A Simple Solution for the General Fractional Ambartsumian Equation

Fractionalisation and solution of the Ambartsumian equation is considered. The general approach to fractional calculus suitable for applications in physics and engineering is described. It is shown that Liouville-type derivatives are the necessary ones, because they fully preserve backward compatibility with classical results. Such derivatives are used to define and solve the fractional Ambartsumian equation. First, a solution in terms of a slowly convergent fractional Taylor series is obtained. Then, a simple solution expressed in terms of an infinite linear combination of Mittag-Leffler functions is deduced. A fast algorithm, based on a bilinear transformation and using the fast Fourier transform, is described and demonstrated for its approximate numerical realisation.

Automated summary

This paper discusses the fractionalization and solution of the Ambartsumian equation. It describes a general approach to fractional calculus suitable for applications in physics and engineering. Liouville-type derivatives are shown to be necessary as they preserve backward compatibility with classical results. These derivatives are used to define and solve the fractional Ambartsumian equation. The paper presents both a solution in terms of a slowly convergent fractional Taylor series and a simple solution expressed as an infinite linear combination of Mittag-Leffler functions. It also introduces a fast algorithm using a bilinear transformation and the fast Fourier transform for numerical approximation.