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Accurate Approximate Solution of Ambartsumian Delay Differential Equation via Decomposition Method

Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
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Math. Comput. Appl. 2019, 24(1), 7; https://doi.org/10.3390/mca24010007
Received: 11 November 2018 / Revised: 19 December 2018 / Accepted: 26 December 2018 / Published: 9 January 2019
The Ambartsumian delay equation is used in the theory of surface brightness in the Milky way. The Adomian decomposition method (ADM) is applied in this paper to solve this equation. Two canonical forms are implemented to obtain two types of the approximate solutions. The first solution is provided in the form of a power series which agrees with the solution in the literature, while the second expresses the solution in terms of exponential functions which is viewed as a new solution. A rapid rate of convergence has been achieved and displayed in several graphs. Furthermore, only a few terms of the new approximate solution (expressed in terms of exponential functions) are sufficient to achieve extremely accurate numerical results when compared with a large number of terms of the first solution in the literature. In addition, the residual error using a few terms approaches zero as the delay parameter increases, hence, this confirms the effectiveness of the present approach over the solution in the literature. View Full-Text
Keywords: Adomian decomposition method; Ambartsumian equation; Milky way; series solution Adomian decomposition method; Ambartsumian equation; Milky way; series solution
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Ebaid, A.; Al-Enazi, A.; Albalawi, B.Z.; Aljoufi, M.D. Accurate Approximate Solution of Ambartsumian Delay Differential Equation via Decomposition Method. Math. Comput. Appl. 2019, 24, 7.

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