This paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity k⁢(x)k(x) and radiative coefficient q⁢(x)q(x) in the 1D heat equation ut=(k⁢(x)⁢ux)x-q⁢(x)⁢uu_{t}=(k(x)u_{x})_{x}-q(x)u from the most available Dirichlet and Neumann boundary measured outputs. The Neumann-to-Dirichlet and Neumann-to-Neumann operators Φ⁢[k,q]⁢(t):=u⁢(ℓ,t;k,q)\Phi[k,q](t):=u(\ell,t;k,q), Ψ⁢[k,q]⁢(t):=-k⁢(0)⁢ux⁢(0,t;k,q)\Psi[k,q](t):=-k(0)u_{x}(0,t;k,q) are introduced, and main properties of these operators are derived. Then the Tikhonov functional J⁢(k,q)=12⁢∥Φ⁢[k,q]-ν∥L2⁢(0,T)2+12⁢∥Ψ⁢[k,q]-φ∥L2⁢(0,T)2J(k,q)=\tfrac{1}{2}\lVert\Phi[k,q]-\nu\rVert^{2}_{L^{2}(0,T)}+\tfrac{1}{2}\lVert\Psi[k,q]-\varphi\rVert^{2}_{L^{2}(0,T)} of two functions k⁢(x)k(x) and q⁢(x)q(x) is introduced, and an existence of a quasi-solution of the inverse coefficient problem is proved. An explicit formula for the Fréchet gradient of the Tikhonov functional is derived through the weak solutions of two appropriate adjoint problems.

中文翻译：

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从Dirichlet和Neumann边界测得的输出同时识别热方程中的热导率和辐射系数

本文研究了一维热方程ut =（k⁢（x）⁢ux的同时确定导热系数k⁢（x）k（x）和辐射系数q⁢（x）q（x）的反系数问题）xq⁢（x）⁢uu_{t} =（k（x）u_ {x}）_ {x} -q（x）u来自最可用的Dirichlet和Neumann边界测得的输出。Neumann-to-Dirichlet和Neumann-to-Neumann算子Φ⁢[k，q]⁢（t）：=u⁢（ℓ，t; k，q）\ Phi [k，q]（t）：= u （\ ell，t; k，q），Ψ⁢[k，q]⁢（t）：=-k⁢（0）⁢ux⁢（0，t; k，q）\ Psi [k，q]（ t）：=-k（0）u_ {x}（0，t; k，q）被引入，并且得出这些算子的主要性质。则Tikhonov函数J⁢（k，q）=12⁢∥Φ⁢∥[k，q]-ν∥L2⁢（0，T）2 +12⁢∥Ψ⁢[k，q]-φ∥L2⁢（ 0，T）2J（k，q）= \ tfrac {1} {2} \ lVert \ Phi [k，q]-\ nu \ rVert ^ {2} _ {L ^ {2}（0，T）} + \ tfrac {1} {2} \ lVert \ Psi [k，q]-\ varphi \ rVert ^ {2} _ {L ^ {2}（0，T）}两个函数k⁢（x）k（ x）和q⁢（x）q（x）被引入，并证明了反系数问题的拟解的存在性。通过两个适当的伴随问题的弱解，得出了Tikhonov泛函的Fréchet梯度的显式公式。