A Fractional Equation with Left-Sided Fractional Bessel Derivatives of Gerasimov–Caputo Type

Abstract

In this article we propose and study a method to solve ordinary differential equations with left-sided fractional Bessel derivatives on semi-axes of Gerasimov–Caputo type. We derive explicit solutions to equations with fractional powers of the Bessel operator using the Meijer integral transform.

Fractional Newton–Raphson Method Accelerated with Aitken’s Method

“…and, from the Theorem 1, the Newton-Raphson method has an order of convergence at least quadratic, that is, it fulfills Equation (15) with p = 2. On other hand, the second derivative of the iteration function of Newton-Raphson method takes the following form:…”

“…Currently, the fractional calculus does not have a unified definition of what is considered a fractional derivative because one of the conditions required to consider an expression as a fractional derivative is to recover the results of conventional calculus when the order α → n, with n ∈ N [12]; among the most common definitions of fractional derivatives are the Riemann-Liouville (R-L) fractional derivative and the Caputo fractional derivative [13][14][15], the latter is usually the most studied since the Caputo fractional derivative allows us a physical interpretation to problems with initial conditions; this derivative fulfills the property of the classical calculus that the derivative of a constant is null regardless of the order α of the derivative; however, this does not occur with the R-L fractional derivative, and this characteristic can be used to solve nonlinear systems [4,16,17].…”

“…This fractional derivative is of the utmost importance since it allows us to give a physical interpretation of the initial value problems, moreover being used to model fractional time. In some texts, it is known as the fractional derivative of Gerasimov-Caputo [15].…”

Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems

“…We are going to establish solvability conditions of problem (1)- (4). In recent decades, fractional integro-differential calculus has become one of the most important tools for solving mathematical modeling problems [4][5][6][7][8]. On the other hand various issues of the control theory, including unique solvability, are of interest to many authors.…”

On two classes of generalized fractional operators (with short historical survey of fractional calculus)

“…Аналогично можно ввести дробные производные Герасимова и для других аналогов дробных операторов Римана-Лиувилля, например, для дробных производных Бесселя, см. недавнюю работу авторов [61].…”