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In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: $\partial^\beta_tu-\hbox{div}(a\nabla u)=f$, $1<\beta<2$. We first construct a difference formula to approximate $\partial^\beta_tu$ by using an interpolation of derivative type. The truncation error of this formula is of $O(\triangle t^{2+\delta-\beta})$-order if function $u(t)\in C^{2,\delta}[0,T]$ where $0\leq\delta\leq 1$ is the H\"older continuity index. This error order can come up to $O(\triangle t^{3-\beta})$ if $u(t)\in C^3[0,T]$. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the $H^1$-norm and $L_2$-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.
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참고문헌 (26)
참고문헌 신청
P. L. Butzer / 2000 / Applications of fractional calculus in physics / World Sci. Publ :1~85
Lokenath Debnath / 2003 / Recent applications of fractional calculus to science and engineering / International Journal of Mathematics and Mathematical Sciences 2003(54):3413~3442
Richard E. Ewing / 2002 / On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials / SIAM Journal on Numerical Analysis 39(6):1865~1888
Guang-hua Gao / 2014 / A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications / Journal of Computational Physics 259:33~50
Yingjun Jiang / 2011 / High-order finite element methods for time-fractional partial differential equations / Journal of Computational and Applied Mathematics 235(11):3285~3290
Lokenath Debnath / 2003 / Recent applications of fractional calculus to science and engineering / International Journal of Mathematics and Mathematical Sciences 2003(54):3413~3442
Richard E. Ewing / 2002 / On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials / SIAM Journal on Numerical Analysis 39(6):1865~1888
Guang-hua Gao / 2014 / A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications / Journal of Computational Physics 259:33~50
Yingjun Jiang / 2011 / High-order finite element methods for time-fractional partial differential equations / Journal of Computational and Applied Mathematics 235(11):3285~3290
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