Space-time Fractional Diffusion Equation Research Articles

ON BLOW-UP FOR A TIME–SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATION WITH EXPONENTIAL KERNEL IN TEMPORAL DERIVATIVE

In this paper, we study the blow-up of the solution to a semilinear time–space fractional diffusion equation, where the time derivative is the Caputo–type derivative with the exponential kernel (“the exponential Caputo derivative” for brevity) of order \(\alpha \in (0,1)\) and the spatial derivative is the fractional Laplacian of order \(s\in (0, 1)\). We first define a mild solution of the considered equation in terms of convolution form, where the fundamental solutions are denoted by Fox H-functions. The local existence and uniqueness of a mild solution are further obtained by using a fixed point argument. Then, a weak solution is defined by the test function and it is proved to be a mild solution. Finally, the finite time blow-up is shown. Besides, the global existence of the solution is shown too where the critical index is determined.

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.

An efficient computational approach for the solution of time-space fractional diffusion equation

ABSTRACT The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson’s equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson’s equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method.

Error analysis of spectral approximation for space–time fractional optimal control problems with control and state constraints

In this paper, spectral discretization of optimal control problem governed by the space–time fractional diffusion equation with integral control and state constraints is investigated. The study of the optimality conditions is very important for analyzing the optimal control problem. Introducing the auxiliary equations and some important operators are to obtain a priori error estimates of spectral approximation for state, adjoint state and control variable rigorously. Additionally, reliable a posteriori error estimates are also proved for the optimal control problem carefully. The analysis results indicate that the errors decay exponentially when the data is smooth.

A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations

In this paper, a fast algorithm is proposed for solving distributed-order time-space fractional diffusion equations. Integral terms in time and space directions are discretized by the Gauss-Legendre quadrature formula. The Caputo fractional derivatives are approximated by the exponential-sum-approximation method, and the finite difference method is applied for spatial approximation. The coefficient matrix of the discretized linear system is symmetric positive definite and possesses block-Toeplitz-Toeplitz-block structure. The preconditioned conjugate gradient method with a block-circulant-circulant-block preconditioner is employed to solve the linear system. Theoretically, the stability and convergence of the proposed scheme are discussed. Numerical experiments are carried out to demonstrate the effectiveness of the scheme.

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.

A Tikhonov regularization method for solving a backward time–space fractional diffusion problem

In this paper, a backward problem for a time–space fractional diffusion equation is considered, which is to determine the initial data from a noisy final data. To deal with this ill-posed problem, combined the ideas of the Tikhonov regularization in Hilbert Schales proposed by Natterer in 1984 and the fractional Tikhonov method given by Hochstenbach–Reichel in 2011, a Tikhonov regularization method is constructed. Based on the Fourier transform, an a-priori and an a-posteriori regularization parameter choice rules are used to guarantee the order optimal convergence rates. By the finite difference methods and the Discrete Fourier transform, numerical examples in one-dimensional and two-dimensional cases are given for the a-posteriori regularization parameter choice rule. Theoretical and numerical results show that the proposed method works well for both the smooth and the non-smooth functions.

Subordination Formulae for Space-time Fractional Diffusion Processes via Mellin Convolution

Fundamental solutions of space-time fractional diffusion equations can be interpret as probability density functions. This fact creates a strong link with stochastic processes. Recasting probability density functions in terms of subordination laws has emerged to be important to built up stochastic processes. In particular, for diffusion processes, subordination can be understood as a diffusive process in space, which is called parent process, that depends on a parameter which is also random and depends on time, which is called directing process. Stochastic processes related to fractional diffusion are self-similar processes. The integral representation of the resulting probability density function for self-similar stochastic processes can be related to the convolution integral within the Mellin transform theory. Here, subordination formulae for space-time fractional diffusion are provided. In particular, a noteworthy new formula is derived in the diffusive symmetric case that is spatially driven by the Gaussian density. Future developments of the research on the basis of this new subordination law are discussed.

Two‐dimensional collocation method for generalized partial integro‐differential equations of fractional order with applications

In this paper, we present a new generalized model of partial integro‐differential equation of fractional order (PIDEFO). The new model of the PIDEFO is defined in terms of the operator presented recently. The operator takes form of the Riemann‐Liouville and Caputo derivatives in special case. The generalized model of the PIDEFO reduces to (1) linear space‐time fractional reaction‐diffusion equation (FRDE), (2) space‐time fractional diffusion equations (STFDE), and (3) time fractional telegraph equations (TFTE) in particular cases. The considered model is solved with collocation method using Jacobi polynomials. For theoretical accuracy, we prove that the Jacobi approximation converges to the exact solution uniformly. Two test examples are considered to verify the accuracy of the proposed method. Further, we present two applications of the proposed model and solve it using the presented method. The obtained numerical results are compared with the existing results. Finally, the numerical stability of the presented method is analyzed under the effect of the random noise in the force term.

