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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 aim to present a novel n-point composite fractional formula for approximating a Riemann–Liouville fractional integral operator. With the use of the definite fractional integral's definition coupled with the generalized Taylor's formula, a novel three-point central fractional formula is established for approximating a Riemann–Liouville fractional integrator. Such a new formula, which emerges clearly from the symmetrical aspects of the proposed numerical approach, is then further extended to formulate an n-point composite fractional formula for approximating the same operator. Several numerical examples are introduced to validate our findings. [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]

In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function's behavior according to different fractional-order values of α numerically. 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]