Space-fractional Wave Equation Research Articles

Energy-preserving splitting finite element method for nonlinear stochastic space-fractional wave equations with multiplicative noise

It is well-known that numerical methods that preserve physical quantities of original system often yield physically reasonable results and possess better numerical stability in long-time simulations. In this paper, we propose a methodology of constructing efficient fully-discrete method preserving averaged energy evolution law for nonlinear stochastic space-fractional wave equations with multiplicative noise. Specifically, the space variable is firstly discretized by Galerkin finite element method, and then the splitting technique and averaged vector field method are employed for the time integration of the resulting spatial semi-discrete scheme. We prove that the proposed fully-discrete method preserves the averaged energy evolution law in the discrete level. Numerical experiments are given to verify the theoretical result.

Unraveling the vector nature of generalized space-fractional Bessel beams

We introduce an exact analytical solution of the homogeneous space-fractional Helmholtz equation in cylindrical coordinates. This solution, called the vector space-fractional Bessel beam (SFBB), has been established from the Lorenz gauge condition and Hertz vector transformations. We perform scalar and vector wave analysis focusing on electromagnetics applications, especially in cases where the dimensions of the beam are comparable to its wavelength $({k}_{r}\ensuremath{\approx}k)$. The propagation characteristics such as the diffraction and self-healing properties have been explored with particular emphasis on the polarization states and transverse propagation modes. Due to continuous order orbital angular momentum dependence, this beam can serve as a bridge between the ordinary integer Bessel beam and the fractional Bessel beam and, thus, can be considered as a generalized solution of the space-fractional wave equation that is applicable in both integer and fractional dimensional spaces. The proposed SFBBs provide better control over the beam characteristics and can be readily generated using digital micromirror devices, spatial light modulators, metasurfaces, or spiral phase plates. Our findings offer insights on electromagnetic wave propagation, thus paving a route towards applications in optical tweezers, refractive index sensing, optical trapping, and optical communications.

Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain

(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives.

On a nonlinear energy-conserving scalar auxiliary variable (SAV) model for Riesz space-fractional hyperbolic equations

In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.

Higher Order Approximation for Stochastic Space Fractional Wave Equation Forced by an Additive Space-Time Gaussian Noise

The infinitesimal generator (fractional Laplacian) of a process obtained by subordinating a killed Brownian motion catches the power-law attenuation of wave propagation. This paper studies the numerical schemes for the stochastic wave equation with fractional Laplacian as the space operator, the noise term of which is an infinite dimensional Brownian motion or fractional Brownian motion (fBm). Firstly, we establish the regularity of the mild solution of the stochastic fractional wave equation. Then a spectral Galerkin method is used for the approximation in space, and the space convergence rate is improved by postprocessing the infinite dimensional Gaussian noise. In the temporal direction, when the time derivative of the mild solution is bounded in the sense of mean-squared $L^p$-norm, we propose a modified stochastic trigonometric method, getting a higher strong convergence rate than the existing results, i.e., the time convergence rate is bigger than $1$. Particularly, for time discretization, the provided method can achieve an order of $2$ at the expenses of requiring some extra regularity to the mild solution. The theoretical error estimates are confirmed by numerical experiments.

Fractional Klein-Gordon equation with singular mass

We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of δ2 is observed.

Highly efficient difference methods for stochastic space fractional wave equation driven by additive and multiplicative noise

A new fully discrete scheme of stochastic space-fractional nonlinear damped wave equations respectively driven by additive and multiplicative noise is presented. Based on the spatial discretization done via a fourth-order central difference scheme, the semi-implicit Crank–Nicolson scheme is developed for the temporal approximation. The trace formulas for the energy are given for both additive noise and multiplicative Stratonovich noise. Furthermore, the discrete energy trace formula of numerical scheme observed is consistent with behavior of theoretical analysis results. Otherwise, the convergence of order for numerical scheme in time is calculated. Some numerical experiments are performed to verify theory in long time computations at last.

