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Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

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  • Essam R. El-Zahar

Abstract

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.

Suggested Citation

  • Essam R. El-Zahar, 2016. "Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-12, November.
    • Handle: RePEc:hin:jnijde:1015634
    DOI: 10.1155/2016/1015634
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    Cited by:

    1. Essam R. El-Zahar & Ghaliah F. Al-Boqami & Haifa S. Al-Juaydi, 2024. "Approximate Analytical Solutions for Strongly Coupled Systems of Singularly Perturbed Convection–Diffusion Problems," Mathematics, MDPI, vol. 12(2), pages 1-24, January.
    2. Essam R. El-Zahar & José Tenreiro Machado & Abdelhalim Ebaid, 2019. "A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations," Mathematics, MDPI, vol. 7(12), pages 1-18, December.

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