Block splitting preconditioner for time-space fractional diffusion equations

<abstract><p>For solving a block lower triangular Toeplitz linear system arising from the time-space fractional diffusion equations more effectively, a single-parameter two-step split iterative method (TSS) is introduced, its convergence theory is established and the corresponding preconditioner is also presented. Theoretical analysis shows that the original coefficient matrix after preconditioned can be expressed as the sum of the identity matrix, a low-rank matrix, and a small norm matrix. Numerical experiments show that the preconditioner improve the calculation efficiency of the Krylov subspace iteration method.</p></abstract>

On a time-space fractional diffusion equation with a semilinear source of exponential type

<abstract><p>In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some $ L^p-L^q $-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in $ L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N) $, the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space.</p></abstract>

Numerical solution of the conformable fractional diffusion equation

In this paper, a numerical approach for solving space-time fractional diffusion equation with variable coefficients is proposed. The fractional derivatives are described in the conformable sense. The numerical approach is based on shifted Chebyshev polynomials of the second kind. The space-time fractional diffusion equation with variable coefficients is reduced to a system of ordinary differential equations by using the properties of Chebyshev polynomials. The finite difference method is applied to solve this system of equations. Numerical results are provided to verify the accuracy and efficiency of the proposed approach.

Cauchy problem for nonlocal diffusion equations modelling Lévy flights

In the present paper, we study the time-space fractional diffusion equation involving the Caputo differential operator and the fractional Laplacian. This equation describes the Lévy flight with the Brownian motion component and the drift component. First, the asymptotic behavior of the fundamental solution of the fractional diffusion equation is considered. Then, we use the fundamental solution to obtain the representation formula of solutions of the Cauchy problem. In the last, the L 2 -decay estimates for solutions are proved by employing the Fourier analysis technique.

The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative

<p style='text-indent:20px;'>In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.</p>

Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions: operational calculus approach

Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions: operational calculus approach

A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time–space fractional diffusion equation

In this analysis, a numerical algorithm in the RKHS approach is utilized to the inverse source problem for the diffusion equation in a time–space fractional sense, where determinations of state variable and source parameter subject to initial–boundary and overdetermination conditions are the main goal. Consequently, specifics theoretical demonstrations are presented to interpret the NPSs to such a fractional problem. In this direction, convergence analysis and error estimates of the developed approach are studied and analyzed as well. Concerning the considered equation, specific unidirectional physical experiments are given in a finite compact regime to confirm the theoretical aspects and to demonstrate the superiority of the utilized approach. Some representative results are presented in two-dimensional graphs, whilst dynamic behaviors of fractional parameters are reported for several fixed α,β values. From the practical viewpoint, the archived simulations and consequences justify that the iterative approach is a straightforward and appropriate tool with computational efficiency for numeric solutions of the inverse source problem.

A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grünwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O(ταh2+τ4), where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.

Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case

Numerous anomalous diffusion processes are characterized by crossovers of the scaling exponent in the mean squared displacement at some correlations time. The bi-fractional diffusion equation containing two time-fractional derivatives is a versatile mathematical tool describing specifically retarded subdiffusive transport, in which the scaling exponents acquires a smaller value, i.e., the diffusion becomes even slower after the crossover. We here derive closed-form multi-dimensional solutions for this integro-differential equation in n spatial dimensions by generalizing the classical Schneider-Wyss solution of the fractional diffusion equation with a single fractional derivative. In the two-dimensional case we develop a limiting approach based on the solution of the space-time fractional diffusion equation. The probabilistic interpretation in higher dimensions is discussed. The asymptotic long- and short-time behaviors are derived. It is shown that the solution of the bi-fractional diffusion equation can be interpreted in terms of the Fox H-transform of the Gaussian distribution.

A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

• Two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space are given. • Two types of inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at a final time is given. • A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. • An error estimates between the exact and it regularize solutions are obtained. In this research, we deal with two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The first one is to identify the source term and the second one is to identify the initial value along with the solution in both cases. These inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. Under the a-priori and the a-posteriori parameter choice rules, the error estimates between the exact and its regularized solutions are obtained. To illustrate the validity of our study, we give numerical examples. A final note is utilized in the ultimate section.

Inverse source problem for a space-time fractional diffusion equation

Inverse source problem for a space-time fractional diffusion equation