Multi-term Time-fractional Diffusion-wave Equations Research Articles

On the solution of multi-term time fractional diffusion-wave equation involving ultra-hyperbolic operator

A diffusion-wave equation with multi-term Hilfer fractional derivatives (HFDs) in time and ultra-hyperbolic operator (UHO) in space has been considered. Fundamental solution of the fractional diffusion-wave equation is obtained by using Laplace and Fourier transform with Mellin-Barnes integral representation. The solution obtained involved the Fox H-function. In addition, we provide some special cases of diffusion-wave equation.

An efficient spectral collocation method based on the generalized Laguerre polynomials to multi-term time fractional diffusion-wave equations

In this study, a spectral collocation method is proposed to solve a multi-term time fractional diffusion-wave equation. The solution is expanded by a series of generalized Laguerre polynomials, and then, by imposing the collocation nodes, the equation is reduced to a linear system of algebraic equations. The coefficients of the expansion can be determined by solving the resulting system. The convergence of the method is proved, and some numerical examples are presented to demonstrate the accuracy and efficiency of the scheme. Finally, conclusions are given.

An efficient alternating segment parallel finite difference method for multi-term time fractional diffusion-wave equation

The multi-term time fractional diffusion-wave equation is of important physical meaning and engineering application value. In order to meet the needs of fast solving multi-term time fractional diffusion-wave equation, an efficient difference algorithm with intrinsic parallelism is proposed in this paper. The alternating segment Crank–Nicolson (ASC-N) parallel difference scheme is constructed with four kinds of Saul’yev asymmetric schemes and the classical Crank–Nicolson (C–N) scheme, based on alternating segment technology. The theoretical analysis shows that the ASC-N scheme is second-order convergence in space and $$3-\alpha $$ order convergence in time.The computing efficiency of the ASC-N scheme can save about 80% for C–N scheme when the number of space grids is large. The theoretical analysis and numerical experiments show that the ASC-N method is effective for solving multi-term time fractional diffusion-wave equation.

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.

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.

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.

Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes

The paper mainly focuses on studying nonconforming quasi-Wilson finite element fully-discrete approximation for two dimensional (2D) multi-term time fractional diffusion-wave equation (TFDWE) on regular and anisotropic meshes. Firstly, based on the Crank–Nicolson scheme in conjunction with L1-approximation of the time Caputo derivative of order α ∈ (1, 2), a fully-discrete scheme for 2D multi-term TFDWE is established. And then, the approximation scheme is rigorously proved to be unconditionally stable via processing fractional derivative skillfully. Moreover, the superclose result in broken H1-norm is deduced by utilizing special properties of quasi-Wilson element. In the meantime, the global superconvergence in broken H1-norm is derived by means of interpolation postprocessing technique. Finally, some numerical results illustrate the correctness of theoretical analysis on both regular and anisotropic meshes.

Sixth-order non-uniform combined compact difference scheme for multi-term time fractional diffusion-wave equation

In this paper, we propose a high-order scheme for the numerical solution of multi-term time fractional diffusion-wave (FDW) equation in one and two-dimensional on non-uniform grids. Based on the sixth-order non-uniform combined compact difference (NCCD) scheme in the space directions on non-uniform grids, an alternating direction implicit (ADI) method is proposed to split the equation into two separate one dimensional equations. The multi-term time fractional derivation is described in the Caputo's sense with scheme of order O(τ3−α), 1<α<2. A numerical analysis of Fourier analysis completed by stability calculations in terms of semi-discrete eigenvalue problems are proposed. The advantage of the non-uniform combined compact difference (NCCD) scheme is that it can decrease the CPU time in comparison with the uniform combined compact difference (CCD) scheme. The sixth-order accuracy in the space directions on non-uniform grids has not been achieved in previous studies.

A wavelet approach for the multi-term time fractional diffusion-wave equation

ABSTRACTIn this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method.

A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation

In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O(ź3źź), 1<ź < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method.

Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O(τ3−α),1<α<2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L∞-convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes.

Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

AbstractSome efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an L1 approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited.In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.