Space-time Fractional Diffusion Research Articles

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

The solution of the time-space fractional diffusion equation based on the Chebyshev collocation method

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Finite Difference Approximation for the Space-Time Fractional Linear Diffusion Equation Involving the Caputo-Hadamard Fractional Derivative

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

On the Solution of Mathematical Model Including Space-Time Fractional Diffusion Equation in Conformable Derivative, Via Weighted Inner Product

This research aims to accomplish an analytic solution to mathematical models involving space-time fractional differential equations in the conformable sense in series form through the weighted inner product and separation of variables method. The main advantage of this method is that various linear problems of any kind of differential equations can be solved by using this method. First, the corresponding eigenfunctions are established by solving the Sturm-Liouville eigenvalue problem. Secondly, the coefficients of the eigenfunctions are determined by employing weighted inner product and initial condition. Thirdly, the analytic solution to the problem is constructed in the series form. Finally, an illustrative example is presented to show how this method is implemented for fractional problems and exhibit its effectiveness and accuracy.

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Numerical solution of the space-time fractional diffusion equation based on fractional reproducing kernel collocation method

Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation

We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the solution does not depend continuously on the measured data. In this paper, the existence and uniqueness of the solution and the conditional stability for the inverse problem are given and proven. Based on the least squares technique, we construct a Galerkin regularization method to overcome the ill-posedness of the considered problem. Under a priori and a posteriori selection rules for the regularization parameter, the Hölder-type convergence results of optimal order for the proposed method are derived. Meanwhile, we verify the regularized effect of our method by carrying out some numerical experiments where the initial value function is a smooth function or a non-smooth one. Numerical results show that this method works well in dealing with the backward problem of the time-space fractional parabolic equation.

The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time–Space Fractional Diffusion Equation

In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time–space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.

On new aspects of Chebyshev polynomials for space-time fractional diffusion process

Abstract Chebyshev collocation scheme and Finite difference method plays central roles for solving fractional differential equations (FDE). Therefore purpose of this paper is to solve fractional mathematical problem of diffusion by Chebyshev collocation method which turns the original problems into the system of fractional ordinary differential and algebraic equations by imposing orthogonality property. This system is solved by implementing Finite difference method. The numerical illustrations confirm that the combination of these two methods allow us to establish one of the best truncated solution in the series form.

The non-uniform L1-type scheme coupling the finite volume method for the time–space fractional diffusion equation with variable coefficients

In this paper, we consider the time–space fractional diffusion equation on a bounded interval with the Caputo time fractional derivative and the two-sided Riemann–Liouville spatial fractional derivative. |The diffusion coefficients are variable with respect to x and t. The temporal non-uniform L1-type discretization and the finite volume method based on the nodal basis functions are adopted to discrete this fractional partial derivative equation. We strictly prove that our scheme is conditionally stable with convergence rate Oh2+N−min{2−α,rα}, where h is the spatial step and N,α,r are the temporal nodal numbers, the fractional parameter and the graded parameter respectively. Finally, some numerical results are presented to validate the theoretical analysis.

Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions

Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions

An Inverse Source Problem for A One-dimensional Time-Space Fractional Diffusion Equation

For the fractional diffusion equation, it has received extensive study from various angles on the inverse problem. This paper is dedicated to determining a source term with time-dependence of the time-space fractional diffusion equation with additional observation data. First, the implicit difference scheme and the matrix transfer technique are used to solve an initial boundary value direct problem, in which the time-space fractional diffusion equation for the homogeneous Dirichlet boundary condition is considered. For the conclusive solution of the given inverse source problem, a numerical method is proposed based on the optimal perturbation algorithm with optimized Tikhonov regularization. The numerical examples illustrate that our proposed numerical method is highly effective and relatively stable.

Numerical Solution of Backward Problem of a Multi-term Time-space Fractional Diffusion Equation

There have been many studies from a variety of perspectives about the fractional diffusion equation for the backward problem. This work looks at the backward problem in a multi-term time-space fractional diffusion equation (TSFDE), which uses the observational final data. This paper proposes a numerical solution based on a Tikhonov regularization technique combined with an optimal perturbation algorithm (OPA) to take care of this poorly posed problem. An implicit finite difference technology is introduced by the matrix transfer technique as a numerical approach to solving the problem of the initial boundary value. It is both highly effective and relatively stable, as our suggested numerical method’s numerical results illustrate.

An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion

Abstract We study the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index H ∈ ( 0 , 1 ) {H\in(0,1)} . With the aid of a novel estimate, by using the operator approach we propose regularity analyses for the direct problem. Then we provide a reconstruction scheme for the source terms f and g up to sign. Next, combining the properties of Mittag-Leffler function, the complete uniqueness and instability analyses are provided. It is worth mentioning that all the analyses are unified for H ∈ ( 0 , 1 ) {H\in(0,1)} .

Solution of the backward problem for the space-time fractional diffusion equation related to the release history of a groundwater contaminant

Abstract Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.

An [formula omitted]-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution

A fast algorithm is proposed for solving two-dimensional distributed-order time–space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss–Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be α-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz–Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.

Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations

<abstract><p>In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $ \delta t $ the recommended method converges at a rate of $ O(\delta t^2 + h^2) $. Based on graphic results and numerical examples, the application of the model is also examined.</p></abstract>

Numerical solution of Atangana–Baleanu–Caputo time-space fractional diffusion equation

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

On spectral polar fractional Laplacian

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer’s algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form.

Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession

This paper considers a two-dimensional tempered time–space fractional diffusion equation with a reaction term on convex domains. Firstly, the analytical solution to a one-dimensional tempered time–space diffusion equation is derived in terms of the Fox H function. However, for a two-dimensional counterpart, an explicit expression for its analytical solution does not seem tractable. This motivates us to resort to numerical methods. Next, the L1 formula on a graded mesh is modified to approximate the Caputo tempered time-fractional derivative. A fast evaluation for the tempered time-fractional operator is developed based on a sum-of-exponentials approximation, reducing the computational work and storage significantly. Furthermore, the Galerkin finite element method based on an unstructured mesh is utilised to solve the problem. Its stability and convergence are established. Finally, two numerical examples in different convex domains are investigated to demonstrate the effectiveness of the numerical method. As an application, the tempered fractional Bloch–Torrey equation retaining Larmor precession in a human brain-like domain is illustrated to observe the evolution of the transverse magnetisation. An interesting finding is that the tempered parameter has a significant impact on the decay of magnetisation.

Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities \(\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,\) involving time-fractional Caputo derivative \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of \(\gamma ,\mu ,m\) and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.