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Asymptotic Initial-Value Method for Second-Order Singular Perturbation Problems of Reaction-Diffusion Type with Discontinuous Source Term

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Abstract

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.

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Authors and Affiliations

  1. Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamilnadu, India

    T. Valanarasu & N. Ramanujam

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  1. T. Valanarasu
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  2. N. Ramanujam
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Correspondence to T. Valanarasu.

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Communicated by F.E. Udwadia.

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Valanarasu, T., Ramanujam, N. Asymptotic Initial-Value Method for Second-Order Singular Perturbation Problems of Reaction-Diffusion Type with Discontinuous Source Term. J Optim Theory Appl 133, 371–383 (2007). https://doi.org/10.1007/s10957-007-9167-3

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