The Study Of A Class Of Inverse Point Heat Source Problem Of Parabolic Type

In this thesis we present a class of inverse point-wise heat source problem of parabolic partial differential equation, whose source term is the linear combination of some Dirac distributions. Our ultimate goal is to study thoroughly the uniqueness and stability of reconstruction of the position of each point heat source, of its combination coefficient, and as well of the number of all the point heat sources. And then we begin to try our best to search for some possible numerical solution to this inverse source problem. But it is so boring a hard work to tackle this inverse problem owing to the presence of Dirac function, which has meaning only as a functional over a proper space, and this is nearly impossible for the treatment, by way of general methods, of this inverse problem, and so at present these is no general way to deal with. This is apparently the reason why this class of inverse point source problem in resent 20 years is interesting to very fewer mathematical researchers. But here, we commit to trying the most to tackle this hard work.As far as this class of inverse problem is concerned, the first natural question to which we must give answer is whether our inverse problem has definite meaning, or in another word, is well defined under some conditions (because inverse problems are always ill defined, as contrast to direct problems), in other word, whether or not the additional condition that we specified to the model can determine uniquely the positions, the whole number, and the combination coefficient of the point heat sources altogether. Moreover, whether the perturbation of these data above can be continuously related correspondingly to the perturbation of the additional condition. After the first question's answer were arrived, the natural thing we must go under way to discuss is how to get the numerical result, which means we must give some effective means to inverse the positions, the whole number, and the combination coefficient of the point heat sources numerically.In order to give answers to these two questions, we at first convert the initial model into a standard parabolic inverse source problem by way of a linear transform, so as to accommodate our discussion that followed. Then, according to the linearity of the problem, we separate it into a parabolic direct problem, which can be solved with ease, as long as some conditions are specified to the boundary and initial values of the model, and a parabolic inverse point heat source problem with homogeneous initial-boundary values, which has a slightly simple structure and is favorable for our following treatment. After the simplified model was presented, we relate it to a kind of inverse point wave source problem through applying heat transform to it, which isequivalent to the former model, and we can achieve the uniqueness and the stability of our initial model as long as the similar assertion is derived for the latter. So it suffices to focus our full attention on the treatment of this inverse point wave source problem.To doing so, we firstly define the weak solution of the corresponding direct problem by use of the method of transposition. And then, we simplify it into an inverse initial speed problem, after applying weak Duhamers principle and some result on integral equation of Volterra type to it. When so much preliminary steps have been finished, we discuss thoroughly the unique determination through the additional condition of the initial speed, which enjoys some important information of the inverse multi-point wave source problem. Applying the result of functional analysis and Fourier analysis, we accomplished this. At last, we dealt with the stability result according to the above several equivalent inverse problems, and arrived at the stability of the inverse problem under some proper a priori estimates on the multi-point wave sources, and applying to it the Roth theorem of Diophantine approximation as well. Observing the equivalent conversions that we have sequentially applied to, the similar assertion is true to our former model.The...