Time fourth-order energy-preserving AVF finite difference method for nonlinear space-fractional wave equations

In this paper, we develop and analyze a new time fourth-order energy-preserving average vector field (AVF) finite difference method for the nonlinear fractional wave equations with Riesz space-fractional derivative. To the corresponding Hamiltonian system of the nonlinear fractional wave equations, the fourth-order weighted and shifted Lubich difference operator in space and the fourth-order AVF method in time are used to develop the time fourth-order energy-preserving AVF finite difference method for the nonlinear fractional wave equations. The energy conservation in the discrete form and unique solvability of the proposed scheme are proved and error estimates of the scheme are further proved to be order of O((Δt)4+h4) in the discrete L2- norm. Numerical experiments confirm energy conservation and high-order accuracy of the proposed scheme.

Two novel energy dissipative difference schemes for the strongly coupled nonlinear space fractional wave equations with damping

In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional convergence and stability of the proposed schemes are exhibited rigidly. By the discrete energy analysis method, it is rigidly shown that the proposed schemes achieve the unconditional convergence rates of O(Δt2+h2) in the discrete L∞-norm for the associated numerical solutions. At last, some numerical results are provided to illustrate the dynamical behaviors of the damping terms and unconditional energy stability of the suggested schemes, and testify the efficiency of theoretical results.

Efficient linear energy dissipative difference schemes for the coupled nonlinear damped space fractional wave equations

In this paper, several efficient energy dissipative linear difference schemes are presented and analyzed for solving the coupled nonlinear damped fractional wave equations. First, the weighted shifted Grünwald difference formula is used to approach the considered fractional system in space direction. Then, we apply second-order centered difference scheme and invariant energy quadratization Crank-Nicolson scheme to discrete the resulting system in time direction, respectively. Subsequently, the convergence and stability of the proposed schemes are discussed. By using the discrete energy method and a ‘cut-off’ function technique, it is proven that the suggested schemes attain the convergence orders of O(Δt2+h2) in the discrete L2- and L∞- norms, without any time step restrictions dependent on the spatial mesh size. Finally, some numerical results are provided to elucidate the physical behavior and unconditional energy stability of the proposed schemes, and confirm the correctness of theoretical results.

Exact and approximate analytical time-domain Green's functions for space-fractional wave equations.

The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α0|ω|y. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $$s \in (0,1)$$ , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.

Approximate analytical time-domain Green’s functions for space-fractional wave equations

Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the power law exponent and attenuation constant for breast and for liver. The results for both breast and liver converge very close to the source, and similar performance is observed in time-domain Green’s functions computed for linear with frequency attenuation. Despite minor differences in the arguments, the approximate analytical time-domain Green’s functions derived for dispersive time-fractional and space-fractional wave equations are also quite similar. [Work supported in part by NIH Grants EB023051 and EB012079.] Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the po...

Analysis of dispersion and propagation properties in a periodic rod using a space-fractional wave equation

This study explores the use of fractional calculus as a possible tool to model wave propagation in complex, heterogeneous media. While the approach presented could be applied to predict field transport in a variety of inhomogeneous systems, we illustrate the methodology by focusing on elastic wave propagation in a one-dimensional periodic rod. The governing equations describing the wave propagation problem in inhomogeneous systems typically consist of partial differential equations with spatially varying coefficients. Even for very simple systems, these models require numerical solutions which are computationally expensive and do not provide the valuable insights associated with closed-form solutions. We will show that fractional calculus can provide a powerful approach to develop comprehensive mathematical models of inhomogeneous systems that can effectively be regarded as homogenized models. Although at first glance the mathematics might appear more complex, these fractional order models can allow the derivation of closed-form analytical solutions that provide excellent estimations of the systems' dynamic responses. Equally important, these solutions are valid in a frequency range that goes largely beyond the well-known homogenization limit of traditional integer order approaches, therefore providing a possible route to high-frequency homogenization. More specifically, this study focuses on the analyses of the dispersion and propagation properties of a periodic medium under single-tone harmonic excitation and illustrates the methodology to obtain a space-fractional wave equation capable of capturing the behavior of the physical system. The fractional wave equation and its analytical solution are compared with numerical results obtained via a traditional finite element method in order to assess their validity and evaluate their performance. It is found that the resulting fractional differential models are, in their most general form, of complex and frequency-dependent order.

Time-domain analysis of power law attenuation in space-fractional wave equations.

