本文選題：時(shí)間分?jǐn)?shù)階波動方程 + 時(shí)間分?jǐn)?shù)階Fokker-Planck方程　； 參考：《湘潭大學(xué)》2017年碩士論文

【摘要】：分?jǐn)?shù)階積分微分方程是指含有分?jǐn)?shù)階積分或分?jǐn)?shù)階導(dǎo)數(shù)的方程,是傳統(tǒng)的微積分方程的推廣.主要包括分?jǐn)?shù)階波動方程,分?jǐn)?shù)階Fokker-Planck方程等.由于解析地求解此類分?jǐn)?shù)階偏微分方程很困難甚至不可能,所以研究這類方程的數(shù)值求解方法是重要的,有價(jià)值的.本文主要是利用Jacobi譜配置法求解時(shí)間分?jǐn)?shù)階波動方程和時(shí)間分?jǐn)?shù)階Fokker-Planck方程.先利用Caputo, Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù)的定義及其相關(guān)性質(zhì)將原問題轉(zhuǎn)化為求解帶弱奇異核的第二類Volterra型積分方程,然后利用適當(dāng)?shù)木�性變換將方程轉(zhuǎn)化為新的Volterra型積分方程,使得方程具有更好的正則性,再分別從時(shí)間,空間上采用Jacobi譜配置法,即以Jacobi-Gauss點(diǎn)為配置點(diǎn),用高斯積分公式逼近積分項(xiàng),得到全離散格式進(jìn)行求解.最后從理論上嚴(yán)格證明了,在L∞和L2ω范數(shù)意義下,原方程的真解與數(shù)值解之間的誤差均具有指數(shù)收斂性.同時(shí),我們也給出了具體的數(shù)值算例,數(shù)值結(jié)果證實(shí)了譜配置法求解這兩類方程的有效性以及理論結(jié)果的正確性.
[Abstract]:Fractional integrodifferential equation refers to the equation with fractional integral or fractional derivative. It is a generalization of traditional calculus equation. It mainly includes fractional wave equation, fractional Fokker-Planck equation and so on. It is very difficult or even impossible to solve this kind of fractional partial differential equation analytically, so it is important and valuable to study the numerical solution of this kind of equation. In this paper, Jacobi spectrum collocation method is used to solve time fractional wave equation and time fractional Fokker-Planck equation. Firstly, by using the definition of Caputo, Riemann-Liouville fractional derivative and its related properties, the original problem is transformed into the Volterra type integral equation of the second kind with weak singular kernels, and then the equation is transformed into a new Volterra integral equation by proper linear transformation. The method of Jacobi spectrum collocation is used in time and space, that is, the Jacobi-Gauss point is used as collocation point, the integral term is approximated by Gao Si integral formula, and the full discrete scheme is obtained. Finally, it is strictly proved theoretically that the errors between the true solution and the numerical solution of the original equation are exponentially convergent in the sense of L 鈭，

本文編號：2017943