Distributed Order Time-fractional Diffusion Equations Research Articles

Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.

An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability

This article proposes a numerical method to find the numerical solutions of the time-fractional diffusion equations involving fractional distributed order operator of Caputo type. Using the finite difference approach, we solve these equations by applying the semi-discrete method regarding the time variable and the fully-discrete method regarding the spatial variable. For the distributed integral part with respect to time, the Gauss–Legendre quadrature formula is applied and to estimate the multi-term time-fractional operator, including the Caputo fractional derivative, the L2 -1 approach is utilized. In addition, the error analysis and stability of the proposed numerical method are studied in this work. Finally, some numerical examples are provided to demonstrate the accuracy and efficiency of the suggested method. These examples are compared to several numerical previous methods stated in the articles, and the results show that the accuracy of our method is superior to these methods.

Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

This article investigates the inverse problem of estimating the weight function using boundary observations in a distributed-order time-fractional diffusion equation. We propose a method based on L2 regularization to convert the inverse problem into a regularized minimization problem, and we solve it using the conjugate gradient algorithm. The minimization functional only needs the weight to have L2 regularity. We prove the weak closedness of the inverse operator, which ensures the existence, stability, and convergence of the regularized solution for the weight in L2(0,1). We propose a weak source condition for the weight in C[0,1] and, based on this, we prove the convergence rate for the regularized solution. In the conjugate gradient algorithm, we derive the gradient of the objective functional through the adjoint technique. The effectiveness of the proposed method and the convergence rate are demonstrated by two numerical examples in two dimensions.

A computational approach based on the fractional Euler functions and Chebyshev cardinal functions for distributed-order time fractional 2D diffusion equation

In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonality, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative operational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss–Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.

Identifying a space-dependent source term in distributed order time-fractional diffusion equations

<p style='text-indent:20px;'>The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.</p>

Distributed order fractional diffusion equation with fractional Laplacian in axisymmetric cylindrical configuration

In this paper, we consider the powers (fractional and natural) of the Laplacian operator in axisymmetric cylindrical geometry. We get the fundamental solution of the distributed order time-fractional diffusion equation with this type of fractional Laplacian and the fractional moment of solution is given. Our study is considered in the plane and half space cases and the roles of the Mittag-Leffler and Wright functions in the structures of solutions are also discussed.

The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed order

This article is devoted to obtain the numerical solution for a class of nonlinear two-dimensional distributed-order time-fractional diffusion equations. We discretize the problem by using a finite difference scheme in the time direction. Then, we solve the discretized nonlinear problem by a collocation approach based on the Legendre polynomials. The numerical algorithm is fully described and convergence analysis of the scheme is evaluated. Finally, few numerical implementations are presented to highlight the flexibility and the convergence rate of this method.

A fast linearized numerical method for nonlinear time-fractional diffusion equations

In this paper, we study a fast linearized numerical method for solving nonlinear time-fractional diffusion equations. A new weighted method is proposed to construct linearized approximation, which enables the unconditional convergence to be established when the nonlinearity f(u) is only locally Lipschitz continuous. In order to reduce the computational cost, the sum-of-exponentials (SOE) technique is employed to evaluate the kernel function in the Caputo derivative. By using the complementary discrete kernels for the coefficients of the refined fast weighted discretization, the proposed method is shown to be unconditionally convergent with respect to the discrete H1-norm. The fast linearized method can also be extended to nonlinear multi-term and distributed-order time-fractional diffusion equations. Numerical examples with different types of nonlinear functions are provided to demonstrate the behavior of proposed methods for both smooth and weakly singular solutions.

An inverse source problem for distributed order time-fractional diffusion equation

We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slow diffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability in suitable function space with a weak norm. By a variational identity connecting the unknown time-dependent source and the interior measurement data, the conjugate gradient method is also introduced to construct the inversion algorithm under the framework of regularizing scheme. We show the validity of the proposed scheme by several numerical examples.

Solving a backward problem for a distributed-order time fractional diffusion equation by a new adjoint technique

Abstract This paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series representation of the regularized solution, we give convergence rates under an a-priori and an a-posteriori regularization parameter choice rule. With a new adjoint technique to compute the gradient of the functional, the conjugate gradient method is applied to reconstruct the initial condition. Numerical examples in one- and two-dimensional cases illustrate the effectiveness of the proposed method.

Uniqueness in the inversion of distributed orders in ultraslow diffusion equations

This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Neumann) boundary condition. We first prove the wellposedness of the forward problem by means of eigenfunction expansion, which ensures that the weak solution has the classical derivatives. We next give a Harnack type inequality of the solution in the frequency domain under the Laplace transform. Finally we show a uniqueness result for the inverse problem in determining the weight function in the distributed order time derivative from one point observation.

Initial-boundary value problem for distributed order time-fractional diffusion equations

We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.

Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grunwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.

Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

In this work we present a new numerical method for the solution of the distributed order time-fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed.

Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem

This paper deals with an initial–boundary value problem for a distributed order time-fractional diffusion equation with the fractional derivative in the Caputo sense. The method of the eigenfunctions expansion in combination with the Laplace transform is first employed to prove the uniqueness and existence of the solution to the initial–boundary value problem and then to show its analyticity in time. As an application of the analyticity of the solution, a uniqueness result for an important inverse problem of determination of the weight function in the distributed order derivative contained in the time-fractional diffusion equation from one interior point observation of its solution is obtained.

Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Abstract This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1. Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.

Exact solutions to some models of distributed-order time fractional diffusion equations via the Fox H functions

In this article, three types of time-fractional diffusion equation of distributed order are introduced and some aspects of these equations are discussed. Using the appropriate joint integral transforms, fundamental solutions of these equations are obtained through the Fox H functions. The Mellin transform is an approach to change the fundamental solutions into the Fox H functions.

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).