Space Fractional Derivatives Research Articles

A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch–Torrey equation with fractional Laplacian

We investigate the suitability of a fractional Laplacian in the construction of a three-dimensional (3D) multi-term time–space fractional Bloch–Torrey equation. Instead of using the Riesz fractional derivative as in most previous research, we consider the fractional Laplacian in space and multi-term Caputo fractional derivative in time. A vertex-centred finite volume method is firstly derived to discretise this 3D fractional dynamic system in combination with the matrix transfer technique and the weighted and shifted Grünwald–Letnikov formula. Since computing the fractional power of the matrix representation m(−Δ) of the Laplacian operator results in a highly dense matrix, we use matrix function approximations developed in a Krylov subspace framework and exploit the sparsity of m(−Δ) to reduce the computational overhead. Then the Lanczos method is employed to approximate vector products of matrix functions. A preconditioner is applied to accelerate the convergence process, which shows excellent performance and reduces the memory considerably. In particular, we introduce block matrix functions to deal with the coupled system. Furthermore, the analytical solution of the two-term time–space fractional diffusion equation is derived and expressed by the multinomial Mittag-Leffler function. The feasibility and effectiveness of the proposed numerical techniques are verified by some numerical examples. Compared with two-dimensional fractional models or single-term time fractional dynamic systems, this multi-term time–space fractional model provides more flexibility for the characterisation of anomalous diffusion in biological tissues.

Numerical simulation of high-dimensional two-component reaction–diffusion systems with fractional derivatives

ABSTRACT Two-component reaction–diffusion models in high dimensions involving space-fractional derivatives are used as a powerful modelling approach for understanding several aspects of spatial heterogeneity and nonlocality. In this paper, we propose an accurate, unconditionally stable, and maximum principle preserving fourth-order method both in space and time variables to study the complex dynamical processes of two-component nonlinear space-fractional reaction–diffusion systems posed in high dimensions. To achieve a fast fourth-order accurate method, we adapt the matrix transfer technique with a fast Fourier transform-based implementation in space and an exponential integrator in time. The main advantage of the method is that it avoids storing the large dense matrix resulting from discretizing the fractional operator with the matrix transform approach and significantly reduces the computational costs. Some numerical experiments are carried out in two and three space dimensions to demonstrate the accuracy and computational efficiency of the method.

The similarity method and explicit solutions for the fractional space one-phase Stefan problems

In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space Stefan problem in terms of the three parametric Mittag-Leffler function \(E_{\alpha ,m,l}(z)\). We consider Dirichlet and Neumann conditions at the fixed face, involving Caputo fractional space derivatives of order \(0<\alpha <1\). We recover the solution for the classical one-phase Stefan problem when the order of the Caputo derivatives approaches one.

Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case

Anomalous diffusion with a power-law time dependence of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents α i . Here we consider the case when such a crossover occurs from a first regime with α 1 to a second regime with α 2 such that α 2 > α 1, i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann–Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green’s function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Lévy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.

Space-fractional heat transfer analysis of hybrid nanofluid along a permeable plate considering inclined magnetic field

In this study, the Caputo space-fractional derivatives of energy equation are used to model the heat transfer of hybrid nanofluid flow along a plate. The plate is considered permeable and affected by an inclined magnetic field. We use the space-fractional derivative of Fourier’s law to communicate between the nonlocal temperature gradient and heat flux. The hybrid nanofluid is formed by dispersing graphene oxide and silver nanoparticles in water. The new fractional integro-differential boundary layer equations are reduced to ordinary nonlinear equations utilizing suitable normalizations and solved via a novel semi-analytical approach, namely the optimized collocation method. The results reveal that the increment of the order of space-fractional derivatives and the magnetic inclination angle increase the Nusselt number. Also, an increase in the order of space-fractional derivatives leads to a thicker thermal boundary layer thickness resulting in a higher temperature. It is also found that the temperature of the fluid rises by changing the working fluid from pure water to single nanofluid and hybrid nanofluid, respectively. What is more, the proposed semi-analytical method will be beneficial to future research in fractional boundary layer problems.

An implicit wavelet collocation method for variable coefficients space fractional advection-diffusion equations

This paper studies a class of variable coefficients fractional advection-diffusion equations with two-sided space fractional derivatives in Caputo sense. By utilizing cubic semi-orthogonal B-spline wavelet collocation method (CSOBSWCM) in space combined with implicit Euler technique in time, an implicit wavelet collocation method is proposed to solve the fractional advection-diffusion equations in one and two dimensions. Next, we discuss a concrete realization of the novel method in which the operational matrix form of CSOBSWCM at each time step is provided. Moreover, we derive the unconditional stability and the convergence order of this numerical method. Finally, numerical examples are provided to verify the theoretical analysis of the present method.

