. Amiraliyev. A uniform convergent method for singularly perturbed nonlinear differential-difference equation.Journal of Informatics and Mathematical Sciences, 9(1):191–199, 2017.
[6] E. Cimen and M. Cakir. Numerical treatment of nonlocal boundary value problem with layer behaviour. Bulletin of the Belgian Mathematical Society - SimonStevin, 24(3):339–352, 2017.
[7] M.W. Derstein, H.M. Gibbs, F.A. Hopf and D.L. Kaplan. Bifurcation gap in a hybrid optically bistable system. Physical Review A, 26(6):3720–3722, 1982. https://doi.org/10.1103/PhysRevA.26.3720
[8] E.R. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
[9] L. ́E. Elsgolc. Qualitative Methods in Mathematical Analysis, in: Translations of Mathematical Monographs, vol.12. AMS, Providence, RI, 1964.
[10] T. Erneux. Applied Delay Differential Equations.Surveys and Tuto-rials in the Applied Mathematical Sciences. Springer, New York, 2009. https://doi.org/10.1007/978-0-387-74372-1
[11] P. Farrell, A. Hegarty, J.H. Miller, E. O’Riordan and G.I. Shishkin. Robust Computational Techniques for Boundary Layers. Applied Mathematics. Chap-man Hall/CRC, New York, 2000.
[12] F.Z. Geng and S.P. Qian. Modified reproducing kernel method for singularly perturbed boundary value problems with a delay.Applied Mathematical Modelling, 39(18):5592–5597, 2015. https://doi.org/10.1016/j.apm.2015.01.021
[13] V.Y. Glizer. Controllability conditions of linear singularly perturbed systems with small state and input delays.Mathematics of Control, Signals, and Systems, 28(1):1–29, 2015. https://doi.org/10.1007/s00498-015-0152-3
[14] M.K. Kadalbajoo and V.P. Ramesh. Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Applied Mathematics and Computation, 188(2):1816–1831, 2007. https://doi.org/10.1016/j.amc.2006.11.046
[15] A. Keane, B. Krauskopf and C.M. Postlethwaite. Climate models with delay differential equations.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(11):1–15, 2017. https://doi.org/10.1063/1.5006923
[16] C.G. Lange and R.M. Miura. Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM Journal on Applied Mathematics, 42(3):502–531, 1982. https://doi.org/10.1137/0142036
[17] C.G. Lange and R.M. Miura. Singular perturbation analysis of boundary value problems for differential-difference equations. IV. A nonlinear example with layer behavior. Studies in Applied Mathematics, 84(3):231–273, 1991. https://doi.org/10.1002/sapm1991843231
[18] L. B. Liu and Y. Chen. Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay. Applied Mathematics and Computation, 227(15):801–810, 2014. https://doi.org/10.1016/j.amc.2013.10.085
[19] A. Longtin and J. Milton. Complex oscillations in the human pupil light reflex with mixed and delayed feedback.Mathematical Biosciences, 90(1-2):183–199,1988. https://doi.org/10.1016/0025-5564(88)90064-8
[20] M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197(4300):287–289, 1977. https://doi.org/10.1126/science.267326
[21] J. J. H. Miller, E. O’Riordan and G. I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. Revised Edition. World Scientific, Singapore, 2012.
[22] H.G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd Edt. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-34467-4
[23] P.A. Selvi and N. Ramanujam. An iterative numerical method for singularly perturbed reaction–diffusion equations with negative shift .Journal of Computational and Applied Mathematics, 296:10–23, 2016. https://doi.org/10.1016/j.cam.2015.09.003
[24] R.B. Stein. Some models of neuronal variability. Biophysical Journal, 7(1):37–68,1967. https://doi.org/10.1016/S0006-3495(67)86574-3
[25] V. Subburayan and N. Ramanujam. Asymptotic initial value technique for singularly perturbed convection-diffusion delay problems with boundary and weak interior layers.Applied Mathematics Letters, 25(12):2272–2278, 2012. https://doi.org/10.1016/j.aml.2012.06.016
[26] H. Zarin. On discontinuous Galerk in finite element method for singularly perturbed delay differential equations.Applied Mathematics Letters, 38(1):27–32, 2014. https://doi.org/10.1016/j.aml.2014.06.013
[2] V.B. Andreev. The green function and a priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes.Differential Equations, 37(7):923–933, 2001. https://doi.org/10.1023/A:1011949419389
[3] M. Cakir.A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. Mathematical Modelling and Analysis,21(5):644–658,2016. https://doi.org/10.3846/13926292.2016.1201702
[4] E. Cimen. A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term.Journal of Mathematical Analysis, 8(1):202–211, 2017.
