Variable-order Time-fractional Diffusion Equations Research Articles

A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.

Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: regularity analysis and uniqueness of determining variable order

Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: regularity analysis and uniqueness of determining variable order

Coefficient inverse problem for variable order time-fractional diffusion equations from distributed data

Coefficient inverse problem for variable order time-fractional diffusion equations from distributed data

Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation

This paper presents an effective numerical method for multi-term variable-order time fractional diffusion equations with the variable-order fractional derivative. The local discontinuous Galerkin method and the finite difference method are used in the spatial and temporal directions, respectively. We prove that the scheme is unconditional stable and convergent with O(hs+1+(Δt)2−r), where r=max{ɛ(t)}. s, h, Δt are the degree of piecewise polynomials, the space step sizes, and the time step sizes, respectively. Some numerical experiments are used to illustrate the effectiveness and applicability of the scheme.

Discretization and Analysis of an Optimal Control of a Variable-Order Time-Fractional Diffusion Equation with Pointwise Constraints

Discretization and Analysis of an Optimal Control of a Variable-Order Time-Fractional Diffusion Equation with Pointwise Constraints

Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations

Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations

Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order

Variable-order time-fractional diffusion equations (tFDEs), in which the variable fractional order accommodates the fractal dimension change of the surrounding medium via the Hurst index, provide a competitive instrument to describe anomalously diffusive transport of particles through deformable heterogeneous materials. We analyze the mapping properties of the fractional integral with a general time-dependent variable order, based on which we prove the well-posedness and smoothing properties of corresponding variable-order tFDE model in multiple space dimensions. We then derive and analyze a fully discretized finite element approximation, in which we develop a novel decomposition of L-1 discretization coefficients to prove an optimal-order error estimate of the numerical scheme based only on the regularity assumptions on the data. Numerical experiments are performed to substantiate the theoretical findings.

The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method

IntroductionThis study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure. ObjectivesThe present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically. MethodsThe variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for (0<α(t)<1). An efficient and accurate meshfree method based on the singular boundary method (SBM) and dual reciprocity method (DRM) in concomitant with finite difference scheme is proposed on three-dimensional arbitrary geometry. To discrete of the temporal term, the finite diffract method (FDM) is utilized. In the spatial variation domain; the proposal method is constructed two part. To evaluating first part, fundamental solution of (VO-TFDE) is transformed into inhomogeneous Helmholtz-type to implement the SBM approximation and other part the DRM is utilized to compute the particular solution. ResultsThe stability and convergent of the proposed method is numerically investigated on high dimensional domain. To verified the reliability and the accuracy of the present approach on complex geometry several examples are investigated. ConclusionsThe result of study provides a rapid and practical scheme to capture the behavior of diffusion process.

A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions

We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O(MNlog3⁡N) computational complexity and O(MNlog2⁡N) memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grunwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the $$L^2$$ -convergence of the scheme are proved for all variable-order $$\alpha (t)\in (0,1)$$ . The proposed method is of accuracy-order $$O(\tau ^3+h^{k+1})$$ , where $$\tau$$ , h, and k are the temporal step size, the spatial step size, and the degree of piecewise $$P^k$$ polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation

In this paper, an accurate numerical method is presented to solve a class of variable-order fractional diffusion problem. The problem first is discretized by a finite difference method in temporal direction, and then a local discontinuous Galerkin method in space. The stability and L2 convergence of the proposed scheme are derived for all variable-order α(t)∈(0,1). We prove that the scheme is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Some numerical experiments are provided to verify the theoretical analysis and high-accuracy of the proposed method.

Inhomogeneous parametric scaling and variable-order fractional diffusion equations

We discuss the derivation and the solutions of integrodifferential equations (variable-order time-fractional diffusion equations) following as continuous limits for lattice continuous time random walk schemes with power-law waiting-time probability density functions whose parameters are position-dependent. We concentrate on subdiffusive cases and discuss two situations as examples: A system consisting of two parts with different exponents of subdiffusion, and a system in which the subdiffusion exponent changes linearly from one end of the interval to another one. In both cases we compare the numerical solutions of generalized master equations describing the process on the lattice to the corresponding solutions of the continuous equations, which follow by exact solution of the corresponding equations in the Laplace domain with subsequent numerical inversion using the Gaver-Stehfest algorithm.

Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

Abstract We study a fully discretized finite element approximation to variable-order Caputo and Riemann–Liouville time-fractional diffusion equations (tFDEs) in multiple space dimensions, which model solute transport in heterogeneous porous media and related applications. We prove error estimates for the proposed methods, which are discretized on an equidistant or graded temporal partition predetermined by the behavior of the variable order at the initial time, only under the regularity assumptions of the variable order, coefficients and the source term but without any regularity assumption of the true solutions. Roughly, we prove that the finite element approximations to variable-order Caputo tFDEs have optimal-order convergence rates on a uniform temporal partition. In contrast the finite element approximations to variable-order Riemann–Liouville tFDEs discretized on a uniform temporal partition achieve an optimal-order convergence rate if $\alpha (0)=\alpha ^{\prime}(0) = 0$ but a suboptimal-order convergence rate if $\alpha (0)&amp;gt;0$. In the latter case, optimal-order convergence rate can be proved by employing the graded temporal partition. We conduct numerical experiments to investigate the performance of the numerical methods and to verify the mathematical analysis.

Wellposedness and smoothing properties of history-state-based variable-order time-fractional diffusion equations

We prove the wellposedness of history-state-based variable-order linear time-fractional diffusion equations in multiple space dimensions. We also prove that the regularity of their solutions depends on the behavior of the variable order at the initial time $$t=0$$, in addition to the usual smoothness assumptions. More precisely, we prove that their solutions have full regularity (i.e., the solutions can achieve high-order smoothness under high-order regularity assumptions of the data) as their integer-order analogs if the variable order has an integer limit at $$t=0$$ or exhibits singular behaviors at $$t=0$$ like in the case of the constant-order time-fractional diffusion equations if the variable order has a non-integer value at $$t=0$$.

Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

Variable-order time-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena including anomalously subdiffusive transport of solutes in heterogeneous porous media and memory effect as constant-order time-fractional diffusion equations do, while eliminating the nonphysical singularity of the solutions near the initial time of the latter. Moreover, variable-order time-fractional diffusion equations themselves occur in many applications.We study the initial-boundary value problem of variable-order time-fractional diffusion equations and prove the uniqueness of determining the variable order in the initial-boundary value problem, from the observations of its solution on a sufficiently small open spatial interval over a sufficiently small time interval.

Wellposedness and regularity of the variable-order time-fractional diffusion equations

It was proved that the solution to a time-fractional partial differential equation lacks regularity at the initial time t=0, which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t=0 is well captured by a time-fractional partial differential equation. We argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time-fractional partial differential equations and the locality of the classical initial condition at t=0. We show that the variable-order time-fractional partial differential equations, in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t=0, turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the (constant-order) time-fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa. We prove the wellposedness of a variable-order linear time-fractional diffusion partial differential equation in multiple space dimensions. We also prove that the regularity of its solution depends on the behavior of the variable order (and its derivatives) at time t=0, in addition to the usual smoothness assumptions. More precisely, we prove that its solution has full regularity like its integer-order analogue if the variable order has an integer limit at t=0 or exhibits singular behaviors at t=0 like in the case of the constant-order time-fractional partial differential equations if the variable order has a non-integer value at time t=0.

A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the basis functions of the approximate space. The inversion solutions with noisy data which give good approximations to the exact solution demonstrate effectiveness of the inversion algorithm for the simultaneous inversion problem.

变分数阶扩散方程微分阶数的数值反演

对于变分数阶扩散方程，给出一个隐式差分求解格式。进一步讨论由内点观测数据确定微分阶数的一个反问题，应用同伦正则化算法在不同参数取值条件下进行数值反演模拟。数值结果表明当微分阶数接近于1时，数值求解及其参数反演效果较好。 An implicit finite difference scheme is introduced to solve the variable-order time-fractional diffu-sion equation, and an inverse problem of determining the variable fractional order is set forth using the additional measurements at one interior point. The homotopy regularization algorithm is applied to solve the inverse problem, and numerical examples are presented. The computational and inversion results demonstrate that the variable order has important influence on the problem, and that the computations become effective when the variable order goes to 1.

Numerical Solution for the Variable Order TimeFractional Diffusion Equation with Bernstein Polynomials

Numerical Solution for the Variable Order TimeFractional Diffusion Equation with Bernstein Polynomials