A fitted numerical technique for a class of singularly perturbed delay differential equations

This paper examines the applicability of a new exponentially fitted scheme on Shishkin mesh for solving a class of singularly perturbed problems in ordinary differential equations containing a small delayed term. The original delayed problem is reduced to an equivalent new problem using the Taylor series expansion procedure, and then a three-term recurrence relationship is obtained by approximating the first and second order derivatives in the reduced problem by their mixed and central difference analogue, respectively. To avoid the non-uniformity in the solution, a fitting factor is introduced into the obtained tri-diagonal scheme, and its value is determined by the use of singular perturbation theory. Thomas' algorithm is efficiently used to solve tri-diagonal system of equations. Convergence and stability of the scheme are discussed in detail. Richardson's extrapolation technique is employed to improve the accuracy and rate of convergence of the solution produced by the scheme. Four standard example problems are solved, and the computational results are presented in terms of the maximum absolute point-wise errors and rate of convergence. Computational results are compared with some other published results. Theoretical and computational analysis reveal that the fitted scheme is able to produce accurate and uniformly convergent solutions with first-order accuracy, which is further improved to second-order accuracy by Richardson extrapolation. Further, the method is most suitable for all the values of (۱/N) ≫ ε where ε ≤ ۱۰−۶ and N is the number of sub-intervals in which the underlying interval is divided.

کلیدواژه ها:

Singular perturbation problem ، boundary layer ، Differential-difference equation ، Fitting factor ، Thomas Algorithm