Time-fractional Wave Research Articles

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.

Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Abstract In order to explore the fractional differential equations in accounting informatization financial software, the author proposes a system for fractional diffusion wave equations and fractional differential equations, two numerical algorithms with higher precision are given, and the amount of computation is reduced at the same time. First, based on the equivalent integral form of the time fractional diffusion wave equation, using the fractional echelon method and the Crank-Nicolson method, for the time fractional diffusion wave equation, a finite difference scheme is designed, this format has second-order accuracy in both the temporal and spatial directions and is computationally stable. Numerical examples verify the accuracy and effectiveness of this format. Then when dealing with the initial value problem of fractional differential equations with Caputo derivative operator, convert it to the equivalent Voltera integral equation system, an initial approximate solution is obtained by a low-order method, derive the residual and error equations, the idea of applying the stepwise correction of spectral delay correction improves the numerical accuracy of the solution, at the same time, the Richard Askey integral equation is used to reduce the amount of calculation. At last, the high precision and effectiveness of the new method are verified by numerical experiments. Experiments show that: Starting from the equivalent integral form of the fractional diffusion wave equation, a second-order finite-difference scheme of the fractional-order diffusive wave equation is constructed, through numerical experiments, it is verified that the scheme has good accuracy and efficiency. In numerical solution, discrete integral equations have better numerical stability than differential equations, therefore, the format also has better stability. When taking different fractional derivative indices a=1.5 and a=1.8, it can be seen that the difference format constructed by the author, in the time direction, has second-order precision, as expected.

A new fractional derivative for solving time fractional diffusion wave equation

The time fractional diffusion wave equation, which can be used to describe wave diffusion process in this article, was studied. First of all, the diffusion wave equation can be extended to a generalized form in the sense of the regularized version of the ‐Hilfer–Prabhakar ( ‐H‐P) fractional operator involving the ‐Mittag‐ function. Then, the analytical solution can be obtained for this considered equation by using the Laplace transform method and the Fourier transform method. As a result, a novel and general solution have been found. The unconventional solution may show new result and phenomenon to wave diffusion process. Thereby, this research provides a window for discovering new diffusion mechanisms.

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

Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model

<abstract><p>In this article, a kind of nonlinear wave model with the Caputo fractional derivative is solved by an efficient algorithm, which is formulated by combining a time second-order shifted convolution quadrature (SCQ) formula in time and a mixed element method in space. The stability of numerical scheme is derived, and an optimal error result for unknown functions which include an original function and two auxiliary functions are proven. Further, the numerical tests are conducted to confirm the theoretical results.</p></abstract>

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>

Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary

We first consider the damped wave inequality ∂2u∂t2−∂2u∂x2+∂u∂t≥xσ|u|p,t>0,x∈(0,L), where L>0, σ∈R, and p>1, under the Dirichlet boundary conditions (u(t,0),u(t,L))=(f(t),g(t)),t>0. We establish sufficient conditions depending on σ, p, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: g≡0 and g(t)=tγ, γ>−1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality ∂αu∂tα−∂2u∂x2+∂βu∂tβ≥xσ|u|p,t>0,x∈(0,L), where α∈(1,2), β∈(0,1), and ∂τ∂tτ is the time-Caputo fractional derivative of order τ, τ∈{α,β}. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.

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.

Analysis and discretization of a variable-order fractional wave equation

We analyze a variable-order time-fractional wave equation, which models, e.g., the vibration of a membrane in a viscoelastic environment. We prove that the solutions to the variable-order ordinary differential equations in the spectral decomposition of the solution to the fractional wave equation exhibit power-law decaying characteristics and overcome the difficulty that its solution operator does not have an exponential decay in contrast to its variable-order fractional diffusion analogue.We prove an optimal-order error estimate of a numerical discretization of the variable-order fractional wave equation only under regularity assumptions of the data of the model but with no smoothness assumption of its solution. As the solution exhibits initial weak singularity, the local truncation error is suboptimal. A conventional analysis gives a suboptimal-order estimate. We develop a new technique to derive the desired optimal-order convergence rate. We also conduct numerical experiments to substantiate the mathematically proved findings.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size σ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom N is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting σ = O(N−1) leads to O(N3) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only O(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

Simulation of 2D and 3D inverse source problems of nonlinear time-fractional wave equation by the meshless homogenization function method

Simulation of 2D and 3D inverse source problems of nonlinear time-fractional wave equation by the meshless homogenization function method

New Solitary Wave Solutions of the Space–time Fractional Coupled Equal Width Wave Equation (CEWE) and Coupled Modified Equal Width Wave Equation (CMEWE)

New Solitary Wave Solutions of the Space–time Fractional Coupled Equal Width Wave Equation (CEWE) and Coupled Modified Equal Width Wave Equation (CMEWE)