A novel exact solution for the fractional Ambartsumian equation

Advances in Difference Equations

article

Ebaid, Abdelhalim¹  Cattani, Carlo²  Al Juhani, Amnah S.¹  El-Zahar, Essam R.³ 

[1]Department of Mathematics, Faculty of Science, University of Tabuk

[2]Engineering School (DEIM), University of Tuscia

[3]Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University

[4]Department of Basic Engineering Science, Faculty of Engineering, Menofia University

关键词: Ambartsumian equation;    Adomian decomposition method;    Laplace transform;    Fractional calculus;    Mittag-Leffler function;    Exact solution;   

DOI  :  10.1186/s13662-021-03235-w

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY