Time-fractional Wave Research Articles

Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation

Abstract In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.

Group analysis to the time fractional nonlinear wave equation

In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis scheme. First, the symmetry can be obtained make uses of the group analysis to the time fractional nonlinear wave equation. Based on the above-found symmetry, this equation was able to reduce into an ordinary differential equation of fractional order. As a result, some new invariant solutions were also constructed for this considered equation. Second, the scaling transformation was also obtained by introducing new independent and dependent variables. Finally, the conservation laws were also found to satisfy the time fractional nonlinear wave equation with the help of the Ibragimov theorem. These novel results show unique nonlinear phenomena.

Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.

Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method

Very recently, Yang, Abdel-Aty and Cattani (2019) introduced a new and intersting fractional derivative operator with non-singular kernel involving Rabotnov fractional-exponential function. In this paper, we present a general framework of the homotopy perturbation transform method (HPTM) for analytic treatment of time fractional partial differential equations in the sense of Yang-Abdel-Aty-Cattani. As applications, time fractional wave equations involving Yang-Abdel-Aty-Cattani fractional derivatives are solved. The solutions are obtained in the form of series involving Prabhakar functions.

On a backward problem for two-dimensional time fractional wave equation with discrete random data

This paper is concerned with a backward problem for a two- dimensional time fractional wave equation with discrete noise. In general, this problem is ill-posed, therefore the trigonometric method in nonparametric regression associated with Fourier truncation method is proposed to solve the problem. We also give some error estimates and convergence rates between the regularized solution and the sought solution under some assumptions.

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

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. The proposed fast linearized method can be directly extended to solve the nonlinear distributed-order time-fractional wave problem. Numerical examples are provided to justify the efficiency.

Approximate Solutions of Time Fractional Diffusion Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

Superconvergence analysis of FEM for 2D multi-term time fractional diffusion-wave equations with variable coefficient

The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximating in space by linear triangular finite element method and in time by Crank-Nicolson scheme. The stability is firstly proved unconditionally. In the analysis of superclose properties, how to deal with the item for variable coefficient is the main difficulty. In order to do this, a new projection operator is defined and the relationship between the proposed projection operator and interpolation operator about linear triangular finite element is deduced. Consequently, the global superconvergence result is obtained by use of interpolation postprocessing technique. The numerical examples show that the proposed numerical method is highly accurate and computationally efficient.

Fractional Landweber method for an initial inverse problem for time-fractional wave equations

ABSTRACT In this paper, we consider the initial inverse problem (backward problem) for an inhomogeneous time-fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

This paper analyzes the convergence of a Petrov–Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

Analytical solutions of time-fractional wave equation by double Laplace transform method

In this paper, we have considered an analytical solution of the time-fractional wave equation with the help of the double Laplace transform. With the proposed technique the exact solution is obtained. The method is very simple and easy. For an application, an example is provide.

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...

Existence and regularity of final value problems for time fractional wave equations

We study a class of final value problems for time fractional wave equations involvingCaputo’s fractional derivative of order 1<α<2 and a symmetric uniformly elliptic operator in a bounded domain Ω⊂RN, N=1,2,3. Firstly, we analyze the difficulties involved in the final value problem under consideration. Secondly, we establish the existence and uniqueness of the mild solution. Finally, we investigate the regularity of the mild solution in some special spaces.

Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term

Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, which brings much more difficulty. In this paper, based on the mid-point quadrature rule and composite two-point Gauss–Legendre quadrature rule, two finite difference schemes are established. Different from the previous works, which concerned only one- or two-dimensional problems with linear source terms, time-fractional wave equations of distributed-order whose source term is nonlinear in two and even three dimensions are considered. In addition, to improve the computational efficiency, the technique of alternating direction implicit (ADI) decomposition is also adopted. The unique solvability of the difference scheme is discussed, and the unconditional stability and convergence are analyzed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two- and three-dimensional cases.

Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach

Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

Identification of a source in a fractional wave equation from a boundary measurement

We deal with an inverse source problem in a time-fractional wave equation. The time-dependent source term is reconstructed using the additional non-invasive measurement in the form of integral over a part of the boundary. We look for the solution and the source term obeying the variational formulation and the equation obtained from applying the measurement on the equation. Using the Rothe method the existence of the solution is proven. The uniqueness is shown in appropriate spaces and some numerical experiments are presented.

A Time-Spectral Algorithm for Fractional Wave Problems

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

Crank–Nicolson WSGI difference scheme with finite element method for multi-dimensional time-fractional wave problem

In this article, a second-order Crank–Nicolson weighted and shifted Grunwald integral (WSGI) time-discrete scheme combined with finite element method is studied for finding the numerical solution of the multi-dimensional time-fractional wave equation. The time-fractional wave equation with Caputo-fractional derivative is transformed into the time-fractional integral equation by integral technique, then the second-order Crank–Nicolson finite element scheme with a time second-order WSGI discrete approximation for fractional integral is formulated. The analysis of stability is made, then a priori error estimate is given by making use of the conversion technique between the WSGI formula and fractional integral. At the end of the article, some numerical examples covering two two-dimensional cases with rectangular element and triangular element and a three-dimensional case with tetrahedral element are shown to test and verify our theoretical results.

A source identification problem in a time-fractional wave equation with a dynamical boundary condition

We focus on a inverse source problem in a partial differential equation containing a fractional derivative in time of the order 1<β<2. The equation is accompanied with a non-standard boundary condition which consists of the classic flux term and the dynamical time-fractional derivative term on the one part of the boundary. To determine both the solution of the equation and the source term, a measurement in a form of a integral over space domain is considered. Using a time-discretization and Rothe’s method, we prove an existence of the strong solution in the suitable functional spaces, and the error estimate is established; moreover, the uniqueness is addressed. The results are supported by some numerical experiments.