Time-fractional Wave Equation Research Articles

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.

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.

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.

Closed form solutions of two time fractional nonlinear wave equations

Abstract In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized ( G ′ / G ) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.

Recognition of a time-dependent source in a time-fractional wave equation

In the present paper, we deal with an inverse source problem for a time-fractional wave equation in a bounded domain in Rd. The time-dependent source is determined from an additional measurement in the form of integral over the space subdomain. The existence, uniqueness and regularity of a weak solution are obtained. A numerical algorithm based on Rothe's method is proposed, a priori estimates are proved and convergence of iterates towards the solution is established.

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.

Finite Difference Approximation of Fractional Wave Equation with Concentrated Capacity

Abstract We consider the time fractional wave equation with coefficient which contains the Dirac delta distribution. The existence of generalized solutions of this initial-boundary value problem is proved. An implicit finite difference scheme approximating the problem is developed and its stability is proved. Estimates for the rate of convergence in special discrete energetic Sobolev norms are obtained. A numerical example confirms the theoretical results.

Simulation of the approximate solutions of the time-fractional multi-term wave equations

In this paper, simulations of the approximation solutions of time-fractional wave, forced wave (shear wave), and damped wave equations are given. The common finite difference rules besides the backward Grünwald–Letnikov scheme are used to find the approximation solution of these models. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. The Von-Neumann stability conditions are also considered and discussed for these models. Besides the simulations of the time evolutions of the approximation solutions, the stationary solutions are also simulated. The numerical results are obtained by the Mathematica software.

Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations.

The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations. To demonstrate some of the similarities and differences that are observed in three related time-fractional partial differential equations, time-domain Green's functions are calculated numerically for the power law wave equation, the Szabo wave equation, and for the Caputo wave equation. These Green's functions are evaluated for water with a power law exponent of y = 2, breast with a power law exponent of y = 1.5, and liver with a power law exponent of y = 1.139. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident very close to the source and that these causal and noncausal time-domain Green's functions converge to the same result away from the source. When noncausal time-domain Green's functions are convolved with a short pulse, no evidence of noncausal behavior remains in the time-domain, which suggests that these causal and noncausal time-fractional models are equally effective for these numerical calculations.

Estimation of shear modulus in media with power law characteristics

Shear wave propagation in tissue generated by the radiation force is usually modeled by either a lossless or a classical viscoelastic equation. However, experimental data shows power law behavior which is not consistent with those approaches. It is well known that fractional derivatives results in power laws, therefore a time fractional wave equation, the Caputo equation, which can be derived from the fractional Kelvin–Voigt stress and strain relation is tested. This equation is solved using the finite difference method with experimental parameters obtained from the existing literature. The equation is characterized by a fractional order which is also the power law exponent of the frequency dependent shear modulus. It is shown that for fractional order between 0 and 1, the equation gives smaller shear modulus than the classical model. The opposite situation applies for fractional order greater than 1. The numerical simulation also shows that the shear wave velocity method is only reliable for small losses. In our case, this is only for a small fractional order. Based on the published values of fractional order from other studies, there is therefore a chance for biased estimation of the shear modulus.

Shift-and-Invert Krylov Methods for Time-Fractional Wave Equations

The article deals with evolution problems involving time derivatives of fractional order α, with 1 < α ≤2. The solutions are expressed in terms of operator Mittag-Leffler functions. The action of such operator functions is approximated by rational Krylov methods whose convergence features are investigated.

Homotopy perturbation method for two dimensional time-fractional wave equation

The aim of this paper is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for two dimensional time-fractional wave equation (TFWE) with the boundary conditions. The fractional derivative is described in the Caputo sense. The initial approximation can be determined by imposing the boundary conditions. The method provides approximate solutions in the form of convergent series with easily computable components. The obtained results shown that the technique introduced here is efficient and easy to implement.