A numerical technique for solving nonlinear singularly perturbed delay differential equations
Abstract
This paper presents a numerical technique for solving nonlinear singularly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a sequence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.
Keyword : singularly perturbed problems, nonlinear delay problems, quasilinearization, parametric cubic spline
How to Cite
Kanth, A. R., & P. Murali, M. K. (2018). A numerical technique for solving nonlinear singularly perturbed delay differential equations. Mathematical Modelling and Analysis, 23(1), 64-78. https://doi.org/10.3846/mma.2018.005
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
T. Aziz and A. Khan. A spline method for second-order singularly perturbed boundary-value problems. Journal of Computational and Applied Mathematics, 147(2):445–452, 2002. https://doi.org/10.1016/S0377-0427(02)00479-X
Z. Bartoszewski and A. Baranowska. Solving boundary value problems for second order singularly perturbed delay differential equations by ε-approximate fixed-point method. Mathematical Modelling and Analysis, 20(3):369–381, 2015. https://doi.org/10.3846/13926292.2015.1048759
R.E. Bellman and R.E. Kalaba. Quasilinearization and nonlinear boundary-value problems. Modern analytic and computational methods in science and mathematics. American Elsevier Pub. Co., New York, 1965.
G.F. Carrier. Singular perturbation theory and geophysics. SIAM Review, 12(2):175–193, 1970. https://doi.org/10.1137/1012041
S. Cengizci. An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations. International Journal of Differential Equations, 2017, 2017. https://doi.org/10.1155/2017/7269450
P.P. Chakravarthy, S.D. Kumar and R.N. Rao. Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension. International Journal of Applied and Computational Mathematics, 3(3):1703–1717, 2017. https://doi.org/10.1007/s40819-016-0204-5
K.W. Chang and F.A. Howes. Nonlinear singular perturbation phenomena: theory and applications, volume 56. Springer Science & Business Media, New York, 2012.
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform numerical methods for problems with initial and boundary layers. Advances in numerical computation series. Boole Press Ltd., 1980.
T.C. Hanks. Model relating heat-flow values near, and vertical velocities of mass transport beneath, oceanic rises. Journal of Geophysical Research, 76(2):537–544, 1971. https://doi.org/10.1029/JB076i002p00537
F.A. Howes. Singular perturbations and differential inequalities, volume 168. American Mathematical Society, Providence, 1976.
M.K. Kadalbajoo and D. Kumar. A computational method for singularly perturbed nonlinear differential-difference equations with small shift. Applied Mathematical Modelling, 34(9):2584–2596, 2010. https://doi.org/10.1016/j.apm.2009.11.021
M.K. Kadalbajoo and K.K. Sharma. Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift. Nonlinear Analysis: Theory, Methods & Applications, 63(5):e1909–e1924, 2005. https://doi.org/10.1016/j.na.2005.02.098
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
R. Narasimhan. Singularly perturbed delay differential equations and numerical methods. In V. Sigamani, J.J.H. Miller, R. Narasimhan, P. Mathiazhagan and F. Victor(Eds.), Differential Equations and Numerical Analysis: Tiruchirappalli, India, January 2015, pp. 41–62. Springer India, New Delhi, 2016. https://doi.org/10.1007/978-81-322-3598-9 3
R.E. O’Malley. Introduction to singular perturbations. North-Holland Series in Applied Mathematics & Mechanics. Academic Press Inc., New York, 1974.
R.N. Rao and P.P. Chakravarthy. A numerical patching technique for singularly perturbed nonlinear differential-difference equations with a negative shift. Applied Mathematics, 2(2):11–20, 2012. https://doi.org/10.5923/j.am.20120202.04
J.M. Varah. A lower bound for the smallest singular value of a matrix. Linear Algebra and its Applications, 11(1):3–5, 1975. https://doi.org/10.1016/0024-3795(75)90112-3
R. Vulanovic, P.A. Farrell and P. Lin. Numerical solution of nonlinear singular perturbation problems modeling chemical reactions. In Applications of Advanced Computational Methods for Boundary and Interior Layers, pp. 192–213. Press, 1993.
Z. Bartoszewski and A. Baranowska. Solving boundary value problems for second order singularly perturbed delay differential equations by ε-approximate fixed-point method. Mathematical Modelling and Analysis, 20(3):369–381, 2015. https://doi.org/10.3846/13926292.2015.1048759
R.E. Bellman and R.E. Kalaba. Quasilinearization and nonlinear boundary-value problems. Modern analytic and computational methods in science and mathematics. American Elsevier Pub. Co., New York, 1965.
G.F. Carrier. Singular perturbation theory and geophysics. SIAM Review, 12(2):175–193, 1970. https://doi.org/10.1137/1012041
S. Cengizci. An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations. International Journal of Differential Equations, 2017, 2017. https://doi.org/10.1155/2017/7269450
P.P. Chakravarthy, S.D. Kumar and R.N. Rao. Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension. International Journal of Applied and Computational Mathematics, 3(3):1703–1717, 2017. https://doi.org/10.1007/s40819-016-0204-5
K.W. Chang and F.A. Howes. Nonlinear singular perturbation phenomena: theory and applications, volume 56. Springer Science & Business Media, New York, 2012.
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform numerical methods for problems with initial and boundary layers. Advances in numerical computation series. Boole Press Ltd., 1980.
T.C. Hanks. Model relating heat-flow values near, and vertical velocities of mass transport beneath, oceanic rises. Journal of Geophysical Research, 76(2):537–544, 1971. https://doi.org/10.1029/JB076i002p00537
F.A. Howes. Singular perturbations and differential inequalities, volume 168. American Mathematical Society, Providence, 1976.
M.K. Kadalbajoo and D. Kumar. A computational method for singularly perturbed nonlinear differential-difference equations with small shift. Applied Mathematical Modelling, 34(9):2584–2596, 2010. https://doi.org/10.1016/j.apm.2009.11.021
M.K. Kadalbajoo and K.K. Sharma. Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift. Nonlinear Analysis: Theory, Methods & Applications, 63(5):e1909–e1924, 2005. https://doi.org/10.1016/j.na.2005.02.098
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
R. Narasimhan. Singularly perturbed delay differential equations and numerical methods. In V. Sigamani, J.J.H. Miller, R. Narasimhan, P. Mathiazhagan and F. Victor(Eds.), Differential Equations and Numerical Analysis: Tiruchirappalli, India, January 2015, pp. 41–62. Springer India, New Delhi, 2016. https://doi.org/10.1007/978-81-322-3598-9 3
R.E. O’Malley. Introduction to singular perturbations. North-Holland Series in Applied Mathematics & Mechanics. Academic Press Inc., New York, 1974.
R.N. Rao and P.P. Chakravarthy. A numerical patching technique for singularly perturbed nonlinear differential-difference equations with a negative shift. Applied Mathematics, 2(2):11–20, 2012. https://doi.org/10.5923/j.am.20120202.04
J.M. Varah. A lower bound for the smallest singular value of a matrix. Linear Algebra and its Applications, 11(1):3–5, 1975. https://doi.org/10.1016/0024-3795(75)90112-3
R. Vulanovic, P.A. Farrell and P. Lin. Numerical solution of nonlinear singular perturbation problems modeling chemical reactions. In Applications of Advanced Computational Methods for Boundary and Interior Layers, pp. 192–213. Press, 1993.