Order Time Fractional Diffusion Equation Research Articles

Unique recovery of the initial state of distributed order time fractional diffusion equation

We use Phragm\'en-Lindel\"of-Liouville argument to prove the uniqueness for the determining the initial state of solution for the time fractional diffusion equation with distributed order derivative. Several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

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.

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

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>

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

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 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.

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.

Reconstruction of a space-dependent source in the inexact order time-fractional diffusion equation

Abstract An inverse problem to recover a space-dependent factor of a source term in the inexact order time-fractional diffusion equation from final data is considered. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is since small errors in the input data cause large errors in the output solution. To overcome this instability we propose the stable approximation solution via a general modified quasi-boundary value regularization method. Order optimal convergence rates for the worst case error of the method are derived under the usual source condition by using an a-priori and an a-posteriori regularization parameter choice rules, respectively. Finally, several numerical examples are provided to 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.

High Spatial Accuracy Analysis of Linear Triangular Finite Element for Distributed Order Diffusion Equations

In this paper, an effective numerical fully discrete finite element scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known $L1$ formula approximation to the distributed order derivative and linear triangular finite element approach for the spatial discretization, we construct a fully discrete finite element scheme. Based on the superclose estimate between the interpolation operator and the Ritz projection operator and the interpolation post-processing technique, the superclose approximation of the finite element numerical solution and the global superconvergence are proved rigorously, respectively. Finally, a numerical example is presented to support the theoretical results.

A numerical method for distributed order time fractional diffusion equation with weakly singular solutions

A finite difference/spectral method is proposed for the numerical approximation of a distributed order time fractional diffusion equation with initial singularity on two dimensional spatial domain. The L1 scheme on graded mesh for the discretization of time Caputo fractional derivative is used to capture the weak initial singularity and Legendre spectral method is adopted for the spatial discretization. It is proved that with appropriate choice of the grading parameter, the scheme can attain order 2−β convergence in temporal direction, where β(0<β<1) is the upper bound of the order of distributed order fractional derivative. Numerical results confirm the sharpness of the error analysis.

Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation

We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the spatial data at an arbitrarily fixed time and we establish the conditional stability estimate of Hölder type in our inverse problem. Our method is based on the Bukhgeim-Klibanov method by means of the Carleman estimate. We also derive the Carleman estimate for the first and half order time fractional diffusion equation.

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.

An extension of the singular boundary method for solving two dimensional time fractional diffusion equations

In this paper, singular boundary method (SBM) in conjunction with the dual reciprocity method (DRM) is extended to the solution of constant and variable order time fractional diffusion equations (TFDEs). In this procedure, finite difference method breaks down the time domain and reduces the time fractional diffusion equation into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. Then SBM-DRM is applied to space semi-discretization of these types of equations, in a two step process. First, DRM, which is a popular meshless method based on radial basis functions (RBFs), is applied to obtain the particular solution. After evaluating the particular solution, singular boundary method can be employed to evaluate the homogeneous solution. To consider the accuracy and efficiency of the presented method, some benchmark problems subjected to the Dirichlet and Neumann boundaries are examined on regular and irregular geometries.

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.

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.

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

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

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient.