Journal of Mathematics and System Science 8 (2018) 182-186 doi: 10.17265/2159-5291/2018.07.002 New Analytic Solution for Ambartsumian Equation Fahad M. Alharbi and Abdelhalim Ebaid Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia Abstract: Analytical solution is obtained for Ambartsumian equation in this paper. This equation is of application in astronomy. The obtained solution has many advantages over the published one in the literature as shown by several comparisons. Key words: Delay equation; analytic solution. 1. Introduction The current work is devoted to analyzing the first order delay differential equation given by Ref. [1]. 1, > , 1 ) ( ) ( q q t y q t y t y             (1) where q is a constant and the initial condition (IC) is , = (0)  y (2) where  is a further constant. Eq. (1) is called Ambartsumian equation which describes the surface brightness in Astronomy [1, 2]. Uniqueness and existence of this model has been investigated by Kato and McLeod [3]. Patade and Bhalekar [1] solved Eqs. (1) and (2) by using the Daftardar-Gejji method [4]. Their solution was expressed as a power series in the independent variable t and they have also addressed the issue of convergence of such series. However, obtaining analytical solution for the present model is still of practical interest. In the literature [5-12], several analytical methods were used to solve various problems in different areas of researches, however, a direct approach is to be introduced in this paper. In order to obtain the desired solution, an effective ansatz is to be implemented to treat Eq. (1) under the IC Eq. (2). The proposed approach is based on a suitable series Corresponding author: Abdelhalim Ebaid, Dr., associate professor, research fields: applied differential equations with applications in physics and biomathematics, nanofluids, special relativity. substitution in terms of exponential functions with undetermined coefficients. Details of the proposed method are presented in the next section and the advantages over the previous solution by Daftardar-Gejji method [1] will be discussed in a subsequent section. 2. Analysis First, we rewrite Eq. (1) as   . 1 = , ) ( ) ( q t y t y t y        (3) Here, the solution of Eq. (3) is assumed as , ) ( = ) ( 0 = t n c n n e a t y      (4) where c is unknown. Accordingly, we have , ) ( = ) ( 0 = t n c n n n e a c t y         (5) and . ) ( = ) ( 1 0 = t n c n n e a t y        (6) Inserting Eqs. (4)-(6) into Eq. (3), yields =0 1 =0 =0 = , n n c t n n n n c t c t n n n n c ae ae ae                   (7) or D DAVID PUBLISHING