陳虹伶， 李小林

（重慶師范大學(xué) 數(shù)學(xué)科學(xué)學(xué)院，重慶 401331）

引 言

Cable 方程是神經(jīng)元?jiǎng)恿W(xué)模型中最基本的方程之一[1]，分?jǐn)?shù)階Cable 方程從分?jǐn)?shù)階Nernst-Planck 方程導(dǎo)出[2]，是經(jīng)典Cable 方程的推廣，可用于模擬離子在棘狀神經(jīng)元樹突中的異常電擴(kuò)散過程.有限差分法[1,3-5]、有限元法[6]、譜方法[7]、徑向基函數(shù)法[8]和無單元Galerkin 法[9]等數(shù)值技術(shù)已被廣泛應(yīng)用于求解分?jǐn)?shù)階Cable方程.

無網(wǎng)格法[10-12]在過去三十多年中得到了迅速發(fā)展，可有效克服有限差分法和有限元法等經(jīng)典數(shù)值方法中網(wǎng)格單元帶來的困難，同時(shí)具有較高的計(jì)算精度.有限點(diǎn)法[13-15]是基于移動(dòng)最小二乘近似構(gòu)造數(shù)值解和配點(diǎn)技術(shù)形成離散代數(shù)方程組的最流行和最簡單的一種無網(wǎng)格方法，已成功求解了大量科學(xué)工程問題.目前，我們還沒有發(fā)現(xiàn)用無網(wǎng)格有限點(diǎn)法研究分?jǐn)?shù)階Cable 方程的報(bào)道.

本文建立數(shù)值分析含有Riemann-Liouville 時(shí)間分?jǐn)?shù)階導(dǎo)數(shù)的Cable 方程的有限點(diǎn)法.首先，借鑒文獻(xiàn)[5]用中心差分格式離散該方程中的時(shí)間導(dǎo)數(shù)；其次，用有限點(diǎn)法建立線性離散代數(shù)系統(tǒng)；然后，受文獻(xiàn)[16]的啟發(fā)推導(dǎo)求解了分?jǐn)?shù)階Cable 方程的有限點(diǎn)法的理論誤差估計(jì)；最后，給出數(shù)值算例驗(yàn)證了該方法的有效性和理論誤差結(jié)果.

1 數(shù) 值 算 法

考慮以下分?jǐn)?shù)階Cable 方程的初邊值問題[9]：

2 誤 差 分 析

并且由式(17)可以得出

根據(jù)移動(dòng)最小二乘近似的重構(gòu)性質(zhì)[18]和式(13)，我們得到

根據(jù)式(8)，我們得到

將式(20)代入式(23)中，有

其中

式(21)和(24)可組裝為如下矩陣形式：

3 數(shù) 值 算 例

考慮以下分?jǐn)?shù)階Cable 方程：

邊界條件為

初始條件為

圖1 給出了時(shí)間步長τ=1/20和節(jié)點(diǎn)間距h=1/20時(shí)，有限點(diǎn)法的數(shù)值解和誤差.數(shù)據(jù)顯示絕對(duì)誤差小于3×10?4，說明解析解和數(shù)值解吻合得非常好，從而證明本文方法具有較高的計(jì)算精度.

圖1 算例在α=0.2，β=0.8，T =5 ，h=1/20 和τ=1/20時(shí)的數(shù)值解和誤差：(a) 數(shù)值解；(b) 誤差Fig. 1 Numerical solution results and errors gained with α=0.2 , β=0.8, T =5 ，h=1/20 and τ=1/20: (a) numerical solution results; (b) errors

圖2 給出了h=0.01時(shí)，相對(duì)誤差//U?Uh//2///U//2和L∞誤差與時(shí)間步長 τ之間的關(guān)系，圖3 給出了當(dāng)τ=0.000 1時(shí)，誤差與節(jié)點(diǎn)間距h之間的關(guān)系.可以看出，誤差隨著τ和h的減小而減小，且數(shù)值解大約以τ1+min{α,β}和h2的速度收斂于解析解，這與理論結(jié)果一致.

圖2 當(dāng)h=0.01，T=1 時(shí)誤差與時(shí)間步長τ的關(guān)系：(a) 相對(duì)誤差；(b) L∞誤差Fig. 2 The relationship between relative errors and L∞ errors obtained for h=0.01 and T=1 with respect to time-step size τ: (a) relative errors ; (b) L∞ errors

圖3 當(dāng)τ=0.000 1, T =1時(shí)誤差與節(jié)點(diǎn)間距h的關(guān)系：(a) 相對(duì)誤差；(b) L∞誤差Fig. 3 The relationship between relative errors and L∞ errors obtained for τ=0.000 1 and T =1 with respect to nodal spacing h:(a) relative errors ; (b) L∞ errors

以上討論的是α ≠β的情況，接下來討論α=β=γ[8]的情況.圖4 給出了τ=1/20和h=1/20時(shí)，有限點(diǎn)法的數(shù)值解和誤差.圖5(a)給出了h=0.01,T=1時(shí)，誤差與τ之間的關(guān)系，圖5(b)給出了當(dāng)τ=0.000 1時(shí)，誤差與h之間的關(guān)系.從圖中可以看出有限點(diǎn)法獲得了很好的數(shù)值結(jié)果.

圖4 算例在γ=0.4，T =5 ，h=1/20 和τ=1/20時(shí)的數(shù)值解和誤差：(a) 數(shù)值解；(b) 誤差Fig. 4 Numerical solution results and errors gained with γ=0.4, T =5 , h=1/20 and τ=1/20: (a) numerical solution results; (b) errors

圖5 誤差與時(shí)間步長τ和節(jié)點(diǎn)間距h 的關(guān)系：(a) 時(shí)間步長τ；(b) 節(jié)點(diǎn)間距hFig. 5 The relationship between the errors and time-step size τ as well as nodal spacing h: (a) for time-step size τ; (b) for nodal spacing h

表1 比較了有限點(diǎn)法和徑向基函數(shù)法[8]在h=0.1， γ=0.25和 γ=0.3時(shí)的L∞誤差，我們發(fā)現(xiàn)有限點(diǎn)法具有更高的計(jì)算精度，明顯優(yōu)于徑向基函數(shù)法.

表1 有限點(diǎn)法和徑向基函數(shù)法在h=0.1, T =1時(shí)的 L∞誤差Table 1 The L∞-errors of the finite point method and the radial basis function method gained with h=0.1, T =1

4 結(jié) 論

針對(duì)分?jǐn)?shù)階Cable 方程，本文用中心差分格式離散時(shí)間導(dǎo)數(shù)，用有限點(diǎn)法進(jìn)行空間離散，推導(dǎo)了詳細(xì)的數(shù)值計(jì)算公式，詳細(xì)分析了該方法的誤差估計(jì).理論誤差分析表明，數(shù)值解的誤差與時(shí)間步長τ和節(jié)點(diǎn)間距h成正比，并且時(shí)間和空間收斂率分別約為τ1+min{α,β}和h2.數(shù)值算例證實(shí)了求解分?jǐn)?shù)階Cable 方程的有限點(diǎn)法的有效性和收斂性，并驗(yàn)證了理論分析結(jié)果.

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