cranknicholson加權與位移gr252;nwald差分格式的精度與穩(wěn)定性

1、學位論文原創(chuàng)性聲明本論文是我個人在導師指導下進行的工作研究及取得的研究成果. 論文中除了特別加以標注和致謝的地方外, 不包含其他人或其它機構已經發(fā)表或撰寫過的研究成果. 對本文的研究做出貢獻的個人和集體, 均已在論文中以明確方式標明. 本人完全意識到本聲明的法律責任由本人承擔.論文作者簽名: 日期: 年 月 日學位論文使用授權的說明本人授權汕頭大學保存本學位論文的電子和紙質文檔, 允許論文被查閱和借閱; 學?？蓪⒈緦W位論文的全部或部分內

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3、tFractional differential equations (FDEs) emerge from numerous topics suchas turbulent flow, chaotic dynamics of classical conservative systems, ground-water contaminant transport, and applications in physics, finance, b

4、iology, andimage processing. Unlike the second-order diffusion equations, there are fewFDEs whose analytical closed-form solutions are available. As a result, numer-ical methods for FDEs have been developed intensively i

5、n the last decades.In this thesis, we are concerned with numerical methods of the solution ofthe following fractional initial-boundary partial differential equation:?u(x, t)?t = d+(x, t)?αu(x, t)?+xα + d?(x, t)?αu(x, t)?

6、?xα + f(x, t),x ∈ (xL, xR), t ∈ (0, T],u(xL, t) = u(xR, t) = 0, 0 ≤ t ≤ T,u(x, 0) = u0(x), x ∈ [xL, xR],where 1 < α < 2, f(x, t) is the source term, and d±(x, t) ≥ 0.We apply the Crank-Nicholson weighted and s

7、hifted Gr¨ unwald difference(CN-WSGD) scheme proposed in [Mathematics of Computation, 84 (2015),pp. 1703–1727] to discretize the above equation. The CN-WSGD scheme hassecond order accuracy. Moreover, it is unconditi

8、oned stable when the diffusioncoefficients d±(x, t) are constant. In this thesis, we mainly study the case wherethe diffusion coefficients d±(x, t) are not constant. We discuss how to choosesuitable weights suc

9、h that the CN-WSGD scheme is optimal about the accuracyand stability. Numerical results are present to compare the CN-WSGD schemeswith different weights.Key words: Fractional differential equation; Gr¨ unwald-Letnik