Method of Reduction of Order for Solving Singularly Perturbed Delay Differential Equations

M. Adilaxmi

Abstract


In this paper, we have presented and illustrated the method of reduction of order for solving singularly perturbed delay differential equations. The given second order singularly perturbed delay differential equation is replaced by a pair of first order problems. These are in turn solved by initial value solvers. The integration of these initial value problems goes in the opposite direction. The applicability of this method is demonstrated by solving some model problems and the numerical results are compared with the exact solution. From the tables and figures, it is observed that the present method produces satisfactory results.


Keywords


Singular perturbations; Delay differential equations; Reduction of order

Full Text:

PDF

References


M. Adilaxmi, D. Bhargavi and Y. N. Reddy, An initial value technique using exponentially fitted non-standard finite difference method for singularly perturbed differentialdifference equations, Applications and Applied Mathematics 14(1) (2019), 245 – 269, URL: http://www.pvamu.edu/aam/wp-content/uploads/sites/182/2019/06/16_R1137_AAM_Reddy_YR_040918_Posted_052419.pdf.

R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York (1963).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Co., New York (1978).

E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problem With Initial and Boundary Layers, Boole Press, Dublin (1980).

R. D. Driver, Ordinary and Delay Differential Equations, Springer, New York (1977).

L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press (1966).

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary value problems for differential-difference equations. V. Small shifts with layer behaviour, SIAM Journal on Applied Mathematics 54 (1994), 249 – 272, DOI: 10.1137/S0036139992228120.

A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York (1981).

R. E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York (1974).

Y. N. Reddy and P. P. Chakravarthy, Method of reduction of order for solving singularly perturbed two-point boundary value problems, Applied Mathematics and Computation 136 (2003), 27 – 45, DOI: 10.1016/S0096-3003(02)00015-2.




DOI: http://dx.doi.org/10.26713%2Fjims.v12i4.1463

eISSN 0975-5748; pISSN 0974-875X