On The Coefficient Inversion Of A Class Of Parabolic Equation

With the promotion of many applied problems arising in engineering areas, the researches on the inverse problems governed by partial differential equations have developed rapidly in recent years, since most of the phenomena in applied areas can be described by PDEs. The reconstruction of coefficients of parabolic equation is one of the important problems. The parameters to be reconstructed often contain heat conduction coefficient, heat source coefficient and source term and so on. The input data may be boundary measurements, interior points measurements, or measurement data at given moment and so on. Generally speaking, they are ill-posed problems.In this thesis, for the following initial boundary value problem of parabolic equationwe consider the inverse problem of reconstructing the coefficient q(x) from the final over-determination u(x,T) = Z_t{x). This inverse problem is nonlinear and ill-posed due to the solution u(x,t) of direct problem depending on the coefficient q(x). On the other hand, the information of q(x) contained in u(x,T) is very weak due to the exponentially decay of u(x,t) with respect to time t. Furthermore, to be of uniqueness for this inverse problem, some a-priori restrictions should be given for initial boundary data.This paper proposes a new iteration scheme to solve this inverse problem. Different from the traditional optimization-based iteration inversion scheme, which must solve the direct problem at each iteration step simultaneously, our inversion scheme firstly convert the original problem into the following nonlinear problemby introducing the transform v(x,t) =u_t(x,t)/u(x,t)and then reconstruct q(x) from v(x,t) in terms ofTaking time t₁ ∈ (0, T) enough near to T, the above formula is approximated by...