Monotonicity of input–output mappings in inverse coefficient and source problems for parabolic equations

Abstract

This article presents a mathematical analysis of input–output mappings in inverse coefficient and source problems for the linear parabolic equation u t = ( k ( x ) u x ) x + F ( x , t ) , ( x , t ) ∈ Ω T : = ( 0 , 1 ) × ( 0 , T ] . The most experimentally feasible boundary measured data, the Neumann output (flux) data f ( t ) : = − k ( 0 ) u x ( 0 , t ) , is used at the boundary x = 0 . For each inverse problems structure of the input–output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input–output mappings. Some numerical applications are presented.