On an explicit finite difference method for fractional diffusion equations

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.Comment: 22 pages, 6 figure

Properties of the reaction front in a reaction-subdiffusion process

We study the reaction front for the process $A+B\to C$ in which the reagents move subdiffusively. We propose a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. Scaling solutions to these equations are presented and compared with those of a direct numerical integration of the equations. We find that for reactants whose mean square displacement varies sublinearly with time as $\sim t^\gamma$, the scaling behaviors of the reaction front can be recovered from those of the corresponding diffusive problem with the substitution $t\to t^\gamma$Comment: Errata corrected, one reference update

Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design

In the paper, we futher consider a fractional-order system from a modified Chua’s circuit system with the smooth degree of 3 proposed by Fu et al. Bifurcation analysis, multistability and coexisting attractors in the the fractional-order modified Chua’s circuit are studied. In addition, microcontroller-based circuit was implemented in real digital engineering applications by using the fractional-order Chua’s circuit with the piecewise-smooth continuous system

Stability and Convergence of Explicit Difference Method for Solving the 3-Dimensional Two-Sided Fractional Diffusion Equation

In this paper, a numerical solution of the 3-dimensional two-sided fractional diffusion equation has been presented. The algorithm for the numerical solution for this equation is based on explicit finite difference method. The consistency, conditional stability, and convergence of the fractional order numerical method are described. The numerical method has been applied to solve a practical numerical example and the results have been compared with exact solution. The results were presented in tables using the MathCAD 12 software package when it is needed. The explicit finite difference method appeared to be effective and reliable in solving the 3-dimensional two-sided fractional Diffusion equatio

Study of reactor constitutive model and analysis of nuclear reactor kinetics by fractional calculus approach

The diffusion theory model of neutron transport plays a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems. The neutrons are here characterized by a single energy or speed, and the model allows preliminary design estimates. The mathematical methods used to analyze such a model are the same as those applied in more sophisticated methods such as multi-group diffusion theory, and transport theory. The neutron diffusion and point kinetic equations are most vital models of nuclear engineering which are included to countless studies and applications under neutron dynamics. By the help of neutron diffusion concept, we understand the complex behavior of average neutron motion. The simplest group diffusion problems involve only, one group of neutrons, which for simplicity, are assumed to be all thermal neutrons. A more accurate procedure, particularly for thermal reactors, is to split the neutrons into two groups; in which case thermal neutrons are included in one group called the thermal or slow group and all the other are included in fast group. The neutrons within each group are lumped together and their diffusion, scattering, absorption and other interactions are described in terms of suitably average diffusion coefficients and cross-sections, which are collectively known as group constants. We have applied Variational Iteration Method and Modified Decomposition Method to obtain the analytical approximate solution of the Neutron Diffusion Equation with fixed source. The analytical methods like Homotopy Analysis Method and Adomian Decomposition Method have been used to obtain the analytical approximate solutions of neutron diffusion equation for both finite cylinders and bare hemisphere. In addition to these, the boundary conditions like zero flux as well as extrapolated boundary conditions are investigated. The explicit solution for critical radius and flux distributions are also calculated. The solution obtained in explicit form which is suitable for computer programming and other purposes such as analysis of flux distribution in a square critical reactor. The Homotopy Analysis Method is a very powerful and efficient technique which yields analytical solutions. With the help of this method we can solve many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. It does not require enough memory space in computer, free from rounding off errors and discretization of space variables. By using the excellence of these methods, we obtained the solutions which have been shown graphically