分数阶偏微分方程的基于二阶时间逼近格式的有限元方法

发布时间：2018-04-24 22:02

本文选题：非线性时间分数阶Cable方程 + 时空分数阶偏微分方程　；参考：《内蒙古大学》2017年硕士论文

【摘要】：本文主要利用两类二阶时间逼近格式结合有限元方法数值求解含有时间分数阶导数的非线性Cable方程和含有时空分数阶导数的一类非线性偏微分方程.在此前的研究中,对于含有分数阶导数的Cable方程,通常采用L1逼近离散Riemann-Liouville分数阶导数,并得到2-α阶的近似精度.在本文中,我们利用具有2阶近似精度的一类加权离散公式逼近Riemann-Liouville分数阶导数,进而形成二阶离散格式.另外我们利用有限元方法结合WS G D算子的离散技巧数值求解一类时空分数阶偏微分方程,并得到时间二阶逼近结果.首先,对于非线性时间分数阶Cable方程,在tk-α/2时间点处,我们用一个带有参数α的两步格式来近似整数阶时间导数,采用具有2阶精度的加权离散格式来近似Riemann-Liouville分数阶导数,并利用Galerkin有限元法进行空间方向离散,形成时间二阶逼近的有限元数值方法.进一步,我们给出了一些数值理论,证明了格式的稳定性,并得到基于L2模意义下的带有O(△t2 +(1+△t-α)hm+1 精度的误差估计结果.为了说明理论结果的正确性,我们通过几个算例给出数值验证.其次,利用二阶WSGD逼近结合有限元数值方法数值求解含有时空分数阶导数的一类偏微分方程,给出了详细的数值计算过程,并通过三类数值例子对格式进行有效验证.数值结果显示我们的数值格式具有二阶近似精度.
[Abstract]:In this paper, two kinds of second-order time approximation schemes combined with finite element method are used to numerically solve the nonlinear Cable equations with time fractional derivatives and a class of nonlinear partial differential equations with space-time fractional derivatives. In previous studies, for Cable equations with fractional derivatives, L1 is usually used to approximate discrete Riemann-Liouville fractional derivatives, and the approximate accuracy of order 2- 伪 is obtained. In this paper we use a class of weighted discrete formulas with 2-order approximation accuracy to approximate the fractional derivative of Riemann-Liouville and then form a second-order discrete scheme. In addition, the finite element method combined with the discrete technique of WS G D operator is used to numerically solve a class of fractional partial differential equations in space and time, and the time second order approximation results are obtained. Firstly, for nonlinear time fractional order Cable equation, we use a two-step scheme with parameter 伪 to approximate the integer order time derivative at tk- 伪 / 2 time point, and use a weighted discrete scheme with second-order accuracy to approximate the Riemann-Liouville fractional derivative. The Galerkin finite element method is used to discretize the spatial direction to form the time second order approximation finite element numerical method. Furthermore, we give some numerical theories, prove the stability of the scheme, and obtain the error estimation results with the accuracy of O (t ~ (2)) ~ (1) t ~ (-1) ~ (-1) based on L _ 2 norm. In order to illustrate the correctness of the theoretical results, numerical verification is given through several numerical examples. Secondly, a class of partial differential equations with space-time fractional derivatives is numerically solved by using second-order WSGD approximation and finite element numerical method. The detailed numerical calculation process is given, and the scheme is validated by three numerical examples. The numerical results show that our numerical scheme has a second order approximate accuracy.
【学位授予单位】：内蒙古大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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