A novel exact solution for the fractional Ambartsumian equation

Fractional calculus (FC) is useful in studying physical phenomena with memory effect. In this paper, a fractional form of Ambartsumian equation is considered utilizing the Caputo fractional derivative. The Heaviside expansion formula in classical calculus (CC) is extended/developed in view of FC. Then, the extended Heaviside expansion formula is applied to obtain the exact solution in a simplest form. Several theorems and lemmas are proved to facilitate the evaluation of the inverse Laplace transform of specific expressions in fractional forms. The exact solution is established in terms of a one-parameter Mittag-Leffler function which is provided for the first time for the Ambartsumian equation in FC. The present solution reduces to the corresponding one in the relevant literature as the fractional order tends to one. Moreover, the convergence of the obtained solution is theoretically proved. Comparisons with another approach in the literature are performed. The advantage of the present analysis over the existing one in the relevant literature is discussed and analyzed.