本文選題：Runge-Kutta-Chebyshev方法 + 時間分數(shù)階反應(yīng)擴散方程��； 參考：《吉林大學(xué)》2015年博士論文

【摘要】：本文研究了求解第二類非線性剛性Volterra積分方程的顯式Pouzet-Runge-Kutta-Chebyshev (PRKC)方法和求解Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程的顯式Abel-Runge-Kutta-Chebyshev (ARKC)方法.Runge-Kutta (RK)方法是一類經(jīng)典的用于求解常微分方程的單步數(shù)值解法.顯式Rvnge-Kutta-Chebyshev (RKC)方法是一類穩(wěn)定的RK方法,具有一階和二階,且其在復(fù)平面上沿負實軸方向的絕對穩(wěn)定區(qū)域長度與s2(s為級數(shù))成正比.因此,顯式RKC方法能夠用于求解非線性大維數(shù)剛性常微分方程組.其良好的穩(wěn)定性質(zhì)來自于以第一類Chebyshev多項式為基礎(chǔ)構(gòu)造方法的穩(wěn)定性函數(shù).通過空間離散, Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程初邊值問題能夠轉(zhuǎn)化為第二類非線性剛性且具有弱奇性核的Volterra積分方程.因此,兩類問題的共同特點就是都具有非線性和剛性.由于隱格式的方法不容易實現(xiàn),所以我們利用顯式RKC方法的思想,構(gòu)造求解這兩類問題的數(shù)值方法.本文分四章,最后一章是總結(jié),前三章的內(nèi)容包括：第一章是緒論,介紹了相關(guān)內(nèi)容的背景和理論基礎(chǔ),即第二類Volterra積分方程,時間分數(shù)階反應(yīng)擴散方程,求解常微分方程的RK方法和顯式RKC方法,以及求解第二類非奇性和弱奇性Volterra積分方程的RK方法.第二章,我們研究了求解第二類非線性剛性Volterra積分方程的顯式PRKC方法.因為求解第二類Volterra積分方程的Pouzet類Volterra-Runge-Kutta (Pouzet-Volterra-Runge-Kutta, PVRK)方法與對應(yīng)求解常微分方程的RK方法不但具有相同的絕對穩(wěn)定區(qū)域,而且具有相同的階數(shù),所以我們選擇PVRK方法作為我們的研究對象.首先,我們分析了PVRK方法關(guān)于基本測試方程的絕對穩(wěn)定性；接著,根據(jù)顯式RKC方法的思想,我們利用第一類Chebyshev多項式構(gòu)造顯式PVRK方法的穩(wěn)定性函數(shù),推導(dǎo)出求解第二類非線性剛性Volterra積分方程的顯式二階PRKC方法.然后,討論了顯式PRKC方法關(guān)于卷積型測試方程的絕對穩(wěn)定性,以及相關(guān)的絕對穩(wěn)定性的數(shù)值研究,包括顯式三級PRKC方法絕對穩(wěn)定性和穩(wěn)定區(qū)域與算法參數(shù)ε和s數(shù)量之間的關(guān)系.最后,通過數(shù)值實驗證明了PRKC方法能夠有效性地求解第二類非線性剛性Volterra積分方程,以及通過調(diào)整s和ε的數(shù)量能夠改變穩(wěn)定區(qū)域.第三章,我們研究了求解Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程的顯式ARKC方法.因為通過空間離散,Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程初邊值問題能夠轉(zhuǎn)化為第二類非線性剛性且具有弱奇性核的Volterra積分方程組,所以這章我們求解的問題就是后者.首先,我們找到了ARK方法和Volterra-Runge-Kutta (VRK)方法系數(shù)之間的關(guān)系,根據(jù)這個關(guān)系能夠用VRK方法的系數(shù)構(gòu)造一階的ARK方法的系數(shù).接著,利用前面的關(guān)系,以PRKC方法為基礎(chǔ)構(gòu)造了求解Caputo導(dǎo)數(shù)意義下的時間分數(shù)階反應(yīng)次擴散方程(即第二類非線性剛性且具有弱奇性核的Volterra積分方程)的顯式Abel-Runge-Kutta-Chebyshev (ARKC)方法.然后,通過數(shù)值的方法分析了顯式ARKC方法關(guān)于測試方程的絕對穩(wěn)定性,推出了方法的參數(shù)s,ε與測試方程參數(shù)λ,α之間的關(guān)系.最后,通過數(shù)值實驗驗證了顯式ARKC方法能夠有效地求解Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程初邊值問題,以及方法的參數(shù)s,ε與方程參數(shù)α之間的關(guān)系.1.求解第二類剛性Volterra積分方程的PRKC方法第二類剛性Volterra積分方程為其中(?)K/(?)y和(?)2K/(?)t(?) y皆為絕對值很大的負數(shù).取則求解第二類剛性Volterra積分方程(1)的s級顯式Pouzet-Runge-Kutta-Chebyshev (PRKC)方法為：其中Yni,yn+1分別為y(tn+cih)和y(tn+h)近似值,且滿足的近似值.其系數(shù)定義為：其中2.求解時間分數(shù)階反應(yīng)擴散方程的顯性RKC方法通過空間離散,Caputo導(dǎo)數(shù)意義下時間分數(shù)階反應(yīng)次擴散方程能夠轉(zhuǎn)化為第二類非線性剛性且弱奇性的Volterra積分方程取求解第二類非奇性剛性且弱奇性Volterra積分方程(3)的s級顯式Abel-Runge-Kutta-Chebyshev (ARKC)方法為其中的近似值.系數(shù)定義為：其中aij,bj,ci是s級顯式PRKC方法(2)的系數(shù).
