Numerical Methods For Solving Two Types Of Inverse Problems In Mathematical Physics

In recent years, inverse problem of mathematical physics is a very active researchbranch, and it has a very important application of felds in the earth physics, materialsscience, engineering control, and biomedical, and so on. Since the wide application andthe complexity of solving this problem, it becomes a hot research topic in many disciplinesover domestic and foreign, therefore, much attention have been payed to. In this paper,diferent numerical methods have been used to solve the two types of concrete inverseproblems in mathematical physics, and numerical examples applied show that our methodsare feasibility and efectiveness.In the introduction of frst chapter, we briefy outline the inverse problems of math-ematical physical and ill-posed problem, giving several examples of inverse problem, andintroducing the present situation of the research and development of inverse problems inmathematical physics.In the second chapter, considering an inverse heat conduction problem with nonho-mogeneous and variable coefcient term and unknown Robin boundary value condition,the problem combined with the additional condition is divided into two sections-directproblem and inverse problem, using the fnite volume method and the weighted coefcientmethod to solve the two problems,respectively. The advantages of this methods lie in thedevelopment of the fnite diference method and remain some characteristics of physicalproblem, especially, it has obvious advantages in the treatment of the source term andthe unknown boundary condition in partial diferential equations.In the third chapter, considering a backward heat conduction problem with nonlinearsource term, restrained optimal perturbation method is proposed to study backward heatconduction problem, which is a brand new thought by fnding the optimal perturbationto determine unknown initial value. In order to overcome the ill-posedness of problem, aregularization is introduced into the object function, spectral projected gradient method isused to solve the optimization problem, in this paper, we analyse the sensitivity of initialvalue problem and numerical examples are given to verify the stability of this method. In the fourth chapter, the main work of this paper is summarized, and we give abrief introduction of our ongoing work and the research will be carried out in the nearfuture.