Time-fractional Wave Equation Research Articles

Initial boundary value problem for a time fractional wave equation on a metric graph

Initial boundary value problem for a time fractional wave equation on a metric graph

Numerical solution of distributed-order time-fractional diffusion-wave equations using Laplace transforms

In this paper, we consider the numerical inverse Laplace transform for distributed order time-fractional equations, where a discontinuous Galerkin scheme is used to discretize the problem in space. The success of Talbot’s approach for the computation of the inverse Laplace transform depends critically on the problem’s spectral properties and we present a method to numerically enclose the spectrum and compute resolvent estimates independent of the problem size. The new results are applied to time-fractional wave and diffusion-wave equations of distributed order.

Distributed-order time-fractional wave equations

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the distributed-order constitutive law. In this work we propose to take positive Radon measures for the constitutive "functions". First, we derive a thermodynamical restriction on the constitutive measures which is easy to check, and therefore suitable for applications. Then we prove that the setting with measures in combination with the derived thermodynamical restriction guarantee existence and uniqueness of solutions for the distributed-order fractional wave equation. We further discuss the support and regularity of the fundamental solution, and conclude with a discussion on wave velocities.

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and Lp−Lq estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results.

An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations

An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis function, where N can be any positive even number. As has been shown, convergence order of these approximations can be N. The original fractional wave equation is transformed into a time Volterra-type integro-differential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform. Then, an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integro-differential equation. Eventually the nonlinear fractional wave equation can be readily and accurately solved. As examples, this method is applied to solve several fractional wave equations with various nonlinearities. Results show that the proposed method can successfully avoid difficulties in the treatment of singularity associated with fractional derivatives. Compared with other existing methods, this method not only has the advantage of high-order accuracy, but it also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution, which significantly reduces the storage and computation cost.

L3 approximation of Caputo derivative and its application to time-fractional wave equation-(I)

In this manuscript, first of all, we have developed two novel numerical approximations (called L3 and ML3 approximation) of the Caputo fractional derivative of order α∈(1,2). We have used the cubic interpolating polynomial on uniform grid points [(tj−2,Uj−2), (tj−1,Uj−1), (tj,Uj), (tj+1,Uj+1)] for 2≤j≤k−1 while the quadratic interpolating polynomial is applied on the first interval [t0,t1]. We have modified the L3 approximation by using cubic Hermite interpolation in the sub-interval [t0,t2]. The novel L3 and ML3, both approximation are second order accurate for all α. Both approximations are tested on various examples and gives highly accurate results. Later, using this L3 approximation, a difference scheme is proposed to solve the time-fractional wave equation (TFWE). The proposed difference scheme is second order accurate in space and time for all α. The scheme is again tested on three numerical problems of TFWE, and the comparative study of the numerical results by the proposed scheme with some existing schemes is also provided to show the effectiveness and accuracy of our scheme.

A novel spectral technique for 2D fractional telegraph equation models with spatial variable coefficients

In this paper, a class of the two-dimensional (2D) mathematical models of the time fractional telegraph equation with spatial variable coefficients is considered. These models include the 2D time fractional telegraph equation (2D-TFTE), the 2D time fractional diffusion wave equation with damping (2D-TFDWED) and the 2D time fractional wave equation (2D-TFWE). The fractional derivatives are described in the Caputo sense. A novel spectral technique is used to solve the aforementioned models. The technique is based on the operational matrices of the shifted Gegenbauer polynomials in conjunction with the spectral tau method. The product of operational matrices of the space vectors is constructed in a simple way to remove the obstacle that facing the use of the tau method in solving the variable coefficients fractional/integer order PDEs. Miscellaneous numerical experiments are offered to verify the accuracy and rapid rate of convergence of the presented numerical technique.

Space-dependent variable-order time-fractional wave equation: Existence and uniqueness of its weak solution

The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coefficients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional differential operator originates in the modelling of viscoelastic materials. The global in time existence of a unique weak solution to the model problem has been proved under appropriate conditions on the data. Rothe’s time discretization method is applied to achieve that purpose.

