Parameter uniform numerical method for a boundary-value problem for singularly perturbed nonlinear delay differential equation of neutral type

Abstract

The boundary-value problems (BVPs) for singularly perturbed nonlinear delay differential equations are studied. In this article, we present a parameter uniform numerical scheme based on the fitting mesh technique in the boundary layer region to solve such BVPs. To tackle the nonlinearity, we use the quasilinearization process [1], and to handle the delay term, we use the Taylor series [2]. In the limit, the solution of the linear BVPs obtained after quasilinearization converges quadratically to the solution of the nonlinear BVP for a judicious initial approximation. The sequence of linear problems so obtained after quasilinearization is analyzed for convergence, and error estimates for the solution of the discretized problem are derived. The method is compared with a standard upwind difference scheme and fitted operator method on uniform mesh by carrying out several numerical experiments.

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Keywords:

• Nonlinear boundary-value problem
• Quasilinearization
• Singularly perturbed
• ϵ-Uniform error estimate
• Delay differential equation

Notes

E-mail: [email protected]