Space-time Fractional Diffusion Equation Research Articles

A fast solver for multidimensional time–space fractional diffusion equation with variable coefficients

In this paper, we study a discretization scheme and the corresponding fast solver for multi-dimensional time–space fractional diffusion equation with variable coefficients, in which L1 formula and shifted Grünwald formula are employed to discretize the temporal and spatial derivatives, respectively. A divide-and-conquer strategy is applied to the large linear system assembling discrete equations of all time levels, which in turn requires to solve a series of multidimensional linear systems related to the spatial discretization. Preconditioned generalized minimal residual method is employed to solve the spatial linear systems resulting from the spatial discretization. The discretization is proven to be unconditionally stable and convergent in the sense of infinity norm for general nonnegative coefficients. Numerical results are reported to show the efficiency of the proposed method.

On a time–space fractional backward diffusion problem with inexact orders

In this paper, we focus on the backward diffusion problem with the Caputo fractional derivative operator in time and a general spatial nonlocal operator. For T>0 and s∈[0,T), we consider the problem (Ps) of recovering the distribution u(x,s) from a measure of the final data u(x,T) for the following non-homogeneous time–space fractional diffusion equation Dtαu(x,t)+KβLγu(x,t)=f(x,t)inRn×(0,T)subject to the final condition u(x,T)=uT(x)inRn. The derivative orders and the nonlocal operator are perturbed with noises. Firstly, for 0<s<T, we prove the well-posedness of Problem (Ps) by studying the unique existence and continuity with respect to the derivative orders, the source term as well as the final value of the solution. Secondly, for s=0, we verify the ill-posedness of Problem (P0) and use the method of modified iterated Lavrentiev to construct a regularization solution from inexact data and inexact derivative orders. We apply a modified form of the discrepancy principle to choose regularization parameter and establish new optimal convergence estimates between the exact solution and its regularized approximation.

Subordination principle for space-time fractional evolution equations and some applications

ABSTRACTThe abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order and operator , , is considered, where generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a -semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation.

Well-posedness for a nonlocal nonlinear diffusion equation and applications to inverse problems

ABSTRACT This article investigates a space–time fractional diffusion equation with a nonlinear source and a space nonlocal operator. The well-posedness is demonstrated from using contraction mapping theorem and generalized Gronwall's inequality firstly. Based on the forward problem, we prove that a nonlinear source term and a fractional order of time derivative are uniquely determined by data which can be observed at one fixed spatial point.

From Power Laws to Fractional Diffusion Processes with and Without External Forces, the Non Direct Way

Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch–Torrey equations on irregular convex domains

Models based on partial differential equations containing time–space fractional derivatives have attracted considerable interest in the past decade because of their ability to model anomalous transport phenomena. These phenomena are strongly connected to the interactions within complex and non-homogeneous media exhibiting spatial heterogeneity. The class of equations with multi-term time–space derivatives of fractional orders has been found to be very useful in the description of such interactions. This motivates the extension of the classical Bloch–Torrey equation through the application of the operators of fractional calculus to new multi-term time–space fractional Bloch–Torrey equations with Riesz fractional operators.In this paper, we firstly propose an unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional diffusion equation with Riesz fractional operators on irregular convex domains. Secondly, we rigorously establish the stability and convergence of the numerical scheme. Thirdly, we extend the computational model to solve a system of coupled two-dimensional multi-term time–space fractional Bloch–Torrey equations. Finally, some numerical results are given to demonstrate the versatility and application of the models.

Difference numerical solutions for time-space fractional advection diffusion equation

In this paper, a time-space fractional advection diffusion equation is considered for the natural extension of the convection diffusion equation. An explicit difference scheme and an implicit difference scheme are presented. The stability and convergence of the two difference schemes are discussed. It is shown that the explicit difference scheme is conditionally stable and convergent, and the implicit difference scheme is unconditionally stable and convergent. The convergence order of the two methods is O (tau+h).

Describe NMR relaxation by anomalous rotational or translational diffusion

This paper employs the general time-space fractional diffusion equation {0 < α, β ≤ 2} to derive a correlation time function for analyzing nuclear magnetic resonance (NMR) relaxation. Both the anomalous rotational and translational diffusion are treated. NMR relaxation time affected by various Hamilton interactions such as dipolar or quadrupolar couplings can be calculated from the obtained Mittag-Leffler type time correlation and their corresponded spectral density functions. Additionally, to verify the results, the theoretical expressions are applied to fit reported experimental data of NMR quadrupolar coupling relaxation of head-to-head poly(propylene) (hhPP) in a polymer blend. The fitting is excellent and more convenient than the fitting utilising the traditional modified Kohlrausch-Williams-Watts (KWW) formalism. Further, it is found that the temperature dependence behavior of the segmental dynamics in anomalous diffusion may obey a different Vogel-Tamman-Fulcher (VTF) expression. The paper proposes new, general formalisms for analyzing various NMR relaxation experiments in macromolecular systems.

Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments

In the Fourier heat conduction equation, when the flux definition is expressed as the product of a constant diffusivity and the temperature gradient, the characteristic length scale evolves as the square root of time. However, if we replace the 1st order transient and gradient terms in the Fourier equation with fractional derivatives and/or define a non-linear spatially dependent diffusivity, it is possible to generate an anomalous space-time scaling, i.e., a scaling where the time exponent differs from the expected value of 1/2. To compare and contrast the possible consequences of using fractional calculus along with a non-linear flux, we investigate a space-time fractional heat diffusion equation that involves a non-linear diffusivity. Following presentation of the governing non-linear fractional equation, we arrive at a space-time scaling that accounts for the combined anomalous contributions of memory (fractional derivative in time), non-locality (fractional derivative in space), and a non-linear diffusivity. We demonstrate how this scaling can manifest in a physical setting by considering the analytical solution of a non-linear fractional space-time diffusion equation, a limit case Stefan problem related to moisture infiltration into a porous media. A direct physically realizable simulation of this process shows how the anomalous space-time scaling is explicitly related to measures of both the memory and non-linearity in the system. Overall, the findings from this work clearly show how the definition of a non-linear diffusivity might contribute to anomalous diffusion behavior and suggests that, in modeling a particular observation, the roles of fractional derivatives and a suitably defined non-linear diffusivity are interchangeable.

Space–time finite element method for the multi-term time–space fractional diffusion equation on a two-dimensional domain

In this paper, we propose a space–time finite element method for the multi-term time–space fractional diffusion equation. First, some fractional derivative spaces are listed, some properties of the fractional derivatives and fractional derivative spaces are given, some definitions and properties of finite element spaces are introduced, and a space–time finite element fully discrete scheme for the considered problem is developed. Second, the existence, uniqueness and stability of the obtained numerical scheme are discussed. Third, under the hypothesis about singular behavior of exact solution near t=0, the convergence is investigated in detail based on the suitable graded time mesh. At last, some numerical tests are given to verify the rationality and effectiveness of our method.

Life span of blowing-up solutions to the Cauchy problem for a time–space fractional diffusion equation

In this paper, we study the blow-up, and global existence of solutions to the following time–space fractional diffusion problem 0CDtαu(x,t)+(−Δ)β∕2u(x,t)=0Jtγ|u|p−1u(x,t),x∈RN,t>0,u(x,0)=u0(x)∈C0(RN),where 0<α<1−γ, 0<γ<1, 0<β≤2, p>1, 0Jtγ denotes the left Riemann–Liouville fractional integral of order γ, 0CDtα is the Caputo fractional derivative of order α and (−Δ)β∕2 stands for the fractional Laplacian operator of order β∕2. We show that if p<1+β(α+γ)∕αN, then every nonnegative solution blows up in finite time, and if p≥1+β(α+γ)∕αN and ‖u0‖Lqc(RN) is sufficiently small, where qc=αN(p−1)∕β(α+γ), then the problem has global solutions. Finally, we give an upper bound estimate of the life span of blowing-up solutions.

Fractionally delineate the neuroprotective function of calbindin-D28k in Parkinson’s disease

Neuron is a fundamental unit of the brain, which is specialized to transmit information throughout the body through electrical and chemical signals. Calcium ([Formula: see text]) ions are known as second messengers which play important roles in the movement of the neurotransmitter. Calbindin-[Formula: see text] is a [Formula: see text] binding protein which is involved in regulation of intracellular [Formula: see text] ions and maintains [Formula: see text] homeostasis level, it also alters the cytosolic calcium concentration ([[Formula: see text]]) in nerve cells to keep the cell alive. Parkinson’s disease (PD) is a chronic progressive neurodegenerative brain disorder of the nervous system. Several regions of the brain indicate the hallmark of the PD. The symptoms of PD are plainly linked with the degeneration and death of dopamine neurons in the substantia nigra pars compacta located in midbrain which is accompanied by depletion in calbindin-[Formula: see text]. In the present paper, the neuroprotective role of calbindin-[Formula: see text] in the cytoplasmic [[Formula: see text]] distribution is studied. The elicitation in [[Formula: see text]] is due to the presence of low amount of calbindin-[Formula: see text] which can be portrayed and is a hallmark of PD. A one-dimensional space time fractional reaction diffusion equation is designed by keeping in mind the physiological condition taking place inside Parkinson’s brain. Computational results are performed in MATLAB.

Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains

In recent years, the fractional differential equations have attracted a lot of attention due to their interested characteristics. Meshfree methods are highly accurate and have been extensively explored in engineering and mechanics fields. However, there is few research to develop the reproducing kernel particle method (RKPM), one of the widely used meshfree approach, for fractional partial differential equations. In this work, we solve time-space fractional diffusion equations in 2D regular and irregular domains. The temporal Caputo fractional derivatives are discretized by the L1 finite difference scheme and the spatial Laplacian fractional derivatives are discretized by RKPM based on the matrix transfer method. Especially, the corrected weighted shifted Grünwald–Letnikov scheme is utilized for temporally non-smooth solutions. Numerical examples in rectangular, circular, sector and human brain-like irregular domains are given to assess the efficiency and accuracy of the proposed numerical scheme. The spatial Laplacian fractional derivatives discretized by conventional finite difference method in the rectangular domain are also presented for comparison. The results indicate that RKPM is very effective for analyzing the considered fractional equations in various domains, which lays a concrete foundation for our further research of real application of human brain modeling.