[5] E. Cimen and G.M. Amiraliyev. A uniform convergent method for singularly perturbed nonlinear differential-difference equation.Journal of Informatics and Mathematical Sciences, 9(1):191–199, 2017.
[6] E. Cimen and M. Cakir. Numerical treatment of nonlocal boundary value problem with layer behaviour. Bulletin of the Belgian Mathematical Society - SimonStevin, 24(3):339–352, 2017.
[7] M.W. Derstein, H.M. Gibbs, F.A. Hopf and D.L. Kaplan. Bifurcation gap in a hybrid optically bistable system. Physical Review A, 26(6):3720–3722, 1982. https://doi.org/10.1103/PhysRevA.26.3720
[8] E.R. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
[9] L. ́E. Elsgolc. Qualitative Methods in Mathematical Analysis, in: Translations of Mathematical Monographs, vol.12. AMS, Providence, RI, 1964.
[10] T. Erneux. Applied Delay Differential Equations.Surveys and Tuto-rials in the Applied Mathematical Sciences. Springer, New York, 2009. https://doi.org/10.1007/978-0-387-74372-1
[11] P. Farrell, A. Hegarty, J.H. Miller, E. O’Riordan and G.I. Shishkin. Robust Computational Techniques for Boundary Layers. Applied Mathematics. Chap-man Hall/CRC, New York, 2000.
[12] F.Z. Geng and S.P. Qian. Modified reproducing kernel method for singularly perturbed boundary value problems with a delay.Applied Mathematical Modelling, 39(18):5592–5597, 2015. https://doi.org/10.1016/j.apm.2015.01.021
[13] V.Y. Glizer. Controllability conditions of linear singularly perturbed systems with small state and input delays.Mathematics of Control, Signals, and Systems, 28(1):1–29, 2015. https://doi.org/10.1007/s00498-015-0152-3
[14] M.K. Kadalbajoo and V.P. Ramesh. Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Applied Mathematics and Computation, 188(2):1816–1831, 2007. https://doi.org/10.1016/j.amc.2006.11.046
[15] A. Keane, B. Krauskopf and C.M. Postlethwaite. Climate models with delay differential equations.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(11):1–15, 2017. https://doi.org/10.1063/1.5006923
[16] C.G. Lange and R.M. Miura. Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM Journal on Applied Mathematics, 42(3):502–531, 1982. https://doi.org/10.1137/0142036
[17] C.G. Lange and R.M. Miura. Singular perturbation analysis of boundary value problems for differential-difference equations. IV. A nonlinear example with layer behavior. Studies in Applied Mathematics, 84(3):231–273, 1991. https://doi.org/10.1002/sapm1991843231
[18] L. B. Liu and Y. Chen. Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay. Applied Mathematics and Computation, 227(15):801–810, 2014. https://doi.org/10.1016/j.amc.2013.10.085
[19] A. Longtin and J. Milton. Complex oscillations in the human pupil light reflex with mixed and delayed feedback.Mathematical Biosciences, 90(1-2):183–199,1988. https://doi.org/10.1016/0025-5564(88)90064-8
[20] M. C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197(4300):287–289, 1977. https://doi.org/10.1126/science.267326
[21] J. J. H. Miller, E. O’Riordan and G. I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. Revised Edition. World Scientific, Singapore, 2012.
[22] H.G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd Edt. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-34467-4
[23] P.A. Selvi and N. Ramanujam. An iterative numerical method for singularly perturbed reaction–diffusion equations with negative shift .Journal of Computational and Applied Mathematics, 296:10–23, 2016. https://doi.org/10.1016/j.cam.2015.09.003
[24] R.B. Stein. Some models of neuronal variability. Biophysical Journal, 7(1):37–68,1967. https://doi.org/10.1016/S0006-3495(67)86574-3
[25] V. Subburayan and N. Ramanujam. Asymptotic initial value technique for singularly perturbed convection-diffusion delay problems with boundary and weak interior layers.Applied Mathematics Letters, 25(12):2272–2278, 2012. https://doi.org/10.1016/j.aml.2012.06.016
[26] H. Zarin. On discontinuous Galerk in finite element method for singularly perturbed delay differential equations.Applied Mathematics Letters, 38(1):27–32, 2014. https://doi.org/10.1016/j.aml.2014.06.013