A numerical technique for solving nonlinear singularly perturbed delay differential equations
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.
This work is licensed under a Creative Commons Attribution 4.0 International License.
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.