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This paper considers a coupled hybrid thermostat system driven by the Ã-Caputo fractional derivative in a Banach algebra. We employ a version of Darbo’s fixed-point theorem combined with the measure of noncompactness (MNC) technique to establish certain existence results. Additionally, an example is provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

This paper investigates the neutron diffusion model with two energy groups in spherical reactors. In particular, the integer-order two energy groups neutron diffusion model in spherical reactors is resolved using the Laplace transform method by regarding the spherical radius r as a time domain. Next, we transform the neutron diffusion model into fractional-order versions using the Caputo differentiator, resulting in what is referred to as the fractional-order two-energy-group neutron diffusion model. To address this fractional-order system, we introduce a novel approach to reduce a system of 2a -order to a duplicated system of a -order, where 0 <α 1. This converted system is then solved using one of the recent modifications of the fractional Euler method called the Modified Fractional Euler Method (MFEM). Several numerical simulations are depicted to verify our findings. [ABSTRACT FROM AUTHOR]

This paper aims to propose a numerical approach to deal with systems of fractional stochastic differential equations. This approach, which is regarded as a modification of the fractional Euler method (FEM), is called the modified fractional Euler method (MFEM). A susceptible infected recovered model (SIR model) is taken into consideration as an example of a system of fractional stochastic differential equations. Such a model is solved with the use of the MFEM in its deterministic and stochastic cases for the purpose of comparing the fulfilled results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel fractional numerical modification of the 5-point classical central formula; called the modified 5-point fractional formula for approximating the first fractional-order derivative in the sense of the Caputo operator. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability. [ABSTRACT FROM AUTHOR]

In this paper, we recall some basic facts and preliminaries related to fractional calculus by making a literature review for its definitions and properties. This would provide sufficient knowledge about this important topic. Several lemmas and theorems are shown in detail for completeness. [ABSTRACT FROM AUTHOR]

A study of a class of nonlinear differential equations involving the φ -Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results. A study of a class of nonlinear differential equations involving the -Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims first to recall the generalized Mittag-Leffler function and propose several properties of the generalized matrix Mittag-Leffler function. Afterward, we set a definition for a further extension of the generalized matrix Mittag-Leffler function and then show that this function is absolutely convergent under a certain condition. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate certain type superlinear nonlocal problems that are a class of parabolic equations with the second-type integral condition. We use the Fadeo-Galarkin method to establish the existence of the weak solution and we prove the uniqueness of this solution for the problem by using an a priori estimate. In addition, we study the theoretical blow-up solution and perform several numerical simulations of finite-time blow-up of a particular example of the main problem. [ABSTRACT FROM AUTHOR]