分数阶积分微分方程的Bernoulli小波数值解法

发布时间：2018-11-13 07:57

【摘要】：信号处理、流体力学、控制理论等很多领域中的现象都能用分数阶积分微分方程描述,但此类方程解析解的求解非常困难,因此相关领域研究者们将目光投向了对其数值解的研究上.目前求解分数阶积分微分方程的数值解法有很多,如有限元法、同伦摄动法、Adomain分解法等,而将小波方法应用于求解分数阶积分微分方程的文献则相对较少.本文考虑用Bernoulli小波方法求解几类分数阶积分微分方程(组)的数值解.本文共分为六章.第一章对分数阶微积分的研究意义及分数阶积分微分方程数值解法的国内外研究现状进行了概述.第二章简要介绍了分数阶微积分和Bernoulli小波的基本理论,推导了Bernoulli小波的乘积算子矩阵和分数阶积分算子矩阵.第三章利用Bernoulli小波的分数阶积分算子矩阵分别求解了非线性分数阶Fredholm积分微分方程、线性和非线性分数阶Fredholm积分微分方程组的数值解并证明了其解的存在唯一性.另外,从理论上证明了 Bernoulli小波方法求解此类方程的收敛性.第四章利用Bernoulli小波的分数阶积分算子矩阵求解了线性和非线性分数阶Fredholm-Volterra积分微分方程以及弱奇异分数阶积分微分方程,数值算例说明了此方法求解这几类方程的可行性.第五章利用Bernoulli小波的分数阶积分算子矩阵求解了微分项中阶数不固定且满足一定初始条件的非线性分数阶Volterra积分微分方程、分数阶Volterra积分微分方程组并对其收敛性给出了证明,数值算例说明了该方法的有效性和准确性.第六章对全文所做的工作进行了总结并对今后进一步的研究提出展望.
[Abstract]:The phenomena in signal processing, fluid mechanics, control theory and many other fields can be described by fractional integro-differential equations, but it is very difficult to solve the analytical solutions of such equations. Therefore, researchers in related fields have focused on the study of its numerical solution. At present, there are many numerical methods for solving fractional integrodifferential equations, such as finite element method, homotopy perturbation method, Adomain decomposition method, etc. In this paper, Bernoulli wavelet method is used to solve several kinds of fractional integrodifferential equations (systems). This paper is divided into six chapters. In the first chapter, the significance of fractional calculus and the present situation of numerical solution of fractional integrodifferential equation are summarized. In chapter 2, the basic theories of fractional calculus and Bernoulli wavelet are briefly introduced, and the product operator matrix and fractional integral operator matrix of Bernoulli wavelet are derived. In chapter 3, by using the fractional integral operator matrix of Bernoulli wavelet, the numerical solutions of nonlinear fractional Fredholm integrodifferential equations, linear and nonlinear fractional Fredholm integrodifferential equations are solved, and the existence and uniqueness of the solutions are proved. In addition, the convergence of Bernoulli wavelet method for solving this kind of equation is proved theoretically. In chapter 4, the linear and nonlinear fractional Fredholm-Volterra integrodifferential equations and weakly singular fractional integral differential equations are solved by using the fractional integral operator matrix of Bernoulli wavelet. Numerical examples show that this method is feasible to solve these kinds of equations. In chapter 5, we use the fractional integral operator matrix of Bernoulli wavelet to solve the nonlinear fractional Volterra integrodifferential equations with uncertain order and satisfying certain initial conditions. The convergence of fractional Volterra integro-differential equations is proved. Numerical examples show the effectiveness and accuracy of the method. The sixth chapter summarizes the work done in this paper and puts forward the prospect of further research in the future.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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