三类变分数阶微积分方程的数值解法

发布时间：2018-04-05 21:11

本文选题：Chebyshev多项式　切入点：算子矩阵　出处：《燕山大学》2016年硕士论文

【摘要】：变分数阶微积分是分数阶微积分理论的拓展。与分数阶微积分的发展历程一样,被提出之后的许多年只是用于理论性研究。近些年来,随着科学技术的发展,一些工科领域开始涉及变分数阶微积分,因为变分数阶微积分的特性之一就是可以较好地描述较大频率范围内材料的遗传特性与记忆特性。然而变分数阶微积分相关问题的研究文献相对匮乏,并不能满足实际问题需要。本文将给出一种求解三类变分数阶微积分方程数值解的算法,包括一维线性、一维非线性变分数阶微积分方程以及二维变时间分数阶扩散方程,旨在对该领域的研究有所帮助。主要包括以下几方面内容:首先,本论文基于移位Chebyshev多项式对一维解函数进行逼近,之后给出移位Chebyshev多项式逼近函数的误差估计及其收敛性分析。根据变分数阶微积分定义的特点推导出变分数阶微分算子矩阵。通过算子矩阵使一维线性变分数阶微积分方程转化为代数方程,再通过离散点的带入得到代数方程组,求解代数方程组即得到原问题的数值解。其次,对于一维非线性变分数阶微积分问题,首先给出移位Chebyshev多项式的一阶积分算子矩阵和微分算子矩阵。利用算子矩阵将一维非线性变分数阶微分方程转化成代数方程,通过离散变量将原问题转化成代数方程组,进而求得原方程的数值解。最后通过非线性算例将本文提出的算法与差分法进行比较,体现出本文提出算法的优势。最后,论文把Chebyshev多项式逼近的方法推广到二维变分数阶微分方程的数值求解中,推导算法过程,最后通过实例与差分法进行比较,证明本文提出算法的有效性和高效性。
[Abstract]:Variational fractional calculus is an extension of fractional calculus theory.As with the development of fractional calculus, it has been used for theoretical research for many years since it was proposed.In recent years, with the development of science and technology, some engineering fields begin to involve variable fractional calculus, because one of the characteristics of variable fractional calculus is that it can describe the genetic and memory characteristics of materials in a large frequency range.However, the literature on the related problems of variational fractional calculus is relatively scarce and can not meet the needs of practical problems.In this paper, we present an algorithm for solving three kinds of variational fractional calculus equations, including one dimensional linear, one dimensional nonlinear variational fractional order differential equations and two dimensional variable time fractional diffusion equations. The purpose of this paper is to be helpful to the study of this field.The main contents are as follows: firstly, this paper approximates the one-dimensional solution function based on the shift Chebyshev polynomial, and then gives the error estimate and convergence analysis of the shift Chebyshev polynomial approximation function.According to the characteristics of the definition of variational fractional calculus, the variational fractional differential operator matrix is derived.The one-dimensional linear variational fractional calculus equation is transformed into an algebraic equation by operator matrix, and then the algebraic equations are obtained through the introduction of discrete points, and the numerical solution of the original problem is obtained by solving the algebraic equations.Secondly, the first order integral operator matrix and differential operator matrix of shift Chebyshev polynomials are given for one dimensional nonlinear variational fractional order calculus problems.The nonlinear fractional differential equations of one dimension are transformed into algebraic equations by operator matrix, the original problems are transformed into algebraic equations by discrete variables, and the numerical solutions of the original equations are obtained.Finally, a nonlinear example is given to compare the proposed algorithm with the difference method, which shows the advantages of the proposed algorithm.Finally, the Chebyshev polynomial approximation method is extended to the numerical solution of two-dimensional variational fractional differential equations, and the algorithm process is deduced. Finally, the effectiveness and efficiency of the proposed algorithm are proved by comparing the algorithm with the difference method.
【学位授予单位】：燕山大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.8

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