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山东大学学报（理学版） ›› 2018, Vol. 53 ›› Issue (10): 61-71.doi: 10.6040/j.issn.1671-9352.0.2018.093

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基于变分理论与时间相关的抛物型反源问题

甄苇苇,曾剑,任建龙   

  1. 兰州交通大学数理学院, 甘肃 兰州 730070
  • 收稿日期:2018-03-09 出版日期:2018-10-20 发布日期:2018-10-09
  • 作者简介:甄苇苇(1994— ),女,硕士研究生,研究方向为数学物理反问题. E-mail:1549070004@qq.com
  • 基金资助:
    国家自然科学基金资助项目(11261029,11461039);甘肃省自然科学基金资助项目(145RJZA124)

Time dependent parabolic inverse source problem based on variational theory

ZHEN Wei-wei, ZENG Jian, REN Jian-long   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Received:2018-03-09 Online:2018-10-20 Published:2018-10-09

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摘要： 研究了一类变系数抛物型方程的源项重构问题,这里的源项仅与时间相关。与以往工作不同,文中的附加条件是关于空间变量积分后得到的,这种类型的附加条件有利于消除随机选择所带来的误差,但同时会导致很多分析方法不可用。基于变分理论,首先给出了变分公式,并利用变分公式证明了解的唯一性;其次给出了时间离散模型和基于线性离散化的变分形式,导出了一系列先验估计,并证明了弱解的存在性。

关键词: 反源问题, 唯一性, 弱解存在性, 变分理论

Abstract: The problem of source term reconstruction for a class of parabolic equation with variable coefficients is studied, and the source term here is only time dependent. Different from the previous work, the additional condition in this paper is about the integration of the spatial variable, this type of additional conditions can help to eliminate the errors caused by random selection. At the same time, it will lead to many analysis methods unavailable. Based on the variational theory, firstly, the variational formula is given and the uniqueness of the solution is proved by the variational formula; Secondly, the time discrete model is given, and based on the variational form of linear discretization, a series of priori estimates are derived and the existence of weak solutions is proved.

Key words: uniqueness, variational theory, existence of weak solution, inverse source problem

中图分类号: 

  • O175.26
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