[Abstract]:In this paper, the explicit Pouzet-Runge-Kutta-Chebyshev (PRKC) method for solving second classes of nonlinear rigid Volterra integral equations and the explicit Abel-Runge-Kutta-Chebyshev (ARKC) method of Abel-Runge-Kutta-Chebyshev (ARKC) method for solving the fractional order reaction of time fractional order reaction under the sense of Caputo derivative.Runge-Kutta (RK) are a class of classical solutions for solving ordinary differential equations. The explicit Rvnge-Kutta-Chebyshev (RKC) method is a class of stable RK methods with the first and two orders, and the absolute stability region length along the negative real axis is proportional to the S2 (s Series) in the complex plane. Therefore, the explicit RKC method can be used to solve the nonlinear large dimensional rigid ordinary differential equations. The stability property is derived from the stability function of the construction method based on the first class of Chebyshev polynomials. By space discretization, the initial boundary value problem of the time fractional order reaction of the time fractional order reaction of the time fractional order reaction in the sense of Caputo derivative can be converted to the second class of nonlinear rigid and weakly singular Volterra integral equations. Therefore, the common special characteristics of the two kinds of problems are common. The points are both nonlinear and rigid. Because the implicit method is not easy to implement, we use the idea of explicit RKC to construct numerical methods for solving these two kinds of problems. This paper is divided into four chapters, the last chapter is a summary, the first chapter of the three chapter is the introduction, introducing the background and theoretical basis of the related content, that is, the first chapter. The two kind of Volterra integral equation, the time fractional order reaction diffusion equation, the RK method and explicit RKC method for solving the ordinary differential equation, and the RK method for solving the second kind of non singular and weak singular Volterra integral equation. The second chapter, we study the explicit PRKC method for solving the second class of nonlinear rigid Volterra integral equations. Because the solution second is solved second. The Pouzet class Volterra-Runge-Kutta (Pouzet-Volterra-Runge-Kutta, PVRK) method of the class Volterra integral equation and the RK method corresponding to the ordinary differential equation not only have the same absolute stable region, but also have the same order, so we choose the PVRK method as our research object. First, we analyze the PVRK method. On the basis of the absolute stability of the basic test equation, then, according to the idea of the explicit RKC method, we use the first class of Chebyshev polynomials to construct the stability function of the explicit PVRK method, and derive the explicit two order PRKC method for solving second classes of nonlinear rigid Volterra integral equations. The absolute stability of the equation and the numerical study of the relative absolute stability, including the relationship between the absolute stability of the explicit three stage PRKC method and the stable region and the number of algorithm parameters, and the number of S. Finally, the numerical experiments show that the PRKC method can effectively solve the second class of nonlinear rigid Volterra integral equations and pass through the numerical experiments. The number of adjusting s and epsilon can change the stable region. In the third chapter, we study the explicit ARKC method for solving the time fractional diffusion equation of the time fractional order in the sense of Caputo derivative, because the initial boundary value problem of the time fractional order reaction of the time fractional order reaction in the sense of the Caputo derivative can be converted to the second class of nonlinear rigidity by space discretization. The Volterra integral equations of the weakly singular kernel, so the problem we solve in this chapter is the latter. First, we find the relationship between the ARK method and the coefficient of the Volterra-Runge-Kutta (VRK) method. According to this relation, we can construct the coefficient of the first order ARK method with the coefficient of the VRK method. Then, using the previous relation, the PRKC method is used. The explicit Abel-Runge-Kutta-Chebyshev (ARKC) method for the time fractional order reaction sub diffusion equation (the second class of nonlinear rigid and weakly singular Volterra integral equations) under the Caputo derivative is constructed. Then, the absolute stability of the explicit ARKC method on the test equation is analyzed by the numerical method. The relationship between the parameters s, epsilon and the parameter of the test equation, a and the alpha. Finally, through numerical experiments, it is proved that the explicit ARKC method can effectively solve the initial boundary value problem of the time fractional order rediffusion equation under the Caputo derivative, and the relation between the parameter s, the equation parameter and the equation parameter.1. to solve the second kind of rigid Volterra. The second class rigid Volterra integral equation of the integral equation is a negative number of the absolute values of the (?) K/ (?) y and (?) 2K/ (?) t (?) y. The s explicit Pouzet-Runge-Kutta-Chebyshev (PRKC) method for solving the second class rigid Volterra integral equation (1) is as follows: Yni, and satisfies the approximate value of Pouzet-Runge-Kutta-Chebyshev (PRKC). The coefficient is defined as: 2. of the explicit RKC methods for solving the time fractional order reaction diffusion equation are discretized by space, and the time fractional order rediffusion equation in the sense of Caputo derivative can be converted into the second class of nonlinear rigid and weakly singular Volterra integral equations to solve second kinds of non singular rigid and weak singularity Volte The s level explicit Abel-Runge-Kutta-Chebyshev (ARKC) method of the RRA integral equation (3) is the approximate value. The coefficients are defined as AIJ, BJ, and CI are the coefficients of the s class explicit PRKC method (2).
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O241.8

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