Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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SOLVING SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR VIA INITIAL VALUE METHOD

  • GEBEYAW, WONDWOSEN (Department of Mathematics, College of Natural and Computational Sciences, Dilla University) ;
  • ANDARGIE, AWOKE (Department of Mathematics, College of Sciences, Bahir Dar University) ;
  • ADAMU, GETACHEW (Department of Mathematics, College of Sciences, Bahir Dar University)
  • Received : 2017.11.16
  • Accepted : 2018.03.18
  • Published : 2018.05.30
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Abstract

In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

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References

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