Spectral method for solving the time fractional Boussinesq equation

Zhang, Hui; Jiang, Xiaoyun; Zhao, Moli; Zheng, Rumeng

doi:10.1016/j.aml.2018.06.008

Cited by 13 publications

(4 citation statements)

References 9 publications

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“…Well‐posedness for fractional diffusion‐wave equations was considered in Reference [22] and diffusion‐wave equation with the Riemann–Liouville partial derivative was solved by applying the Laplace and Fourier transforms in Reference [12]. Reference [37] established a spectral method for the time fractional Boussinesq equation and proved the stability and convergence. Reference [33] developed a finite element method for time‐fractional diffusion equations with variable coefficient and gave convergence and superconvergence results.…”

Section: Introductionmentioning

confidence: 99%

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Shi

Zhao

Wang

et al. 2021

Numerical Methods Partial

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A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank-Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time-space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.

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Section: Introductionmentioning

confidence: 99%

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Shi

Zhao

Wang

et al. 2021

Numerical Methods Partial

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“…Xu et al [16] also proposed an iterative method to construct the analytical solution. Recently, a Fourier spectral method [24] was developed to obtain the numerical solutions of the equation.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the time-fractional Boussinesq equation, as a generalization of the Boussinesq equation, can be used to describe the surface water waves with a long memory property. Roughly speaking, two types of methods have been used to solve the fractional differential equation, including analytical [5,6,12,13,16,[18][19][20] and numerical methods [1,3,7,8,11,14,17,[21][22][23][24]. As far as the analytical methods for the time-fractional Boussinesq equations are concerned, the authors in [5] applied the modified Kudryashov method to solve the nonlinear conformable time-fractional Boussinesq equations.…”

Section: Introductionmentioning

confidence: 99%

Homotopy Analysis Method for the Time-Fractional Boussinesq Equation

Yang

2020

Universal Journal of Mathematics and Applications

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In this paper, the exact and approximate analytical solutions to the time-fractional Boussinesq equation are constructed using the homotopy analysis method. Several examples about the fourth-order and sixth-order time-fractional Boussinesq equations show the flexibility and efficiency of the method. Furthermore, by choosing an appropriate value for the auxiliary parameter h, we can obtain the N-term approximate solution with improved accuracy.

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“…The inertial and drag resistances are taken into account in the developed model. In [31], Fourier spectral approximation for the time fractional Boussinesq equation with periodic boundary condition is considered. In [29], longtime dynamics of a damped Boussinesq equation is investigated.…”

Section: Introductionmentioning

confidence: 99%

Active control of an improved Boussinesq system

Yildirim

2020

Math. Model. Nat. Phenom.

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In this paper, optimal control of excessive water waves in a canal system, modeled by a nonlinear improved Boussinesq equation, is considered. For this aim, well-posedness and controllability properties of the system is investigated. Suppressing of the waves in the canal system is successfully obtained by means of optimally determining of canal depth control function via maximum principle, which transforms to optimal control problem to solving an nonlinear initial-boundary-terminal value problem. The beauty of the present paper than other studies existing in the literature is that optimal canal depth control function is analytically obtained without linearization of nonlinear term. In order to show effectiveness and robustness of the control actuation, several numerical examples are given by MATLAB in tables and graphical forms.

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Spectral method for solving the time fractional Boussinesq equation

Cited by 13 publications

References 9 publications

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Novel superconvergence analysis of anisotropic triangular FEM for a multi‐term time‐fractional mixed sub‐diffusion and diffusion‐wave equation with variable coefficients

Homotopy Analysis Method for the Time-Fractional Boussinesq Equation

Active control of an improved Boussinesq system

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