基于时间分数阶扩散方程源项反演的一阶与二阶数值微分方法（英文）  

FIRST AND SECOND ORDER NUMERICAL DIFFERENTIATION BASED ON THE INVERSE SOURCE PROBLEM OF TIME-FRACTIONAL DIFFUSION EQUATION

作　　者：陈树立 阮周生[1,2] 王泽文 张文[1,2] Chen Shuli;Ruan Zhousheng;Wang Zewen;Zhang Wen(School of Science, East China University of Technology, Nanchang 330013;Institute of Science and Engineering Computing, East China University of Technology, Nanchang 330013)

机构地区：[1]东华理工大学理学院,南昌330013 [2]东华理工大学科学与工程计算研究所,南昌330013

出　　处：《高等学校计算数学学报》2019年第3期244-264,共21页Numerical Mathematics A Journal of Chinese Universities

基　　金：Supported by National Natural Science Foundation of China(Grant No.11761007,11661004);Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province(Grant No.20172BCB22019);Ground Project of Science and Technology of Jiangxi Universities(Grant No.KJLD14051);Graduate Innovation Project of East China University of Technology(DHYC-201830)

摘　　要：In this article, we consider the classically ill-posed problem of numerical differentiation in the framework of the PDEs-based numerical differentiation methods. A novel scheme for the first and second order numerical derivatives is proposed by the approach of an inverse source problem for a time-fractional diffusion equation. The numerical differentiation problem is transformed into the inverse source problem which induces a regularized optimization problem. The convergence rates of regularization solutions are derived under the a priori and a posteriori strategies for selecting regularization parameters, respectively. Finally, several examples are given to verify the efficiency and st ability of the proposed scheme.In this article,we consider the classically ill-posed problem of numerical differentiation in the framework of the PDEs-based numerical differentiation methods.A novel scheme for the first and second order numerical derivatives is proposed by the approach of an inverse source problem for a time-fractional diffusion equation.The numerical differentiation problem is transformed into the inverse source problem which induces a regularized optimization problem.The convergence rates of regularization solutions are derived under the a priori and a posteriori strategies for selecting regularization parameters,respectively.Finally,several examples are given to verify the efficiency and stability of the proposed scheme.

关 键 词：numerical DIFFERENTIATION INVERSE source problem time-fractional DIFFUSION EQUATION CONVERGENCE rate 

分 类 号：O242.2[理学—计算数学]