Solving Boundary Value Problems for Second Order Singularly Perturbed Delay Differential Equations by ε-Approximate Fixed-Point Method

Zbigniew Bartoszewski; Anna Baranowska

doi:10.3846/13926292.2015.1048759

Solving Boundary Value Problems for Second Order Singularly Perturbed Delay Differential Equations by ε-Approximate Fixed-Point Method

Abstract

In this paper, the boundary value problem for second order singularly perturbed delay differential equation is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by cubic spline vh defined by its values vi = vh(ti) at grid points ti, i = 0, 1, ... ,N. The necessary for construction the cubic spline and missing the first derivatives at the boundary are replaced by the derivatives of the corresponding interpolating polynomials matching the grid points values nearest to the boundary points. An approximation of the solution is obtained by minimization techniques applied to a function whose arguments are the grid point values of the sought spline. The results of numerical experiments with two boundary value problems for the second order singularly perturbed delay differential equations as well as their comparison with the results of other methods employed by other authors are also provided.

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Authors (2)

Zbigniew Bartoszewski dr hab.
Anna Baranowska dr
- Instytut Oceanologii PAN w Sopocie Instytut Oceanologii

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DOI:: Digital Object Identifier (open in new tab) 10.3846/13926292.2015.1048759
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Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: Mathematical Modelling and Analysis no. 20, pages 369 - 381,
ISSN: 1392-6292
Language:: English
Publication year:: 2015
Bibliographic description:: Bartoszewski Z., Baranowska A.: Solving Boundary Value Problems for Second Order Singularly Perturbed Delay Differential Equations by ε-Approximate Fixed-Point Method// Mathematical Modelling and Analysis. -Vol. 20, nr. 3 (2015), s.369-381
DOI:: Digital Object Identifier (open in new tab) 10.3846/13926292.2015.1048759
Verified by:: Gdańsk University of Technology