Boundary stabilization for time-space fractional diffusion equation

This article is concerned with boundary stabilization for time-space fractional diffusion equation. The classic backstepping method is incapable of designing the boundary controller for such kind of problem due to the difficulty in calculating space fractional derivative of Volterra integral transformation. We overcome this difficulty by decomposing the system into two subsystems: one is a finite dimensional system that is unstable but is shown to be controllable, the other is an infinite dimensional system that is stable. By verifying the Kalman rank condition, we are able to propose a controller to stabilize the unstable finite dimensional system and prove that the proposed controller actually stabilizes the entire system.

Spatiotemporal chaos in diffusive systems with the Riesz fractional order operator

A reliable and efficient numerical technique for the approximation of Riesz fractional partial differential equations with chaotic and spatiotemporal chaos properties is developed in this work. In such a system, the standard anomalous diffusion terms are modeled using the Riesz fractional derivative which can be extended from one to high dimensional cases. The methodology adopts finite difference schemes, as well as the novel Fourier spectral methods for the approximation of the Riesz fractional derivatives in space. The resulting system of ordinary differential equations is advanced in time with the exponential time-differencing Runge–Kutta method. The performance and applicability of these methods were tested with some practical and real-life problems which are of current and recurring interests arising from computational physics, engineering and other areas of applied sciences. Simulation results are given for different instances of fractional parameter α in the intervals (0,1) and (1,2) which correspond to subdiffusion and superdiffusion scenarios, respectively.

Properties of the Katugampola Fractional Operators

Abstract In this work, there are considered higher order fractional operators defined in the sense of Katugampola. There are proved some fundamental properties of the Katugampola fractional operators of any arbitrary real order. Moreover, there are given conditions ensuring existence of the higher order Katugampola fractional derivative in space of the absolutely continuous functions.

Meshfree numerical integration for some challenging multi-term fractional order PDEs

&lt;abstract&gt;&lt;p&gt;Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.&lt;/p&gt;&lt;/abstract&gt;

An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 &lt; β 0 &lt; β 1 &lt; β 2 &lt; ⋯ &lt; β m &lt; 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

Space-time fractional porous media equation: Application on modeling of S&P500 price return.

We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of "fractionalization" are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized q-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that q-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)1063-651X10.1103/PhysRevE.99.062313]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.

A new formulation of finite difference and finite volume methods for solving a space fractional convection\u2013diffusion model with fewer error estimates

Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

Anomalous overland flow on hillslopes: A fractional kinematic wave model, its solutions and verification with data from laboratory observations

This paper analyses a fractional kinematic wave equation (fKWE) for overland flow and evaluates its solutions for applications using data from overland flow flumes with simulated rainfall in the laboratory. Solutions of fKWE presented have been derived for large time or when the Laplace transform variable s→0, which is one of the most important situations for overland flow. The solutions include expressions for the depth, velocity, and unit discharge, of overland flow. Fitting the approximate solution for the depth of overland flow to the data yields the values of the order of space-fractional derivatives, ρ, which is around ρ=1.5. It is found that ρ increases with the slope gradient and the rainfall intensity. The approximate order of ρ=1.5, which is about the average of order 1 for the advection equation and 2 for the diffusion equation, implies that diffusive mechanisms manifest in the overland flow. The findings mean that either an fKWE is used or a diffusion term is needed to account for dynamic forces in overland flow as the fKWE captures more physical mechanisms.

Finite difference method for solving fractional differential equations at irregular meshes

This paper presents a novel meshless technique for solving a class of fractional differential equations based on moving least squares and on the existence of a fractional Taylor series for Caputo derivatives. A “Generalized Finite Difference” approach is followed in order to derive a simple discretization of the space fractional derivatives. Consistency, stability and convergence of the method are proved. Several examples illustrating the accuracy of the method are given.

On a discrete model that dissipates the free energy of a time-space fractional generalized nonlinear parabolic equation

The present work is the first manuscript of the literature in which a numerical model that preserves the dissipation of free energy is proposed to solve a time-space fractional generalized nonlinear parabolic system. More precisely, we investigate an extension of the multidimensional heat equation with nonlinear reaction and fractional derivatives in space and time. The model considers homogeneous Dirichlet boundary conditions and initial data. The temporal fractional derivative is understood in the Caputo sense, and Riesz fractional derivatives are employed in the spatial variables. The system possesses a free energy functional and we prove mathematically that it is a decreasing function of time. Motivated by these facts, we propose a discretization of the space-time-fractional model which dissipates the discrete free energy. The discrete model is obtained using fractional-order centered differences to approximate the Riesz derivatives, and the L1 scheme to estimate the Caputo derivatives. The numerical model is a consistent approximation of the continuous system with quadratic order in space, and at most second order in time. Simulations confirm the capability of the numerical model to dissipate the free energy of the continuous fractional system.

Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation

In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

Fractional Ginzburg-Landau equations as generalizations of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is employed. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate.

The finite volume element method for the two-dimensional space-fractional convection\u2013diffusion equation

We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.