【作者】 张华森；

【导师】 刘斌；

【作者基本信息】 华中科技大学 ， 应用数学， 2011， 硕士

【摘要】 本文讨论了一类Caputo意义下的时间分数阶扩散方程的初边值问题的数值逼近.分数阶偏微分方程是一类将经典整数阶偏微分方程中的导数定义用分数阶导数替换而得到的微分方程.和整数阶偏微分方程相比,分数阶偏微分方程更能准确地描述某些自然现象和物理过程.本文将有限元方法应用于分数阶偏微分方程,采用了一种不同的方法来近似Caputo分数阶导数并且讨论了迭代格式的稳定性,得到了较为满意的结果.首先,文章阐述了分数阶偏微分方程的相关历史背景和研究现状,并提出了本文将要解决的问题.其次,我们利用Caputo分数导数和Riemann - Liouville分数导数的一个关系式,得到了与原问题相等价的一个系统.接着在时间层上作了半离散,并利用Gru¨wald - Letnikov导数定义提出了一个差分算子,用其来近似分数阶微分算子,从而建立起了相应的变分方程,并得到了在α-范数意义下的误差估计.然后,又在空间维上对方程进一步作全离散,得到了全离散方程.对于全离散方程,我们都得到了0-范数意义和α-范数意义的误差估计.为了能应用于实际计算,我们进一步推出了有限元方程的具体表达形式,然后得到了相应刚度矩阵和代数方程.同时证明了迭代格式关于初值的稳定性.在论文的最后,我们给出了两个数值计算的例子和一些图表,数据表明计算结果与理论分析相吻合,进而证明了该方法的可行性.更多还原

【Abstract】 A numerical approximation for a Caputo’s time-fractional diffusion equation withinitial and boundary conditions is discussed in the paper. Fractional differential equa-tions are a type of differential equations in which the definition of classical integer orderderivative is replaced by that of fractional derivative. It is better to model some natu-ral phenomenon and physical processes by fractional differential equations than integerorder equations. The finite element method is applied to fractional partial equations. Adifferent method is proposed for approximating the Caputo’s fractional derivative,andthe stability of iterative scheme is proved.First,recent researches of fractional partial differential equations ,as well as histo-ry background,and the problem to be solved are stated.Secondly,a system equivalentto the original problem is obtained by using a relationship between Riemann-Liouvillefractional derivative and Caputo’s.After that,semiffdiscretization is executed at the timedirection.With the help of Gru¨wald-Letnikov derivative,a variational equation is de-duced by approximating the differential operator with difference operator.The error isestimated in theα-norm sense.Then,the full-discrete equation is derived,and both errorestimation in theα-norm and 0-norm sense are obtained. The finite element equationform is presented for the practical computation.Then, the corresponding stiffness matrixand algebraic equations are derived.Also the stability about initial values of iterationis proved.Last,two numerical examples and some diagrams are given to illustrate thefeasibility of this method.It is indicated that results coincide with theoretical analysis.更多还原