Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula

With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.

A compact scheme for two-dimensional nonlinear time fractional wave equations

Based on the equivalent integro-differential form of the considered problem, a numerical approach to solving the two-dimensional nonlinear time fractional wave equations(NTFWEs) is considered in this paper. To this end, an alternating direction implicit(ADI) numerical scheme is derived. The scheme is established by combining the secondorder convolution quadrature formula and Crank–Nicolson technique in time and afourth-order difference approach in space. The convergence and unconditional stability of the proposed compact ADI scheme are strictly discussed after a concise solvabilityanalysis. A numerical example is shown to demonstrate the theoretical analysis.

Galerkin Finite Element Approximation for Semilinear Stochastic Time-Tempered Fractional Wave Equations with Multiplicative Gaussian Noise and Additive Fractional Gaussian Noise

To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation,it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time.The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise,because of the potential fluctuations of the external sources.The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation.First,the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized,which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense.We further present a complete regularity theory for the regularized equation.A standard finite element approximation is used for the spatial operator,and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established.Finally,numerical experiments are performed to confirm the theoretical analysis.

Hybrid absorbing boundary condition based on transmitting boundary and its application in 3D fractional viscoacoustic modeling

An accurate numerical simulation for wave equations is essential for understanding of wave propagation in the earth's interior as well as full waveform inversion and reverse time migration. However, due to computational cost and hardware capability limitations, numerical simulations are often performed within a finite domain. Thus, an adequate absorbing boundary condition (ABC) is indispensable for obtaining accurate numerical simulation results. In this study, we develop a hybrid ABC based on a transmitting boundary, which is referred to as THABC, to eliminate artificial boundary reflections in 3D second-order fractional viscoacoustic numerical simulations. Furthermore, we propose an adaptive weighted coefficient to reconcile the transmitting and viscoacoustic wavefields in THABC. Through several numerical examples, we determine that the proposed THABC approach is characterized by the following benefits. First, with the same number of absorbing layers, THABC exhibits a better ability in eliminating boundary reflection than traditional ABC schemes. Second, THABC is more effective in computation, since it only requires the wavefields at the current and last time steps to solve the transmitting formula within the absorbing layers. Benefiting from a simple but effective combination between the transmitting equation and the second-order wave equation, our scheme performs well in the 3D fractional Laplacian viscoacoustic numerical simulation.

A family of solutions of the time–space fractional longitudinal wave equation

In this article,we have studied a nonlinear time–space fractional longitudinal wave equation in the context of the conformable fractional derivative.Through the soliton ansatz method and a direct integration approach with the symmetry condition,new soliton and solitary wave solutions are derived.Furthermore,the existing conditions of these obtained solutions are also given in this text.These new results add to the existing literature.We believe that they can provide a new window into the understanding of this model.

New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense.Applying the generalized Kudryashov method through with symbolic computer maple package,numerous new exact solutions are successfully obtained.All calculations in this study have been established and verified back with the aid of the Maple package program.The executed method is powerful,effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.

New analytic solutions of the space-time fractional Broer-Kaup and approximate long water wave equations

In the present paper,the exp(−φ(ξ))expansion method is applied to the fractional Broer-Kaup and approximate long water wave equations.The explicit approximate traveling wave solutions are obtained by using this method.Here,fractional derivatives are defined in the conformable sense.The obtained traveling wave solutions are expressed by the hyperbolic,trigonometric,exponential and rational functions.Simulations of the obtained solutions are given at the end of the paper.

On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method

In this paper,we mainly study the time-space fractional strain wave equation in microstructured solids.He’s variational method,combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation.The main advantage of the variational method is that it can reduce the order of the differential equation,thus simplifying the equation,making the solving process more intuitive and avoiding the tedious solving process.Finally,the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method.The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Backlund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.