### A two-dimensional inverse heat conduction problem for estimating heat source.

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Development of engineering structures and technologies frequently works with advanced materials, whose properties, because of their complicated microstructure, cannot be predicted from experience, unlike traditional materials. The quality of computational modelling of relevant physical processes, based mostly on the principles of classical thermomechanics, is conditioned by the reliability of constitutive relations, coming from simplified experiments. The paper demonstrates some possibilities of...

Modelling of building heat transfer needs two basic material characteristics: heat conduction factor and thermal capacity. Under some simplifications these two factors can be determined from a rather simple equipment, generating heat from one of two aluminium plates into the material sample and recording temperature on the contacts between the sample and the plates. However, the numerical evaluation of both characteristics leads to a non-trivial optimization problem. This article suggests an efficient...

Let $y\left(h\right)(t,x)$ be one solution to$${\partial}_{t}y(t,x)-\sum _{i,j=1}^{n}{\partial}_{j}\left({a}_{ij}\left(x\right){\partial}_{i}y(t,x)\right)=h(t,x),\phantom{\rule{0.166667em}{0ex}}0\<t\<T,\phantom{\rule{0.166667em}{0ex}}x\in \Omega $$with a non-homogeneous term $h$, and ${y|}_{(0,T)\times \partial \Omega}=0$, where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain. We discuss an inverse problem of determining $n(n+1)/2$ unknown functions ${a}_{ij}$ by $\{{\partial}_{\nu}y\left({h}_{\ell}\right){|}_{(0,T)\times {\Gamma}_{0}}$, $y\left({h}_{\ell}\right)(\theta ,\xb7){\}}_{1\le \ell \le {\ell}_{0}}$ after selecting input sources ${h}_{1},...,{h}_{{\ell}_{0}}$ suitably, where ${\Gamma}_{0}$ is an arbitrary subboundary, ${\partial}_{\nu}$ denotes the normal derivative, $0\<\theta \<T$ and ${\ell}_{0}\in \mathbb{N}$. In the case of ${\ell}_{0}={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $({h}_{1},...,{h}_{{\ell}_{0}})$ from a set $\mathscr{H}\subset {\left\{{C}_{0}^{\infty}((0,T)\times \omega )\right\}}^{{\ell}_{0}}$ with an arbitrarily fixed subdomain $\omega \subset \Omega $. Moreover we can take ${\ell}_{0}=(n+3)n/2$ by making special choices for ${h}_{\ell}$, $1\le \ell \le {\ell}_{0}$. The proof is...

Let y(h)(t,x) be one solution to $${\partial}_{t}y(t,x)-\sum _{i,j=1}^{n}{\partial}_{j}\left({a}_{ij}\left(x\right){\partial}_{i}y(t,x)\right)=h(t,x),\phantom{\rule{0.166667em}{0ex}}0<t<T,\phantom{\rule{0.166667em}{0ex}}x\in \Omega $$ with a non-homogeneous term h, and ${y|}_{(0,T)\times \partial \Omega}=0$, where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\{{\partial}_{\nu}y\left({h}_{\ell}\right){|}_{(0,T)\times {\Gamma}_{0}}$, $y\left({h}_{\ell}\right)(\theta ,\xb7){\}}_{1\le \ell \le {\ell}_{0}}$ after selecting input sources ${h}_{1},...,{h}_{{\ell}_{0}}$ suitably, where ${\Gamma}_{0}$ is an arbitrary subboundary, ${\partial}_{\nu}$ denotes the normal derivative, $0<\theta <T$ and ${\ell}_{0}\in \mathbb{N}$. In the case of ${\ell}_{0}={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $({h}_{1},...,{h}_{{\ell}_{0}})$ from a set $\mathscr{H}\subset {\left\{{C}_{0}^{\infty}((0,T)\times \omega )\right\}}^{{\ell}_{0}}$ with an arbitrarily fixed subdomain $\omega \subset \Omega $. Moreover we can take ${\ell}_{0}=(n+3)n/2$ by making special choices for...

In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity....

This paper is concerned with some optimal control problems for the Stefan-Boltzmann radiative transfer equation. The objective of the optimisation is to obtain a desired temperature profile on part of the domain by controlling the source or the shape of the domain. We present two problems with the same objective functional: an optimal control problem for the intensity and the position of the heat sources and an optimal shape design problem where the top surface is sought as control. The problems...