本文選題：分?jǐn)?shù)階對(duì)流次擴(kuò)散方程 + 緊ADI方法　； 參考：《華東師范大學(xué)》2016年博士論文

【摘要】：近年來,隨著分?jǐn)?shù)階偏微分方程在科學(xué)和工程領(lǐng)域的應(yīng)用越來越廣泛,分?jǐn)?shù)階偏微分方程數(shù)值方法的研究正變得越來越重要.本文致力于對(duì)幾類時(shí)間分?jǐn)?shù)階偏微分方程建立有效的高階緊有限差分方法,包括分?jǐn)?shù)階對(duì)流次擴(kuò)散方程,帶Neumann邊界條件的分?jǐn)?shù)階次擴(kuò)散方程,修正反常次擴(kuò)散方程和分?jǐn)?shù)階擴(kuò)散波方程.首先,考慮一類二維的分?jǐn)?shù)階對(duì)流次擴(kuò)散方程的數(shù)值方法,這類方程帶有一個(gè)α(0α1)階的Caputo時(shí)間分?jǐn)?shù)階導(dǎo)數(shù).我們先把原始方程變換為一個(gè)特殊的等價(jià)形式,對(duì)這個(gè)等價(jià)問題在空間方向上用四階緊有限差分逼近并在時(shí)間方向上用交替方向隱格式(ADI)方法處理.所得的緊ADI格式是唯一可解的和無條件穩(wěn)定的.在加權(quán)L∞,H1和L2范數(shù)下,給出了最優(yōu)誤差估計(jì).并且誤差估計(jì)顯示緊ADI格式有min{1+α,2-α)階時(shí)間精度和四階空間精度.通過兩個(gè)數(shù)值算例進(jìn)一步驗(yàn)證了理論分析和所得新格式的有效性.其次,對(duì)一類非齊次Neumann邊界條件分?jǐn)?shù)階次擴(kuò)散方程建立了一個(gè)緊交替方向隱式(ADI)有限差分格式.這類方程所帶的時(shí)間分?jǐn)?shù)階導(dǎo)數(shù)為α(0α1)階Caputo分?jǐn)?shù)階導(dǎo)數(shù).嚴(yán)格證明了格式的無條件穩(wěn)定性和收斂性.在加權(quán)L2和L∞范數(shù)下給出了誤差估計(jì),誤差估計(jì)顯示所得的緊ADI格式有四階空間精度和min{2-α,1+α}階時(shí)間精度.對(duì)α∈(0,1/2)和α∈(1/2,1)分別建立了兩個(gè)Richardson外推算法,把數(shù)值解的全局時(shí)間精度提高到max{2-α,1+α}階.并對(duì)α∈(0,1/2)的外推算法給出了嚴(yán)格的收斂階分析.數(shù)值結(jié)果驗(yàn)證了緊ADI格式的數(shù)值精度和外推算法的有效性.接下來,對(duì)一類二維修正反常次擴(kuò)散方程建立了一個(gè)Crank-Nicolson型的緊局部一維(LOD)格式,這個(gè)方程含有(1-α)和(1-β)(0α,β1)階的兩個(gè)Riemann-Liouville時(shí)間分?jǐn)?shù)階導(dǎo)數(shù).所得格式由三對(duì)角系統(tǒng)組成且所有計(jì)算都像一維問題一樣完全在一個(gè)空間方向上執(zhí)行.這個(gè)性質(zhì)明顯簡化了編程并降低了計(jì)算復(fù)雜度和計(jì)算量.嚴(yán)格證明了格式的無條件穩(wěn)定性和收斂性.在標(biāo)準(zhǔn)H1和H2范數(shù)以及加權(quán)L∞范數(shù)下給出了誤差估計(jì),并且誤差估計(jì)顯示緊LOD格式有2 min{α,β}階時(shí)間精度和四階空間精度.接著構(gòu)造了一個(gè)Richardson外推算法提高格式的時(shí)間精度,如果α≠β,外推算法把時(shí)間精度提高到nin{α+β,4min{α,β}}階,如果α=β則可以提高到min{1+α,4α}階.數(shù)值結(jié)果驗(yàn)證了緊LOD格式的數(shù)值精度和外推算法的高有效性.最后討論一類帶α(1α2)階Caputo分?jǐn)?shù)階導(dǎo)數(shù)的高維分?jǐn)?shù)階擴(kuò)散波方程的數(shù)值方法.對(duì)這類方程建立了一個(gè)緊LOD有限差分格式.所得格式由三對(duì)角系統(tǒng)組成且所有計(jì)算都像一維問題一樣完全在一個(gè)空間方向上執(zhí)行.對(duì)三維情況在H1范數(shù)下嚴(yán)格地證明了格式的無條件穩(wěn)定性和收斂性.誤差估計(jì)顯示提出的緊LOD格式有(3-α)階時(shí)間精度和四階空間精度.數(shù)值結(jié)果驗(yàn)證了我們的理論分析和新格式的有效性.
[Abstract]:In recent years, with the application of fractional partial differential equations in the field of science and engineering, the numerical methods of fractional partial differential equations are becoming more and more important. This paper is devoted to the establishment of efficient higher-order compact finite difference methods for several kinds of fractional partial differential equations, including fractional convection subdiffusion equations, fractional subdiffusion equations with Neumann boundary conditions, Modified anomalous subdiffusion equation and fractional diffusion wave equation. Firstly, a numerical method for a class of two-dimensional fractional convection subdiffusion equations with a Caputo time fractional derivative of order 偽 0 偽 1 is considered. We first transform the original equation into a special equivalent form, and approach the problem in the space direction with the fourth order compact finite difference method and in the time direction with the alternating direction implicit scheme adi. The compact ADI scheme is solvable and unconditionally stable. Under the weighted L 鈭，

本文編號(hào)：2011077