Time-fractional Wave Equation Research Articles

The well-posedness analysis in Besov-type spaces for multi-term time-fractional wave equations

The well-posedness analysis in Besov-type spaces for multi-term time-fractional wave equations

A fast H3N3‐2σ$_\sigma$‐based compact ADI difference method for time fractional wave equations

AbstractIn the present paper, we derive a fast H3N3‐2‐based compact alternating direction implicit (ADI) scheme for the time fractional wave equation in two‐dimensional spatial domains. The time fractional derivative involved in the equation is discretized by the H3N3‐2 formula combined with sum‐of‐exponential technique. The former was first proposed by Du et al., while the latter has become a widely used technique to reduce the computational cost in the processing of convolution integrals. The spatial discretization adopts the standard compact ADI method, which can ensure that the space can reach the fourth‐order accuracy without increasing the amount of computation. The numerical stability and convergence of the difference scheme are analyzed rigorously. As an auxiliary illustration, a numerical example is given to verify the validity of the theoretical findings.

Unique inversion of orders and potential for multi-term time fractional wave equations

The paper investigates an inverse problem of determining the orders and potential function of multi-term time fractional wave equations. By analyzing the asymptotic behavior of the multivariate Mittag–Leffler function, we show the uniqueness of the inverting the orders and potential term using interior domain measurement data. The article also provides numerical simulations, showing how to handle noisy data and reconstruct the parameters of the equation through Tikhonov regularization.

Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation

Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.

Correction of a High-Order Numerical Method for Approximating Time-Fractional Wave Equation

Correction of a High-Order Numerical Method for Approximating Time-Fractional Wave Equation

Boundary value problem for the time-fractional wave equation

Boundary value problem for the time-fractional wave equation

A meshless particle method for solving time-fractional wave equations

A meshless particle method for solving time-fractional wave equations

Application of subordination principle to coefficient inverse problem for multi-term time-fractional wave equation

Application of subordination principle to coefficient inverse problem for multi-term time-fractional wave equation

Finite element method for an optimal control problem governed by a time fractional wave equation

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings.

A fast linearized Galerkin finite element method for the nonlinear multi-term time fractional wave equation

This paper is concerned with a class of nonlinear multi-term time fractional wave equations. Based on the Galerkin finite element method (FEM) in space and the L2-1σ formula in time, combined with the corresponding fast algorithm, an efficient fully discrete scheme is constructed. The unconditional optimal convergence property in H1-norm of the proposed scheme is discussed by utilizing the time-space splitting technique. Furthermore, the optimal L2-norm convergence and the H1-norm superconvergence results of the method are obtained. Finally, several illustrative numerical experiments are performed to verify the theoretical results. Compared with the existing works, studying the convergence of FEMs for the nonlinear fractional wave equation is a great challenging task, and there are significant differences in research approaches.

Determination of initial data in time-fractional wave equation

Determination of initial data in time-fractional wave equation

Unconditionally Stable and Convergent Difference Scheme for Superdiffusion with Extrapolation

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1, 2)$$\\end{document} and the error is shown to have the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big ( d_{3} \ au ^{3- \\alpha } + d_{4} \ au ^{4-\\alpha } + d_{5} \ au ^{5-\\alpha } + \\cdots \\big ) + \\big ( d_{2}^{*} \ au ^{4} + d_{3}^{*} \ au ^{6} + d_{4}^{*} \ au ^{8} + \\cdots \\big ) $$\\end{document} at any fixed time, where τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} denotes the step size and dl,l=3,4,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{l}, l=3, 4, \\dots $$\\end{document} and dl∗,l=2,3,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{l}^{*}, l\\,=\\,2, 3, \\dots $$\\end{document} are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order O(τ3-α),α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O (\ au ^{3- \\alpha }), \\alpha \\in (1, 2)$$\\end{document} and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are O(τ4-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O ( \ au ^{4- \\alpha })$$\\end{document} and O(τ2(3-α)),α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O ( \ au ^{2(3- \\alpha )}), \\alpha \\in (1, 2)$$\\end{document}, respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.

Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann-Liouville time derivative

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Uniqueness of solution with zero boundary condition for time-fractional wave equations

We consider a solution u to an initial boundary value problem in a bounded domain Ω over a time interval (0,T) for a time-fractional wave equation where the order of time derivative is between 1 and 2. We prove that if u|ω×(0,T)=0 for arbitrarily chosen subdomain ω⊂Ω, then u=0 in Ω×(0,T).

Finite element algorithm with a second‐order shifted composite numerical integral formula for a nonlinear time fractional wave equation

AbstractIn this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests.

Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient

In this paper, a fully discrete finite element scheme is established with L1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H, and then a linearized problem is solved in a fine-grid space with mesh size h, H≫h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms.

Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α∈(1,2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.

NONEXISTENCE OF GLOBAL SOLUTIONS FOR TIME FRACTIONAL WAVE EQUATIONS IN AN EXTERIOR DOMAIN

This paper concerns two classes of initial-boundary value problems of semilinear time fractional wave equations in an exterior domain. Based on a special technique of test functions, we determine some critical exponents in the sense of Fujita for a corresponding problem. We also obtain the existence and nonexistence of global weak solutions with certain appropriate conditions.