Computation and Control II pp 247-261 | Cite as
Parameter Estimation in the Stefan Problem
Chapter
Abstract
We consider the problem of estimating an unknown time-dependent diffusion coefficient in a Stefan problem. We shall treat the one-dimensional case here, for which the partial differential equation model is given by
$$
\begin{gathered}
u_t = \left( {a\left( t \right)u_x } \right)_x {\text{ 0 < t}} \leqslant {\text{T,0 < x < s(t),}} \hfill \\
{\text{a}}\left( t \right)u_x \left( {0,t} \right) = g\left( t \right){\text{ 0 < t < T,}} \hfill \\
u = \left( {s\left( t \right),t} \right) = 0{\text{ 0}} \leqslant {\text{t}} \leqslant {\text{T,}} \hfill \\
{\text{u}}\left( {x,0} \right) = \phi \left( x \right){\text{ 0}} \leqslant {\text{x}} \leqslant {\text{b,}} \hfill \\
\end{gathered} $$
(1.1)
$$\begin{gathered}
\dot s\left( t \right) = - \gamma a\left( t \right)u_x \left( {s\left( t \right),t} \right){\text{ 0 < t}} \leqslant {\text{T,}} \hfill \\
{\text{s}}\left( 0 \right) = b. \hfill \\
\end{gathered} $$
(1.2)
Keywords
Integral Equation Initial Guess Fredholm Integral Equation Quadrature Point Stefan Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer Science+Business Media New York 1991