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Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

Author

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  • Weam Alharbi

    (Department of Mathematics, Faculty of Sciences, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia)

  • Sergei Petrovskii

    (Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK)

Abstract

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.

Suggested Citation

  • Weam Alharbi & Sergei Petrovskii, 2020. "Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method," Mathematics, MDPI, vol. 8(12), pages 1-11, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2247-:d:465144
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    References listed on IDEAS

    as
    1. Sayed M. Khaled & Essam R. El-Zahar & Abdelhalim Ebaid, 2019. "Solution of Ambartsumian Delay Differential Equation with Conformable Derivative," Mathematics, MDPI, vol. 7(5), pages 1-10, May.
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