Simultaneous Inversion Of Nonlinear Heat Source Coefficient And Initial Temperature For Heat Equation

The heat conduction process has many important applications in industrial fields,such as the design of furnaces,heat transfer processes for the equipments under high temperature and high pressure(e.g.,waste heat boilers in ammonia synthesis towers and large-scale ethylene plants).For these processes,some physical parameters,such as non-homogeneous terms in the heat conduction equation,thermal conductivity,initial temperature are impossible to be measured directly.People usually apply other related measurable temperature field to firstly invert the unknown ingredients in the system to identify these parameters,and then to get the solution to the heat conduction process,which are typical inverse problems for the heat conduction equations.In natural sciences and engineering,the unknown nonlinear heat source coefficient and initial temperature in the heat conduction equation are important parameters.In this thesis,the nonlinear heat source coefficient and the initial temperature are simultaneously reconstructed from the temperature measurement values specified at two different moments.This thesis considers the transient heat transfer process by the following mathemat-ical model on a bounded domain ? in finite-dimensional spaces:where the bounded conductor S2[RN for N=1,2,3,(?)? ?C2+l(l ?(0,1)),v is the outward unit normal to the boundary(?)?,0<k0?k(x)? C1+l(?)representing the thermal conductivity,and k0 is a given positive constant.0 ?q(x)E Cl(?)and 0??(x)?l+1((?)Q)represent the domain and surface non-linear heat source coefficients,respectively,while and ?(x)represent an internal source,heat flux and initial temperature,respectively.For known functions(k(x),f(x,t),?(x,t)),we consider an inverse problem of simul-taneous reconstruction of q(x)and initial temperature ?(x)together with ?(x)from given measurement of temperature at two different instants.Compared with the existing re-searches,we adopt an alternating iterative scheme to minimize the Tikhonov regularizing functional,which is of fast calculation speed and better reconstruction effect.This thesis consists of the following five sections:·The background for the inverse heat conduction problems,the existing research works and the research contents of our thesis are given in section 1.·Some preliminary knowledge for simultaneously reconstruct the nonlinear heat source coefficient and the initial temperature are stated in section 2.·In section 3,after stating the existence and uniqueness result for the inverse prob-lem,we transform the original inverse problem into an optimization problem for the Tikhonov regularized functional,and prove the existence and uniqueness of the min-imizer of this functional.We then apply the alternative iteration scheme to solve this regularized functional with two unknowns,and the convergence of the iterative scheme is also proved.·The iterative CGM based on the sensitivity equation for the adjoint problem and the Frechet derivative with respect to q(x)and ?(x)are established,respectively,and the numerical results are presented in section 4.·In section 5,we summarize the researches of this article and give some problems that need further research.