Time-fractional Diffusion-wave Equation Research Articles

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

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.

A fast compact finite difference scheme for the fourth-order diffusion-wave equation

In this paper, the H 2 N 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When K 1 < 3 2 , the convergence order is O ( τ 3 − γ + h 4 + ϵ ) , where K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

Computational analysis of time fractional diffusion wave equation via novel B-splines-based technique

&lt;abstract&gt;&lt;p&gt;In this paper, the authors propose and utilize a novel B-splines-based technique to solve time fractional diffusion wave equations numerically. The present approach employs the arbitrary-order fractional derivative in conjunction with the $ \theta $-weighted scheme. The integration of the Atangana-Baleanu fractional derivative into time fractional diffusion wave equations represents a novel advancement within B-spline methodologies. Furthermore, the stability and convergence of the presented schemes are given. The suggested approach not only exhibits unconditional stability, it also attains second-order convergence in both temporal and spatial dimensions. We conclude by applying our theoretical results to certain numerical examples to demonstrate their efficiency and accuracy.&lt;/p&gt;&lt;/abstract&gt;

The quasi-boundary regularization method for identifying the source term of time-fractional diffusion wave equation

The quasi-boundary regularization method for identifying the source term of time-fractional diffusion wave equation

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

Symmetric fractional order reduction method with L1 scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

Symmetric fractional order reduction method with L1 scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

Jacobi polynomials for the numerical solution of multi-dimensional stochastic multi-order time fractional diffusion-wave equations

In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

The quasi-reversibility regularization method for backward problems of the time-fractional diffusion-wave equation

This paper is devoted to two backward problems in a time-fractional diffusion-wave equation, with the aid of extra measurement data at a final time. Since these two problems are ill-posed, a quasi-reversibility method is obtained by employing eigenfunction expansion, and the existence, uniqueness and regularity of the regularized solutions are also proven. We can get order-optimal error estimates from the a-priori parameter choice rule. Finally, several numerical examples of both one- and two-dimension show that the proposed regularization method is effective and stable.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions

We present a linear and nonlinear comprehensive class of variable-order (VO) time-fractional partial differential equations in multi-dimensions with Caputo VO fractional operator. This class comprises, as special cases, several linear and nonlinear equations of important applications in diverse fields, such as, the VO time-fractional telegraph equation, the VO time-fractional diffusion-wave equation and the VO time-fractional Klein–Gordon equation. A novel and accurate spectral tau method based on the shifted Gegenbauer polynomials (SGPs) is presented for the numerical treatment of the aforementioned class of equations. Actually, applying the tau method for solving this class of equations is very difficult, especially in the presence of the nonlinear term of these equations and the nature of the VO derivatives. To overcome this difficulty, new operational matrices for the VO fractional derivative and the approximation of the multiplication of the space–time Kronecker product vectors in multi-dimension are derived. These new matrices have the ability to remove the aforementioned difficulties that facing for applying the tau method. Several linear and nonlinear numerical examples are examined and compared in multi-dimensions with other methods to verify the validity, applicability and accuracy of the proposed methodology.

Unconditional convergence analysis of two linearized Galerkin FEMs for the nonlinear time-fractional diffusion-wave equation

In this paper, we present and analyze two linearized Galerkin finite element schemes, which are constructed by employing the H2N2 formula and its fast version in time direction, for solving the nonlinear time-fractional diffusion-wave equation. By utilizing mathematical induction, the optimal error estimates in H1-norm are derived without any ratio restrictions between the time step size τ and the space mesh size h. The key point in our argument is the application of Sobolev’s embedding inequality to the fully discrete solution uhn. On the other hand, additional time-discrete elliptic system and the inverse inequality, which play a vital role in the temporal–spatial error splitting technique, are avoided in our numerical analysis. Finally, two numerical experiments are given to demonstrate the theoretical findings.

Local error estimates of the fourth-order compact difference scheme for a time-fractional diffusion-wave equation

This paper considers the numerical approximation for the time-fractional diffusion-wave equation with the order α∈(1,2), whose solution behaves a weak singularity at t=0. To construct the high-order scheme, the intermediate variable w=Dtα/2(u−tϕ˜) is introduced, then we can rewrite the original problem as a coupled system, where u is the solution of the time-fractional diffusion-wave equation and ϕ˜=ut(x,0). By using the L1 scheme on graded meshes in temporal direction and the compact difference scheme in spatial direction, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability and the local error estimate of the proposed scheme is given. Finally, numerical examples are presented to verify the accuracy and efficiency of the proposed method.

Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value problem, such as the existence, uniqueness and regularity estimates. Moreover, we apply the obtained theories to an inverse source problem. The uniqueness of this inverse problem under the boundary measurements is proved.

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

Identify the Robin coefficient in an inhomogeneous time-fractional diffusion-wave equation

This paper is devoted to identifying the Robin coefficient in a one-dimensional inhomogeneous time-fractional diffusion-wave equation from the viewpoint of Bayesian inference approach. The priori modeling is implemented by a Markov Random Field (MRF). A hierarchical Bayesian model is adopted to select regularization parameters automatically. The Markov chain Monte Carlo (MCMC) algorithms are conducted to explore the posterior state space. Three numerical examples are carried out to demonstrate the performance of the proposed method.