A Simple Solution for the General Fractional Ambartsumian Equation

Abstract

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.

Anisotropic Fractional Cosmology: K-Essence Theory

“…In the time domain, a non-local description becomes manifest as a memory effect, and in the space domain, it manifests as non-homogeneous similarity structures [1][2][3]. During the last decades, FC has been the subject of intense theoretical and applied research in almost all areas of the sciences and engineering from the point of view of classical and quantum systems [4][5][6][7][8][9][10][11][12][13][14]; recently, new studies on FC have been made [15][16][17][18]. This is because FC describes complex physical systems more accurately, and at the same time investigates more about simple dynamical systems [19,20].…”

Anisotropic Fractional Cosmology: K-Essence Theory

“…In the time domain, a non-local description becomes manifest as a memory effect, and in the space domain, it manifests as non-homogeneous similarity structures [1][2][3]. During the last decades, FC has been the subject of intense theoretical and applied research, almost in all areas of the sciences and engineering, from the point of view of the classical and quantum systems [4][5][6][7][8][9][10][11][12][13][14], recently new studies on FC have been made [15][16][17][18]. This is because, the FC describes more accurately the complex physical systems and at the same time, investigates more about simple dynamical systems [19,20].…”