Stefan problem, inverse

The problem to determine from the motion of the boundary between two phases of some material the change in the boundary conditions or in the coefficients of the differential equation for, e.g., the temperature of the material considered (cf. Stefan problem). For example, find the flow $q(t)=\partial u(0,t)/\partial x$ from the conditions:

$$\frac{\partial u}{\partial t}=a^2\frac{\partial^2u}{\partial x^2},\quad0<x<\xi(t),\quad0<t\leq T,$$

$$u(x,0)=\phi(x),\quad0\leq x\leq\xi_0;\quad u(\xi(t)-0,t)=\mu(t),$$

$$\gamma(t)\frac{d\xi(t)}{dt}=-\frac{\partial u(\xi(t)-0,t)}{\partial x};\quad\xi(0)=\xi_0>0,$$

where $\phi(x)$, $\mu(t)$, $\gamma(t)\geq\gamma_0>0$, and $\xi(t)$ are given functions. For an approximate solution of this problem, the variational method is often used (see [1]).

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The inverse Stefan problem is obviously related with the non-characteristic Cauchy problem for the corresponding parabolic operator. A formula for solutions to inverse Stefan problems was derived in [a1].

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