两类时间分数阶扩散方程不适定问题的正则化方法和算法

发布时间：2018-05-11 21:46

本文选题：时间分数阶扩散方程 + 未知源识别　；参考：《兰州理工大学》2017年硕士论文

【摘要】：本文考虑两类时间分数阶扩散方程反问题,分别是未知源识别问题和反演初值问题.这两类问题都是不适定问题,它们的解(如果存在)不连续依赖于测量数据.本文第二章考虑一般有界区域上变系数时间分数阶扩散方程未知源识别问题,这是一个不适定问题.本文采用Landweber迭代法恢复问题的不适定性,并且给出在先验和后验两种正则化参数选取规则下的收敛性估计.最后,数值实验表明本文所用的Landweber迭代法对解决此类问题是有效的.第三章讨论一般有界区域上变系数齐次时间分数阶扩散方程反演初值问题,这是一个不适定问题.本文应用Landweber迭代法求解该问题,并且证明在先验和后验两种正则化参数选取规则下的收敛性估计.最后,数值结果表明Landweber迭代法对解决此类问题是有效和稳定的.第四章考虑高维变系数非齐次时间分数阶扩散方程反演初值问题,这个问题是不适定的.本文利用拟逆方法得到问题的正则解,并且给出先验和后验两种正则化参数选取规则下精确解与正则解之间的收敛性估计.最后,在一维和二维两种情形下的数值例子表明拟逆方法对解决此类问题是有效的.其中,本文所解决的变系数非齐次时间分数阶扩散方程反演初值问题,是一个比较新颖的不适定问题.本文中的理论结果和数值结果都能充分有效的说明所采用的正则化方法能够很好地求解给定的不适定问题.
[Abstract]:In this paper, we consider two kinds of inverse problems of fractional diffusion equations in time, which are unknown source identification problem and inverse initial value problem, respectively. Both of these problems are ill-posed problems whose solutions (if any) are discontinuous dependent on measured data. In chapter 2, we consider the problem of identifying unknown sources of time fractional diffusion equations with variable coefficients in a general bounded domain, which is an ill-posed problem. In this paper, the Landweber iterative method is used to restore the ill-posed property of the problem, and the convergence estimates are given under the rules of the selection of priori and posteriori regularization parameters. Finally, numerical experiments show that the Landweber iterative method used in this paper is effective in solving such problems. In chapter 3, we discuss the inverse initial value problem of homogeneous time fractional diffusion equations with variable coefficients in a general bounded domain, which is an ill-posed problem. In this paper, the Landweber iterative method is used to solve the problem, and the convergence estimates are proved under the rules of the priori and posteriori regularization parameter selection. Finally, the numerical results show that the Landweber iterative method is effective and stable for solving such problems. In chapter 4, we consider the inverse initial value problem of nonhomogeneous time fractional diffusion equation with high dimensional variable coefficient, which is ill-posed. In this paper, we obtain the regular solution of the problem by using the quasi-inverse method, and give the convergence estimates between the exact solution and the regular solution under the rule of selecting two regularization parameters a priori and a posteriori. Finally, numerical examples in one and two dimensions show that the quasi-inverse method is effective for solving this kind of problem. The inhomogeneous time fractional diffusion equation with variable coefficients is a novel ill-posed problem. The theoretical and numerical results in this paper can fully and effectively illustrate that the regularization method used in this paper can solve the given ill-posed problem well.
【学位授予单位】：兰州理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

【参考文献】