《一类热传导方程的多点源反演研究硕士》由会员分享，可在线阅读，更多相关《一类热传导方程的多点源反演研究硕士（49页珍藏版）》请在金锄头文库上搜索。
1、研 究 生 毕 业 论 文论文题目一类热传导方程的多点源反演研究 / 摘 要本文主要研究热传导方程的多点源反演问题唯一性、稳定性和反演算法, 其中热传导方程的源项与Dirac分布的线性组合有关, 即源项为。首先, 在第一类Dirichlet边界条件下, 研究了由表面热流密度来反演热传导方程多点源的唯一性和稳定性, 即反演点源个数、热源强度组合系数和点源位置的唯一性和稳定性。利用热变换方法将热传导方程反问题转化为等价的双曲型方程反问题,然后通过分析等价的双曲型方程反问题得到原反问题的唯一性和条件稳定性结论。其次,根据本文得到的由热流密度反演多点源的唯一性和稳定性结果,以及已有的由部某个点的温度值
2、和由某个时刻的温度场反演多点源的唯一性和稳定性结果,采用遗传算法对上述三个反问题进行数值模拟。数值模拟结果表明,遗传算法对由某个时刻的温度场多点源的反问题是有效的,而对另外两个反问题则不够理想。 进一步,分析了产生这种现象的原因。再次,利用非数值优化中影响显著的模拟退火算法对本文涉及的三个反问题进行了数值模拟。与利用遗传算法模拟的结果相同,数值模拟结果显示模拟退火算法对由某个时刻的温度场多点源的反问题是有效的,而对另外两个反问题则不够理想。 最后,针对本文的研究容和数值模拟结果,结合热传导反问题的研究现状,提出一些待解决的问题和进一步研究的思路。关键词：热传导方程,源项反问题,遗传算法,模拟退
3、火算法T THESIS：Reconstruction of Point Sources for a Class of Heat Conduction Equation SPECIALIZATION: Computational MathematicsPOSTGRADUATE: WangYanSUPERVISOR: XuDingHua WangZeWenABSTRACTThis thesis mainly studies the uniqueness, conditional stability and inverse algorithms of the inverse point source
4、s problems for a class of heat conduction equation, where the source term is related with the linear combination of the Dirac-distribution, that is, the source term is .First, the uniqueness and stability of the inverse point sources problem, that is to reconstruct the numbers of point sources, the
5、combination coefficients and the locations, is studied for a heat conduction equation with the Dirichlet boundary condition. Using the method of the heat transformation, the inverse problem is transformed into an equivalent problem which is an inverse hyperbolic equation problem. Then, the uniquenes
6、s and conditional stability are obtained by analyzing the inverse hyperbolic equation problem.Secondly, based on the results of the uniqueness and conditional stability for the above three inverse problem, the simulations are given by the method of genetic algorithm. The results of the numerical sim
7、ulation show that genetic algorithm can reconstruct effectively the point sources from the measurements of, but it is not ideal for the other two inverse problems. Furthermore, the reason which causes the above results is given.Thirdly, simulated annealing algorithm, which is a famous non-numerical
8、optimization method, is used to recover the above three inverse point sources problem of heat conduction equation. Finally, in view of the content of the paper and the results of the numerical simulation, which combine with the status quo of the inversion problem of the heat conduction equation, som
9、e conclusions are proposed as to as some problems which need to be researched in the future work.Keywords:Heat conduction equation,Inverse source problem, Genetic Algorithm, Simulated Annealing Algorithm目录摘要IABSTRACTII第一章 引言及问题的提出1 1.1反问题的研究意义11.2 抛物型方程源项反演的研究动态21.3 本文所要研究的主要容21.4 本文的整体框架5第二章 反问题的理论结果62.1反问题的等价转化62.2 反问题的辅助定理82.3 反问题的唯一性112.4 反问题的稳定性12第三章 基于遗传算法的多点源反演153.1 遗传算法153.2 遗传算法的原理153.3 遗传算法的步骤163.4 遗传算法的算法设计163.5 基于遗传算法的多点源反演203.6 结论28第四章 基于模拟退火算法的多点源反演294.1 模拟退火算法的基本理论
