A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative

2018 ◽  
Vol 35 (3) ◽  
pp. 936-954 ◽  
Author(s):  
Haonan Li ◽  
Shujuan Lü ◽  
Tao Xu
Keyword(s):  
Spectral Method ◽  
Fully Discrete ◽  
Fractional Cattaneo Equation

scholarly journals Efficient Galerkin finite element methods for a time-fractional Cattaneo equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
An Chen ◽  
Lijuan Nong
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Numerical Examples ◽  
Fully Discrete ◽  
Backward Euler ◽  
Galerkin Finite Element Methods ◽  
Difference Methods ◽  
Fractional Cattaneo Equation

Abstract In this paper, we develop two efficient fully discrete schemes for solving the time-fractional Cattaneo equation, where the fractional derivative is in the Caputo sense with order in $(1, 2]$ ( 1 , 2 ] . The schemes are based on the Galerkin finite element method in space and convolution quadrature in time generated by the backward Euler and the second-order backward difference methods. Error estimates are established with respect to data regularity. We further compare our schemes with the L2-$1_{\sigma }$ 1 σ scheme. Numerical examples are provided to show the efficiency of the schemes.

scholarly journals A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingfa Fei ◽  
Guoyu Zhang ◽  
Nan Wang ◽  
Chengming Huang
Keyword(s):  
Spectral Method ◽  
Numerical Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Strongly Coupled ◽  
Fully Discrete ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Legendre Spectral Method ◽  
Prior Bound

AbstractIn this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in $L^{\infty }$ L ∞ -norm is obtained, and the spectral scheme is shown to be unconditionally convergent in $L^{2}$ L 2 -norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.

A Fully Discrete Spectral Method for Fisher’s Equation on the Whole Line

2016 ◽  
Vol 6 (4) ◽  
pp. 400-415 ◽  
Author(s):  
Yu-Jian Jiao ◽  
Tian-Jun Wang ◽  
Qiong Zhang
Keyword(s):  
Finite Difference ◽  
Theoretical Analysis ◽  
Asymptotic Solution ◽  
Spectral Method ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fisher’S Equation ◽  
Fully Discrete ◽  
Fisher's Equation ◽  
Solution Behaviour

AbstractA generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

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