An Inverse Boundary Value Problem For The Heat Equation With Robin Boundary Condition

Inverse problems of mathematical physics have become important research topics in the fields of computational mathematics,applied mathematics and system science.They have important and extensive application backgrounds in life science,material science,signal processing and industrial control,etc.The inverse problem for the heat equation is an important class of inverse problems of mathematical physics.The inverse boundary value problems for the heat equation with Dirichlet or Neumann boundary condition have been studied thoroughly.However,the inverse boundary value problems with Robin boundary condition are still in the initial stages of the researches.In this paper,we consider the inverse problem for a mixed boundary problem of the heat conduction equation.We consider a doubly connected bounded domain which the exterior boundary condition is of Neumann type and the interior boundary condition is of Robin type.The exterior boundary is denoted by ?2,and the interior boundary is denoted by ?1 Assumed that ?2 is known,and the Robin coefficients A depends only on the spatial variables.First,we reformulate the inverse problem as a nonlinear ill-posed operator equation.Then,we linearize it by Newton method,and solve it using regularized least square method.In this thesis,we focus on the numerical algorithm and its numerical implementation of the above inverse problem.The thesis is organized as follows:In Chapter 1,we describe the related research background and existing works on inverse boundary value problems for the heat equation,and then summarize our main work.In Chapter 2,the boundary integral equation method is used to solve the direct problem.The solution of the heat equation is expressed in the form of single-layer poten-tials.Then a system of boundary integral equations is obtained by the jump relations of the single-layer potentials at the boundary.The collocation method is used to discrete the system of integral equations,and the density function is obtained by solving this system.Thus the numerical solution of the direct problem is obtained,and the measurement data on the exterior boundary of the inverse problem is simulated numerically.In Chapter 3,a numerical method to solve the inverse problem is established.The inverse boundary value problem is reformulated as a nonlinear ill-posed operator equation,and then solved by the Newton method.In the iteration process,the total differentiation of the operator in the sense of Frechet derivative needs to be calculated.Therefore,we firstly give the theoretical derivation of total differentiation of this operator,and then introduce a numerical scheme for computing the total differentiation of this operator.In Chapter 4,the numerical realization scheme of the inverse problem considered is given,and several numerical examples are provided to illustrate the efficiency and effectiveness of the numerical algorithm.In Chapter 5,the main works and novel contributions of this thesis are summarized,and the related research issues in future are prospected.