Initial-boundary value problems for multi-term time-fractional wave equations

In this paper, we deal with the well-posedness and the long-time behavior of the initial-boundary value problems for the multi-term Caputo time-fractional wave equations. First, by proving a new property concerned with the boundedness of the multivariate Mittag-Leffler functions, the unique existence of the weak solution is established. Also, we show that the solution depends on the parameters of the equation in a continuous way. In addition, the $$H^{2}$$ -decay rate of the solution is represented in terms of some orders of the fractional derivative.

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

This paper is a comprehensive study of Lp estimates for time fractional wave equations of order α∈(1,2) in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.

On a terminal value problem for stochastic space‐time fractional wave equations

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data and the spectrum of the fractional Laplacian operator (in a bounded domain , ). First, we show the existence and uniqueness of a mild solution in , for a suitable sub‐space of . A limitation of this result is the lack of time continuity at . Second, we study the inverse problem (IP) of recovering when the terminal value data and the source are given. We give an explanation why the time continuity of the solution at could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill‐posed, that is, recovery cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.

A second-order L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients

Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α∈(1,2) is proposed. This method is different from the known second-order methods, which is called the L2-1σ Crank-Nicolson method in this paper. Using the L2-1σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L2-norm and H1-norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α∈(1,2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L2-1σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes.

Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group

We first consider a nonlinear time fractional wave equation with a fractional damping posed on the Heisenberg group with Caputo fractional derivatives that interpolate the heat equation and the wave equation with the linear damping. It extends the Caputo–Wisner, the modified Szabo and the fractional Zener models. The non‐linearity accounts for a nonlinear medium. We present the Fujita exponent for blow‐up which sheds light on the admissible nonlinearity in practice. Then, we establish sufficient conditions ensuring non‐existence of local solutions. using the nonlinear capacity method. Furthermore, we extend the analysisto the case of the system to show the flexibility of the method used.

A Numerical Method for the Variable-Order Time-Fractional Wave Equations Based on the H2N2 Approximation

Aiming at the initial boundary value problem of variable-order time-fractional wave equations in one-dimensional space, a numerical method using second-order central difference in space and H2N2 approximation in time is proposed. A finite difference scheme with second-order accuracy in space and 3 − γ ∗ order accuracy in time is obtained. The stability and convergence of the scheme are further discussed by using the discrete energy analysis method. A numerical example shows the effectiveness of the results.

Temporal Second-Order Finite Difference Schemes for Variable-Order Time-Fractional Wave Equations

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 21 November 2019Accepted: 03 November 2020Published online: 05 January 2022Keywordsvariable-order time-fractional wave equation, stability, error estimate, finite difference, ADI schemeAMS Subject Headings65M06, 65M12, 65M15Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam

Phase Portraits and Traveling Wave Solutions of a Fractional Generalized Reaction Duffing Equation

In this paper, we study the traveling wave solutions of the fractional generalized reaction Duffing equation, which contains several nonlinear conformable time fractional wave equations. By the dynamic system method, the phase portraits of the fractional generalized reaction Duffing equation are given, and all possible exact traveling wave solutions of the equation are obtained.

The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $

<abstract><p>This paper is concerned with the semilinear time fractional wave equations on the whole Euclidean space, also known as the super-diffusive equations. Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $ L^2 $-solutions for linear and semilinear problems. The methods of this paper rely upon the relevant wave operators estimates, Sobolev embedding and fixed point arguments.</p></abstract>

On a backward problem for nonlinear time fractional wave equations

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces

Our purpose is to introduce a notion of weak solution for a class of abstract fractional differential equations. We point out that the time fractional derivative occurring in the equations is in the sense of the Caputo derivative. We prove existence results for weak and strong solutions. To justify the abstract theory we develop, we apply two examples of concrete equations: time-fractional wave equations and time-fractional Petrovsky systems. Both these concrete examples are of great interest in the theory of fractional partial differential equations.

The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders.