非线性时间分数阶方程的两种数值方法

发布时间：2018-06-11 12:37

本文选题：修正分数阶扩散方程 + WSGD逼近　；参考：《内蒙古大学》2017年硕士论文

【摘要】：本文讨论了非线性时间分数阶问题的两种数值计算方法,即:修正的非线性时间分数阶扩散方程的WSGD逼近Galerkin有限元方法以及非线性分数阶常微分方程的线性插值多项式法.首先,第一种计算方法是通过WSGD逼近算法和有限元离散的方法构造出非线性时间分数阶扩散方程的数值离散格式.首先,我们形成数值格式,其中时间的整数阶导数部分用二阶向后差分的方法离散,分数阶导数部分采用WSGD算子近似,同时在空间方向上运用Galerkin有限元方法进行离散处理.其次,推导证明了有限元解的存在性和唯一性,给出所求问题在L2模意义下全离散格式的数值解与精确解之间的误差估计.最后,通过数值实验得到近似O(T2)的时间收敛阶,验证理论结论的正确性.通过数值算例也可以发现WSGD逼近算法和Galerkin有限元方法结合能够显著提高收敛精度.其次,第二种计算方法是利用线性插值多项式法构造了一类含有Hadamard部分有限积分的非线性常微分方程的数值离散格式.在时间方向上,分数阶导数利用线性插值多项式方法逼近,整数阶导数通过二阶向后差分格式离散.经过推理得到了收敛精度为O(Tmin{1+α,1+3}的误差估计结果.最后,通过数值结果和理论结果的对比,直观的说明了推理结果是正确的.
[Abstract]:In this paper, we discuss two numerical methods for nonlinear time fractional order problem, that is, WSGD approximation Galerkin finite element method for nonlinear time fractional order diffusion equation and linear interpolation polynomial method for nonlinear fractional ordinary differential equation. Firstly, the first method is to construct the numerical discretization scheme of nonlinear time fractional diffusion equation by WSGD approximation algorithm and finite element discretization method. First, we form a numerical scheme, in which the integral derivative of time is discretized by second-order backward difference, the fractional derivative is approximated by WSGD operator and the Galerkin finite element method is used in the spatial direction. Secondly, the existence and uniqueness of the finite element solution are proved, and the error estimates between the numerical solution and the exact solution of the full discrete scheme in the sense of L2 norm are given. Finally, the order of time convergence is obtained by numerical experiments, which verifies the correctness of the theoretical conclusion. Numerical examples also show that the combination of WSGD approximation algorithm and Galerkin finite element method can significantly improve the convergence accuracy. Secondly, the second method is to construct a numerical discrete scheme for a class of nonlinear ordinary differential equations with Hadamard partial finite integral by means of linear interpolation polynomial method. In time direction, fractional derivative is approximated by linear interpolation polynomial method, integer order derivative is discretized by second-order backward difference scheme. The result of error estimation with convergent accuracy of OTmin {1 伪 1 / 3} is obtained by reasoning. Finally, through the comparison between the numerical results and the theoretical results, it is intuitively proved that the reasoning results are correct.
【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】：O241.8

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