Two dimensional determination of source terms in linear parabolic equation from the final overdetermination

Xiujin Miao,Zhenhai Liu

doi:10.1186/s13662-015-0401-2

Abstract

A case of steady-case heat flow through a plane wall, which can be formulated as \(u_{t}(x,y,t)- \operatorname{div} (k(x,y) \nabla u(x,y,t)) = F(x,y,t)\) with Robin boundary condition \(-k(1,y)u_{x}(1,y,t)= \nu_{1} [u(1,y,t)-T_{0}(t)]\), \(-k(x,1)u_{y}(x,1,t)= \nu_{2} [u(x,1,t)-T_{1}(t)]\), where \(\omega:=\{F(x,y,t);T_{0}(t);T_{1}(t)\}\) is to be determined, from the measured final data \(\mu_{T}(x,y)=u(x,y,T)\) is investigated. It is proved that the Fréchet gradient of the cost functional \(J(\omega )=\|\mu_{T}(x,y)-u(x,y,T;\omega)\|^{2}\) can be found via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove the existence of a quasi-solution of the inverse problem. A steepest descent method with line search, which produces a monotone iteration scheme based on the gradient, is formulated. Some convergence results are given.

Highlights

Consider the one dimensional physical system in Figure, where the left of the solid is full of hot gas
5 Conclusions This paper presents a theoretical study of a case of steady-case heat flow through a plane wall with the two dimensional Robin boundary condition
The inverse problem consists of determining the source terms ω := {F(x, y, t); T (t); T (t)} by using observational measurements of the final state uT (x, y) = u(x, y, T)

Summary

Introduction

Consider the one dimensional physical system in Figure , where the left of the solid is full of hot gas. Whenever a temperature gradient exists in the solid medium, heat will flow from the higher-temperature region to the lower-temperature region. According to Fourier’s law, for a homogeneous, isotropic solid, the following equation holds: q(x, t) = –kuxx(x, t), ). where q(x, t) represents heat flow per unit time, per unit area of the isothermal surface in the direction of decreasing temperature, u(x, t) is the temperature distribution in the solid, and k is called the thermal conductivity. There may be a heat source g(x, t) in the solid. Under these conditions, the physical system can be formulated as

Methods

Conclusion