Variable-order Fractional Diffusion Equation Research Articles

On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

<abstract><p>In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.</p></abstract>

A second-order numerical method for nonlinear variable-order fractional diffusion equation with time delay

In this paper, a linearized numerical scheme of nonlinear variable-order fractional diffusion equation with time delay is constructed. We apply the L2−1σ formula to discretize the temporal derivative and second-order central difference scheme to discretize the spatial derivative. The proposed method is unconditionally stable and convergent with Oτ2+h2, where τ and h are the time and space steps, respectively. Numerical experiment demonstrates the effectiveness and accuracy of the numerical scheme.

Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations

Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations

A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative term

Using the meshless collocation method, a scheme for solving nonlinear variable-order fractional diffusion equations with a fourth-order derivative term is presented. Here the fractional derivative term is approximated by weighted and shifted Grünwald difference (WSGD) approximation formula. The difficulty caused by the nonlinear terms is carefully handled by quasilinearization technique. Using the radial basis functions, the solution of the problem is written in terms of the primary approximation, and the related correcting functions at each time step. Then the approximation is substituted back to the governing equations where the unknown parameters can be determined. Finally, the method is supported by several numerical experiments on irregular domains.

Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework

In this paper, we first study some properties of the variable-order fractional derivative defined by the Caputo fractional derivative and particularly present a Leibniz-type rule, which makes the variable-order fractional derivative to be expressed as an infinite sum of integer-order derivatives. Then we use such properties to build a Lie symmetry framework for a class of scalar variable-order fractional partial differential equations and show that such type of equations has an elegant symmetry structure, which facilitates us to explore the symmetry properties. By means of the Lie symmetry framework and the symmetry structure, we perform a Lie symmetry classification of the variable-order fractional diffusion equations and construct the corresponding symmetry reductions and certain exact solutions.

Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution.

In this paper, we formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells. We find its solution in the long-time limit describing non-uniform distribution due to a space-dependent fractional exponent. In the continuous space limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering of lysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependent fractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomes in living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equation validate our analytical solution. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

A second-order numerical method for space–time variable-order diffusion equation

In this paper, we propose a second-order finite difference scheme to study a class of space–time variable-order fractional diffusion equation, and show that the scheme is not only unconditionally stable but also convergent with the convergence order O(τ2+h2) under certain conditions. Some numerical examples are illustrated which are in good agreement with our theoretical results.

The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for space Riesz variable-order fractional diffusion equations. Based on weighted-shifted-Grunwald-difference (WSGD) operators proposed in Lin and Liu (J. Comput. Appl. Math. 363, 77–91 (2020)) for Riemann-Liouville fractional derivatives, we derive WSGD operators for variable-order ones by using the relation between variable-order fractional derivative and (constant-order) fractional derivative. We then apply Crank-Nicolson-weighted-shifted-Grunwald-difference (CN-WSGD) schemes to the initial-boundary problem for space Riesz variable-order diffusion equations. Theoretical results on the stability and convergence of CN-WSGD schemes are presented and proved. Moreover, we derive a problem-based method to choose suitable CN-WSGD schemes, which leads to unconditioned stable linear systems with optimal upper bound for accuracy. Numerical results show that the proposed schemes are very efficient.

Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation.

We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed by Chechkin, Gorenflo, and Sokolov [J. Phys. A, 38, L679 (2005)JPHAC50305-447010.1088/0305-4470/38/42/L03]. We identify a new advection term that causes ultraslow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms in the long-time limit. This uncovers the anomalous mechanism by which nonuniform distributions can occur. We perform MonteCarlo simulations of the underlying anomalous random walk and find excellent agreement with the asymptotic solution.

Finite difference scheme for multi-term variable-order fractional diffusion equation

In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order alpha(x,t) in(0,1) and the Riesz variable-order space fractional derivatives of order beta(x,t) in (0,1), gamma(x,t)in(1,2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.

Fractional and fractal derivative models for transient anomalous diffusion: Model comparison

Transient anomalous diffusion characterized by transition between diffusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian diffusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional diffusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), breakthrough curves (exhibiting nuance in diffusive state transition), and possible hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures transient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to capture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications.

Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications

Abstract In this paper, we deal with maximum principles for multi-term space-time variable-order Riesz-Caputo fractional differential equations (MT-STVO-RCFDEs, for short). We firstly derive several important inequalities for variable-order fractional derivatives at extreme points. Based on these inequalities, we obtain the maximum principles. Finally, these principles are employed to show that the uniqueness of solutions of the (MT-STVO-RCFDEs) and continuous dependance of solutions on initial-boundary value conditions.

Discrete Fractional Diffusion Equation of Chaotic Order

Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

The inherent heterogeneities of many geophysical systems often give rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with space–time dependent variable coefficients. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcy’s law with variable order Riemann–Liouville fractional derivatives, this leads to a new variable-order fractional diffusion equation with variable coefficients.In this paper we consider a new two-dimensional variable-order fractional percolation equation with variable coefficients. An alternating direct method for the two-dimensional variable-order fractional percolation equation is proposed. Stability and convergence of the implicit alternating direct method are discussed. Finally, some numerical results are given. The numerical results demonstrate the effectiveness of the methods. These techniques can be used to simulate three-dimensional variable-order fractional partial differential equations.

A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures

Abstract This study proposes a new variable-order fractional diffusion equation model to describe the coupled chloride diffusion-binding processes in reinforced concrete, in which the order of fractional derivative term is a variable function instead of a constant in the standard fractional model. The concentration influence coefficient k is introduced to capture the effect of concentration dependency on chloride transport due to the chloride binding behavior. The two parameters in the proposed model can be determined directly by a statistical analysis of measurement data. Four test cases illustrate that the proposed variable-order fractional derivative model agrees significantly better with experimental data than the most commonly used traditional model governed by the classical Fick’s second law, especially when a large concentration coefficient k is involved. That proposed model is also verified by accurately predicting chloride concentration profiles in a period of 200 days.

FINITE DIFFERENCE SCHEMES FOR VARIABLE-ORDER TIME FRACTIONAL DIFFUSION EQUATION

Variable-order fractional diffusion equation model is a recently developed and promising approach to characterize time-dependent or concentration-dependent anomalous diffusion, or diffusion process in inhomogeneous porous media. To further study the properties of variable-order time fractional subdiffusion equation models, the efficient numerical schemes are urgently needed. This paper investigates numerical schemes for variable-order time fractional diffusion equations in a finite domain. Three finite difference schemes including the explicit scheme, the implicit scheme and the Crank–Nicholson scheme are studied. Stability conditions for these three schemes are provided and proved via the Fourier method, rigorous convergence analysis is also performed. Two numerical examples are offered to verify the theoretical analysis of the above three schemes and illustrate the effectiveness of suggested schemes. The numerical results illustrate that, the implicit scheme and the Crank–Nicholson scheme can achieve high accuracy compared with the explicit scheme, and the Crank–Nicholson scheme claims highest accuracy in most situations. Moreover, some properties of variable-order time fractional diffusion equation model are also shown by numerical simulations.