求解第二类刚性Volterra积分方程和时间分数阶反应扩散方程的Runge-Kutta-Chebyshev方法

发布时间：2018-06-14 09:57

本文选题：Runge-Kutta-Chebyshev方法 + 时间分数阶反应扩散方程　；参考：《吉林大学》2015年博士论文

【摘要】：本文研究了求解第二类非线性刚性Volterra积分方程的显式Pouzet-Runge-Kutta-Chebyshev (PRKC)方法和求解Caputo导数意义下时间分数阶反应次扩散方程的显式Abel-Runge-Kutta-Chebyshev (ARKC)方法.Runge-Kutta (RK)方法是一类经典的用于求解常微分方程的单步数值解法.显式Rvnge-Kutta-Chebyshev (RKC)方法是一类稳定的RK方法,具有一阶和二阶,且其在复平面上沿负实轴方向的绝对稳定区域长度与s2(s为级数)成正比.因此,显式RKC方法能够用于求解非线性大维数刚性常微分方程组.其良好的稳定性质来自于以第一类Chebyshev多项式为基础构造方法的稳定性函数.通过空间离散, Caputo导数意义下时间分数阶反应次扩散方程初边值问题能够转化为第二类非线性刚性且具有弱奇性核的Volterra积分方程.因此,两类问题的共同特点就是都具有非线性和刚性.由于隐格式的方法不容易实现,所以我们利用显式RKC方法的思想,构造求解这两类问题的数值方法.本文分四章,最后一章是总结,前三章的内容包括：第一章是绪论,介绍了相关内容的背景和理论基础,即第二类Volterra积分方程,时间分数阶反应扩散方程,求解常微分方程的RK方法和显式RKC方法,以及求解第二类非奇性和弱奇性Volterra积分方程的RK方法.第二章,我们研究了求解第二类非线性刚性Volterra积分方程的显式PRKC方法.因为求解第二类Volterra积分方程的Pouzet类Volterra-Runge-Kutta (Pouzet-Volterra-Runge-Kutta, PVRK)方法与对应求解常微分方程的RK方法不但具有相同的绝对稳定区域,而且具有相同的阶数,所以我们选择PVRK方法作为我们的研究对象.首先,我们分析了PVRK方法关于基本测试方程的绝对稳定性；接着,根据显式RKC方法的思想,我们利用第一类Chebyshev多项式构造显式PVRK方法的稳定性函数,推导出求解第二类非线性刚性Volterra积分方程的显式二阶PRKC方法.然后,讨论了显式PRKC方法关于卷积型测试方程的绝对稳定性,以及相关的绝对稳定性的数值研究,包括显式三级PRKC方法绝对稳定性和稳定区域与算法参数ε和s数量之间的关系.最后,通过数值实验证明了PRKC方法能够有效性地求解第二类非线性刚性Volterra积分方程,以及通过调整s和ε的数量能够改变稳定区域.第三章,我们研究了求解Caputo导数意义下时间分数阶反应次扩散方程的显式ARKC方法.因为通过空间离散,Caputo导数意义下时间分数阶反应次扩散方程初边值问题能够转化为第二类非线性刚性且具有弱奇性核的Volterra积分方程组,所以这章我们求解的问题就是后者.首先,我们找到了ARK方法和Volterra-Runge-Kutta (VRK)方法系数之间的关系,根据这个关系能够用VRK方法的系数构造一阶的ARK方法的系数.接着,利用前面的关系,以PRKC方法为基础构造了求解Caputo导数意义下的时间分数阶反应次扩散方程(即第二类非线性刚性且具有弱奇性核的Volterra积分方程)的显式Abel-Runge-Kutta-Chebyshev (ARKC)方法.然后,通过数值的方法分析了显式ARKC方法关于测试方程的绝对稳定性,推出了方法的参数s,ε与测试方程参数λ,α之间的关系.最后,通过数值实验验证了显式ARKC方法能够有效地求解Caputo导数意义下时间分数阶反应次扩散方程初边值问题,以及方法的参数s,ε与方程参数α之间的关系.1.求解第二类刚性Volterra积分方程的PRKC方法第二类刚性Volterra积分方程为其中(?)K/(?)y和(?)2K/(?)t(?) y皆为绝对值很大的负数.取则求解第二类刚性Volterra积分方程(1)的s级显式Pouzet-Runge-Kutta-Chebyshev (PRKC)方法为：其中Yni,yn+1分别为y(tn+cih)和y(tn+h)近似值,且满足的近似值.其系数定义为：其中2.求解时间分数阶反应扩散方程的显性RKC方法通过空间离散,Caputo导数意义下时间分数阶反应次扩散方程能够转化为第二类非线性刚性且弱奇性的Volterra积分方程取求解第二类非奇性刚性且弱奇性Volterra积分方程(3)的s级显式Abel-Runge-Kutta-Chebyshev (ARKC)方法为其中的近似值.系数定义为：其中aij,bj,ci是s级显式PRKC方法(2)的系数.
[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).
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.8

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