一类热传导方程的多点源反演研究

东华理工大学硕士学位论文ABSTRACTABSTRACTThis thesis mainly studies the uniqueness,conditional stability and inverse algorithmsof the inverse point sources problems for a class of heat conduction equation,where thesource term is related with the linear combination of the Dirac-distribution,that is,thesource term is F(x.((s).First,the uniqueness and stability of the inverse point sources problem,that is toreconstruct the numbers of point sources n,the combination coefficients a,and thelocations s,,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 anequivalent problem which is an inverse hyperbolic equation problem.Then,the uniquenessand conditional stability are obtained by analyzing the inverse hyperbolic equationproblem.Secondly,based on the results of the uniqueness and conditional stability for theabove three inverse problem,the simulations are given by the method of genetic algorithm.The results of the numerical simulation show that genetic algorithm can reconstructeffectively the point sources from the measurements of u(x,T),but it is not ideal for theother two inverse problems.Furthermore,the reason which causes the above results isgiven.Thirdly,simulated annealing algorithm,which is a famous non-numericaloptimization method,is used to recover the above three inverse point sources problem ofheat conduction equation.Finally,in view of the content of the paper and the results of thenumerical simulation,which combine with the status quo of the inversion problem of theheat conduction equation,some conclusions are proposed as to as some problems whichneed to be researched in the future work.Keywords:Heat conduction equation,Inverse source problem,GeneticAlgorithm,Simulated Annealing Algorithm