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency, and in medical ultrasound, power law attenuation is often described by fractional calculus models that contain one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions are known, but similar expressions are not available for the space-fractional models that describe power law attenuation. To address this deficiency, a numerical approach for calculating time-domain Green's functions for the Chen-Holm space-fractional wave equation and Treeby-Cox space-fractional wave equation is introduced, where challenges associated with the numerical evaluation of a highly oscillatory improper integral are addressed with the Filon integration formula combined with the Pantis method. Numerical results are computed for both of these space-fractional wave equations at different distances in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal and that away from the origin, the time-domain Green's function for the Treeby-Cox space-fractional wave equation is very similar to the time-domain Green's function for the time-fractional power law wave equation.

High-order numerical modeling for two-dimensional two-sided space-fractional wave equation based on meshless method

The Grunwald formula is the traditional method to deal with Riemann–Liouville fractional derivative, while its convergence is only [Formula: see text]. In this paper, a high-order polynomial approximation is presented for the Riemann–Liouville fractional derivative. The quadratic polynomial functions and their fractional derivatives with explicit expressions are constructed to approximate the fractional derivative instead of Grunwald formula or shifted Grunwald formula. We proved that this technique has convergence of [Formula: see text]. Based on the MLS approximation and the high-order polynomial approximation and center difference method, a meshless analysis is proposed for the two-dimensional two-sided space-fractional wave equations (SFWE). The SFWE is found to be very adequate in describing anomalous transport and dispersion phenomena. In the meshless method, the trial function for the SFWE is constructed by the MLS approximation and the Riemann–Liouville fractional derivative is approximated by the high-order polynomial approximation, and the essential boundary conditions can be directly and easily imposed on as finite element method. This technique avoids singular integral, and has high accuracy and efficiency. Numerical results demonstrate that this method is highly accurate and computationally efficient for space-fractional wave equations.

Meshless analysis of two-dimensional two-sided space-fractional wave equation based on improved moving least-squares approximation

ABSTRACTThe Space-fractional wave equations (SFWE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Due to the non-local property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model. In this paper, a meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation. The trial function for the SFWE is constructed by the IMLS approximation, where the resulting algebraic equation system to obtain the shape functions is no more ill conditioned and has high computational efficiency. The Riemann–Liouville operator is discretized by the Grünwald formula. The centre difference method and the strong-forms of the SFWE are used to obtain the final fully discrete algebraic equation. And the essential boundary conditions can be directly and easily imposed on as a finite element method. Due to the adoption of IMLS approximation and strong-forms, this method will be highly accurate and efficient. Numerical results demonstrate that this method is highly accurate and computationally efficient for SFWE. Moreover, the convergence and error estimate have been analysed in our study.

Numerical evaluation of time-domain Green's functions for space-fractional wave equations

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency. Several different models for the power law attenuation of medical ultrasound have been developed using fractional calculus, where each contains one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions have been derived and evaluated numerically, but a similar analysis has not yet been performed on the space-fractional models. To address this deficiency, time-domain Green's functions are numerically calculated here for two space-fractional models, namely the Chen-Holm and Treeby-Cox space-fractional wave equations. Numerical results are computed for both of these in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal everywhere. Away from the origin, the time-domain Green's function for the dispersive Treeby-Cox space-fractional wave equation is very similar to the time-domain Green's functions calculated for the corresponding time-fractional wave equations, but the time-domain Green's function for the nondispersive Chen-Holm space-fractional wave equation is quite different. To highlight the similarities and differences between these, time-domain Green's functions are compared and evaluated at different distances for both breast and liver.

Microlocal analysis of fractional wave equations

We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space‐time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non‐characteristic regularity.

The Numerical Analysis of Two-Sided Space-Fractional Wave Equation with Improved Moving Least-Square Ritz Method

A numerical analysis of the space-fractional wave equation is carried out by the improved moving least-square Ritz (IMLS-Ritz) method. The trial functions for the space-fractional wave equation are constructed by the IMLS approximation. By the Galerkin weak form, the energy functional is formulated. Employing the Ritz minimization procedure, the final algebraic equations system is obtained. In this numerical analysis, the applicability and efficiency of the IMLS-Ritz method are examined by some example problems. Comparing the numerical results with the analytical solutions, the stability and accuracy of the IMLS-Ritz method are also presented.