A solution technique for random and nonlinear inverse heat conduction problems
Inverse problems for the one-dimensional, inhomogeneous, nonlinear heat equation \(\partial_ tu(\omega,x,t)=\partial_ x(h(x,u(\omega,x,t))\partial_ xu(\omega,x,t))+ s(\omega,t)f(x)\) are considered in the interval \([0,1]\) (with a Neumann boundary condition at \(x=1\)). The source term \(f\) has compact support in (0,1) (\(h\) a known nonlinearity, \(s(\omega,t)\) random process, \(\omega\) random variable). The author proposes a `spatial' finite-element approach in conjunction with a Hermite expansion technique for the heat flux density to obtain a numerical procedure. The paper deals with two specific inverse questions: (1) reconstruction of temperature \(u(t,0)\) and its first derivative \(\partial_ xu(0,t)\); (2) reconstruction of the heat transfer coefficient \(H_ t\) occurring in the convective boundary condition \(h(0,u(0,t))\partial_ xu(0,t)=H_ t(\omega)(u(0,t)-T_ \infty(t))\), \(T_ \infty(t)\) a known ambient temperature, under the assumption that \(u\) is known at the initial time 0 as well as for all relevent times at a point \(x_ d\in [0,1]\) outside of the support of \(f\). The proposed procedure is supported by several numerical examples and graphs.
reconstruction of the heat transfer coefficient
scientific article; zbMATH DE number 10055
scientific article