Identifying an unknown term in an inverse problem of linear diffusion equations
The purpose of this paper is to identify an unknown function in a linear diffusion equation from overspecified data measured on the boundary. For \(T>0\), let \(Q_ T=\{(x,t):\) \(0<x<1\), \(0<t<T\}\) and consider the problem of determining an unknown function p(s) which is defined on [0,M], \(M>0\), and a function u(x,t) satisfying (1) \(\partial_ tu(x,t)=\kappa \partial_{xx}u(x,t)\) in \(Q_ T\), (2) \(u(x,0)=0\), \(0<x<1\), (3) \(- \partial_ xu(0,t)+p(u(0,t))=0,\) \(0<t<T\), (4) \(\partial_ xu(1,t)=f(t)\), \(0<t<T\), and the overspecified condition (5) \(u(0,t)=g(t)\), \(0<t<T\), where \(\kappa\) is a given constant and f(t) and g(t) are given functions. We study the inverse problem (1)-(5) and prove the existence and uniqueness solution to this problem. We identify the function p when (p,u) is the solution to the inverse problem (1)-(5).
scientific article; zbMATH DE number 4085306