Investigations on several high-order ADI methods for time-space fractional diffusion equation

The paper is devoted to the construction of high-precision unconditionally stable finite difference methods for solving time-space fractional diffusion equation with the Caputo fractional derivative (of order β, with β ∈ (0, 1)) in time and the Rimann-Liouville fractional derivatives (of order α, with α ∈ (1, 2]) in space. Two kinds of difference schemes with the approximation orders O(τ2−β + h3) and O(τ2 + h3) respectively are constructed. The stability and convergence are analyzed in detail. The obtained results are illustrated numerically by some examples, and a comparative study of several high-order schemes is also carried out.

Bayesian approach to a nonlinear inverse problem for a time-space fractional diffusion equation

Inverse problems for fractional differential equations have become a promising research area because of their wide applications in many scientific and engineering fields. In particular, the correct orders of fractional derivatives are hard to know as they are usually determined by experimental data and contain non-negligible uncertainty. Therefore, research on inverse problems involving the orders is necessary. Furthermore, problems involving the inversion of fractional orders are essentially nonlinear. Since classical methods may find it hard to provide satisfactory approximations and fail to capture the relevant uncertainty, a natural way to solve such inverse problems is through a Bayesian approach. In this paper, we consider an inverse problem of simultaneously recovering the source function and the orders of both time and space fractional derivatives for a time-space fractional diffusion equation. The problem will be formulated in the Bayesian framework, where the solution is the posterior distribution incorporating the prior information about the unknown and the noisy data. Under the considered infinite-dimensional function space setting, we prove that the corresponding Bayesian inverse problem is well-defined based on a proof of the continuity of the forward mapping. In addition, we also prove that the posterior distribution depends continuously on the data with respect to the Hellinger distance. Moreover, we adopt the iterative regularizing ensemble Kalman method to provide a numerical implementation of the considered inverse problem for the one-dimensional case. The numerical results shed light on the viability and efficiency of the method.

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

Recently, several problems in mathematics, physics, and engineering have been modeled via distributed-order fractional diffusion equations. In this paper, a new class of time distributed-order and space fractional diffusion equations with variable coefficients on bounded domains and Dirichlet boundary conditions is considered. By performing numerical integration we transform the time distributed-order fractional diffusion equations into multiterm time-space fractional diffusion equations. An implicit difference scheme for the multiterm time-space fractional diffusion equations is proposed along with a discussion about the unconditional stability and convergence. Then, the fast Krylov subspace methods with suitable circulant preconditioners are developed to solve the resultant linear system in light of their Toeplitz-like structures. The aforementioned methods are proved to acquire the capability to reduce the memory storage of the proposed implicit difference scheme from mathcal{O}(M^{2}) to mathcal {O}(M) and the computational cost from mathcal{O}(M^{3}) to mathcal{O}(Mlog M) during iteration procedures, where M is the number of grid nodes. Finally, numerical experiments are employed to support the theoretical findings and show the efficiency of the proposed methods.

INverse Source Problem for a Space-Time Fractional Diffusion Equation

Abstract For a space-time fractional diffusion equation, an inverse problem of determination of a space dependent source term along with the solution is considered. The fractional derivatives in time and space are defined in the sense of Caputo. Due to an over-specified data at final time say T, we proved that there exists a unique solution of the inverse source problem. We use the eigenfunction expansion method to prove our main results. Several special cases of space-time fractional diffusion equations are discussed and results are interpolated from generalized results. Some examples are provided.

Block preconditioning strategies for time–space fractional diffusion equations

We present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time–space fractional diffusion equations. In contrast to the traditional time-marching procedure, the discretization via finite difference is considered in a fully coupled time–space framework. The resulting fully coupled discretized linear system is a summation of two Kronecker products. The four preconditioning methods are based on block diagonal, banded block triangular and Kronecker product splittings of the coefficient matrix. All preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator. Numerical experiments show the efficiency of the four block preconditioners, and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one.

An inverse time-dependent source problem for a time–space fractional diffusion equation

This paper is devoted to identify a time-dependent source term in a time–space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point. The existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous Dirichlet boundary condition are proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Based on the separation of variables, we transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the fist kind. The generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time-dependent source term. Numerical experiments for six examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.

A fast second-order implicit scheme for non-linear time-space fractional diffusion equation with time delay and drift term

In this paper, a second-order accurate implicit scheme based on the L2–1σ formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time-space fractional diffusion equations with time drift term and non-linear delayed source function. The stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions, then we theoretically and numerically show that the proposed scheme converges in the L2-norm with the order O((Δt)2+h2) with time step Δt and mesh size h. Moreover, it finds that the discreted linear systems are symmetric Toeplitz systems. In order to solve these systems efficiently, the conjugate gradient method with suitable circulant preconditioners is designed. In each iterative step, the storage requirements and the computational complexity of the resulting equations are O(N) and O(NlogN) respectively, where N is the number of grid nodes. Numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners.