Time-fractional Diffusion-wave Equation Research Articles

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

In this study, we have proposed an efficient numerical algorithm based on third degree modified extended B-spline (EBS) functions for solving time-fractional diffusion wave equation with reaction and damping terms. The Caputo time-fractional derivative has been approximated by means of usual finite difference scheme and the modified EBS functions are used for spatial discretization. The stability analysis and derivation of theoretical convergence validates the authenticity and effectiveness of the proposed algorithm. The numerical experiments show that the computational outcomes are in line with the theoretical expectations. Moreover, the numerical results are proved to be better than other methods on the topic.

Superconvergence analysis of FEM for 2D multi-term time fractional diffusion-wave equations with variable coefficient

The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by linear triangular finite element method and in time by Crank-Nicolson scheme. The stability is firstly proved unconditionally. In the analysis of superclose properties, how to deal with the item for variable coefficient is the main difficulty. In order to do this, a new projection operator is defined and the relationship between the proposed projection operator and interpolation operator about linear triangular finite element is deduced. Consequently, the global superconvergence result is obtained by use of interpolation postprocessing technique. The numerical examples show that the proposed numerical method is highly accurate and computationally efficient.

Unconditionally convergent numerical method for the two-dimensional nonlinear time fractional diffusion-wave equation

In this paper, we develop a Crank-Nicolson Legendre spectral method for solving the two-dimensional nonlinear time fractional diffusion-wave equation in bounded rectangular domains. In terms of the error splitting argument technique, an optimal error estimate of the numerical scheme is obtained without any time-step size conditions, while the usual analysis for high dimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Some numerical results are given to justify the theoretical analysis.

Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

Determination of the initial data in a time-fractional diffusion-wave problem by a final time data

This paper is to determine an initial value for a time-fractional diffusion-wave equation from the final time data. The regularity of the weak solution for the direct problem is improved. The existence and uniqueness of a weak solution for the adjoint problem are proved by using the Fourier method. The Tikhonov regularization method is applied for finding a stable approximate solution. In order to get the minimizer of the Tikhonov regularization functional, we propose a conjugate gradient algorithm based on the strict deductions of the sensitive problem and the adjoint problem. Four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness and stability of the proposed algorithm.

A meshless local collocation method for time fractional diffusion wave equation

In this manuscript, we present a radial basis function based local collocation method for solving time fractional diffusion-wave equation. The advantage of the local collocation method is only the discretization points located in each sub-domain, surrounding the collocation point, need to be considered. It also avoids the ill-conditioning problem arises due to the use of global collocation method. The theoretical stability and convergence of the proposed time semi-discrete scheme are also discussed. Numerical experiments for one dimension and two dimension problems are carried out. It is shown that the present method is efficient and numerical results have nice agreements with theoretical result.

Coupling of the improved singular boundary method and dual reciprocity method for multi-term time-fractional mixed diffusion-wave equations

The main purpose of this work is to present an impressive numerical scheme to solve two-dimensional multi-term time fractional mixed diffusion-wave differential equations (TFMDWE). The proposed method is based on the compact dual reciprocity method and the meshless improved singular boundary method (ISBM). The most significant privilege of the proposed method is mathematically simple, efficient, quite accurate which requires relatively less computational cost. Singular boundary method is one of truly meshless methods and it does not need any element or mesh for both field interpolation and background integration. For the considered problems, time derivatives are approximated via the time stepping method. Several numerical results are provided to show the accuracy of the present method for multi-term time-fractional mixed diffusion-wave equations.

A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives, the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries. The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.

Inversion of the Initial Value for a Time-Fractional Diffusion-Wave Equation by Boundary Data

Abstract In this study, we consider an inverse problem of recovering the initial value for a multi-dimensional time-fractional diffusion-wave equation. By using some additional boundary measured data, the uniqueness of the inverse initial value problem is proven by the Laplace transformation and the analytic continuation technique. The inverse problem is formulated to solve a Tikhonov-type optimization problem by using a finite-dimensional approximation. We test four numerical examples in one-dimensional and two-dimensional cases for verifying the effectiveness of the proposed algorithm.

RECOVERING A SPACE-DEPENDENT SOURCE TERM IN A TIME-FRACTIONAL DIFFUSION WAVE EQUATION

This work is concerned with identifying a space-dependent source function from noisy final time measured data in a time-fractional diffusion wave equation by a variational regularization approach. We provide a regularity of direct problem as well as the existence and uniqueness of adjoint problem. The uniqueness of the inverse source problem is discussed. Using the Tikhonov regularization method, the inverse source problem is formulated into a variational problem and a conjugate gradient algorithm is proposed to solve it. The efficiency and robust of the proposed method are supported by some numerical experiments.

Generalized diffusion-wave equation with memory kernel

We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.

Hygrothermoelastic response of a hollow cylinder based on a coupled time-fractional heat and moisture transfer model

In this paper, the time-fractional diffusion-wave equations are introduced to describe the coupled heat conduction and moisture diffusion in a hollow pipe. The outer surface of the hollow pipe is assumed to be insulated and damp-proof, while the inner surface is subjected to thermal and moisture shocks. Using the method of separation of variables and the Laplace transform, analytic expressions for transient hygrothermoelastic response of temperature change, moisture distribution, displacement and stress components in the hollow cylinder are obtained in terms of Mittag–Leffler functions, respectively. Numerical results are calculated, and the effects of fractional order on the dynamic response of temperature, moisture and hygrothermal stresses are illustrated graphically. Sub-diffusive and super-diffusive transport coupling behaviors as well as wave-like behaviors are shown. The results in the case of the fractional order $$\alpha =1$$ reduce to those for the classical Fourier heat conduction and the Fick’s moisture diffusion coupling model.

A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations

This study presents a semi-analytical boundary-only collocation technique for solving multi-term time-fractional diffusion-wave equations. In the present collocation technique, the Laplace transformation is first implemented to convert time-fractional diffusion-wave equation to a series of time-independent nonhomogeneous equations in Laplace domain. Then the composite multiple reciprocity method (CMRM) is applied to construct a high-order homogeneous equation, which has the same solution with one of time-independent nonhomogeneous equations in Laplace domain. The collocation Trefftz scheme with high-order T-complete functions is used to obtain the semi-analytical solution of high-order homogeneous equation with boundary-only collocation in Laplace domain. Finally the numerical Laplace inversion scheme is introduced to invert the Laplace domain solutions back into the time-dependent solutions of time-fractional diffusion-wave equations. The proposed method is easy-to-implement and flexible for irregular domain problems. It evades costly convolution integral calculation in time fractional derivation approximation, and avoids the effect of time step on numerical accuracy and stability. Error analysis and numerical investigations show that the proposed collocation method is highly accurate, computationally efficient, and numerically stable for multi-term time fractional diffusion-wave equations.

Reconstruction of a time-dependent source term in a time-fractional diffusion-wave equation

ABSTRACTThis paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is assumed as a multiplication of a temporal function and a spatial function. We try to identify the time-dependent source term according to an additional solute concentration distribution measured at a point on the boundary or an inner point in the solution domain. Firstly, we discuss the continuity of the weak solution for the direct problem. Then, we transform the inverse problem into a first kind of Volterra integral equation and show its ill-posedness. We use a generalized Tikhonov regularization to solve the Volterra integral equation. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Lastly, we test three examples for the inverse time-dependent source problem and show the effectiveness of the proposed regularization method.

Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains

In this work, a novel two-dimensional (2D) multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains will be considered. Different from the general multi-term time-fractional diffusion-wave or sub-diffusion equation, the new equation not only possesses the diffusion-wave and sub-diffusion terms simultaneously but also has a special time-space coupled derivative. Although the analytic solution of this kind of equation can be derived using a multi-level sum of an infinite series and Fox H-functions, it is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. In this paper, we will consider the finite element method for the novel 2D multi-term time fractional mixed diffusion equation. Firstly, we utilise the mixed L schemes to approximate the time fractional sub-diffusion term, diffusion-wave term and the coupled time-space derivative, respectively. Secondly, we establish the variational formulation and use the finite element method to discretise the equation. Then we adopt linear polynomial basis functions on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence analysis of the numerical scheme. To show the effectiveness of our method, three examples are investigated, in which a 2D multi-term time fractional mixed diffusion equation on a circular domain and a 2D generalized Oldroyd-B fluid in a magnetic field are analysed.

New local discontinuous Galerkin method for a fractional time diffusion wave equation

ABSTRACTThis paper is devoted to the study of a new discontinuous finite element idea for the time fractional diffusion-wave equation defined in bounded domain. The time fractional derivatives are described in the Caputo's sense. By applying the sine transform on the time fractional diffusion-wave equation, we make the equation depend on time. Then we use definition of Caputo's derivative and by defining l-degree discontinuous finite element with interpolated coefficients we solve the mentioned equation. Error estimate, existence and uniqueness are proved. Finally, the theoretical results are tested by some numerical examples.

An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is O(N−r+τ2), where N,τ,r are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids.

Spatial High Accuracy Analysis of FEM for Two-dimensional Multi-term Time-fractional Diffusion-wave Equations

In this paper, high-order numerical analysis of finite element method (FEM) is presented for two-dimensional multi-term time-fractional diffusion-wave equation (TFDWE). First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H1−norm and temporal convergence in L2-norm with order $$O(h^{2}+\tau^{3-\alpha})$$ , where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.

A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation

We firstly generalize a multi-term time fractional diffusion-wave equation to the multi-term variable-order time fractional diffusion-wave equation (M-V-TFD-E) by the concept of variable-order fractional derivatives. Then we implement the Chebyshev wavelets (CWs) through the operational matrix method to approximate its solution in the unit square. In fact, we apply the operational matrix of variable-order fractional derivative (OMV-FD) of the CWs to derive the unknown solution. We proceed with coupling the collocation and tau methods to reduce M-V-TFD-E to a system of algebraic equations. The important privilege of method is handling different kinds of conditions, i.e., initial-boundary conditions and Dirichlet boundary conditions, by implementing the same techniques. The convergence and error estimation of the CWs expansion in two dimensions are theoretically investigated. We also examine the applicability and computational efficiency of the new scheme through the numerical experiments.