Numerical methods for singularly perturbed second order ordinary differential equations with a delay. (English) Zbl 1329.65158

Ramanujam, N.

Subrahmanyam, P. V. (ed.) et al., Proceedings of the national symposium on mathematical methods and applications, NSMMA, Chennai, India, December 22, 2010. Invited talks. Chennai: Indian Institute of Technology Madras. 10-22 (2011).

Summary: A numerical method named as Initial Value Technique (IVT) is suggested to solve singularly perturbed boundary value problems for second order ordinary delay differential equations. In this technique, the original problem of solving the second order differential equation is reduced to solving first order differential equations, some of which are singularly perturbed with out delay and others are regular differential equations with a delay term. The singularly perturbed problems are solved by second order hybrid finite difference scheme, where as the delay problems are solved by the fourth order Runge-Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm and it is of order \(O(\varepsilon+N^{-2}\ln^2N)\), where \(N\) and \(\varepsilon\) are discretization parameter and perturbation parameter respectively. Numerical results are provided to illustrate the theoretical results.
For the entire collection see [Zbl 1300.00036].

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
65L11	Numerical solution of singularly perturbed problems involving ordinary differential equations

Keywords:

initial value technique; boundary value problem; singularly perturbed problem; maximum principle; delay; hybrid finite difference scheme; Shishkin mesh