两类分数阶Volterra型积分微分方程的数值解法

发布时间：2018-06-17 04:10

本文选题：分数阶积分微分方程 + 再生核理论　；参考：《哈尔滨工业大学》2015年硕士论文

【摘要】：分数阶积分微分的出现发生在基础物理学中,它的出现所带来的新问题使数学家和物理学家对分数阶微积分理论产生了极大的兴趣。因为分数阶微积分方程能够更准确的描述实际现象的动力学行为,因此在工程学和物理学等其他领域都有着很广泛的应用,例如可以应用非线性分数阶微积分来模拟地震的非线性振动,并且在控制学中我们也可以发现分数阶微分(FDES)的身影。但是通常情况下这类方程很难得到解析解,所以求解其数值解就变得非常重要且具有实际应用价值。近几十年来,学者们已经提出了一些数值方法用于求解分数阶积分微分方程,如Adomain分解法、有限差分法、多项式配置法、小波法等。但理论体系仍需进一步的完善。因此,本文对两类分数阶Volterra型积分微分方程的数值解法进行了探讨,即非线性分数阶Volterra型积分微分方程和分数阶Volterra型人口增长模型。作为预备知识本文第2章节,介绍了分数阶导数基本定义,再生核理论的基本知识,为下面两部分再生核的应用作铺垫。本文的第3章首先对模型进行了解释说明,其次应用再生核理论求解非线性分数阶Volterra型积分微分方程并得到近似解。最后,由具体的数值算例验证该算法的优越性。本文的第4章主要研究的是分数阶Volterra型人口增长模型的数值解法问题。本文基于再生核理论,对再生核方法进行改进求解此类型的方程,并建立了一套完整的理论体系。最后,数值实验的结果表明本文所提出的算法十分有效且易于操作。
[Abstract]:The appearance of fractional integro-differential in basic physics brings about new problems which make mathematicians and physicists have great interest in fractional calculus theory. Because fractional calculus equations can more accurately describe the dynamics of actual phenomena, they are widely used in engineering, physics and other fields. For example, nonlinear fractional calculus can be used to simulate the nonlinear vibration of earthquakes, and the fractional differential FDESs can also be found in control. However, it is very difficult to obtain the analytical solution for this kind of equation, so it is very important to solve its numerical solution and has practical application value. In recent decades, some numerical methods have been proposed to solve fractional integrodifferential equations, such as Adomain decomposition method, finite difference method, polynomial collocation method, wavelet method and so on. However, the theoretical system still needs to be further improved. Therefore, in this paper, the numerical solutions of two kinds of fractional Volterra type integro-differential equations are discussed, that is, nonlinear fractional Volterra integral differential equations and fractional Volterra type population growth models. As the preparatory knowledge, this paper introduces the basic definition of fractional derivative and the basic knowledge of the theory of reproducing kernel, which paves the way for the application of the following two parts of reproducing kernels. In chapter 3, the model is explained firstly, and then the nonlinear fractional Volterra integro-differential equation is solved by using the reproducing kernel theory and the approximate solution is obtained. Finally, the superiority of the algorithm is verified by a numerical example. In chapter 4, the numerical solution of fractional Volterra population growth model is studied. Based on the reproducing kernel theory, the reproducing kernel method is improved to solve the equations of this type, and a complete theoretical system is established. Finally, the numerical results show that the proposed algorithm is very effective and easy to operate.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.6

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