MATHEMATICAL PHYSICS

L. I. RUBINSHTEIN

ON ONE VARIANT OF THE STEFAN PROBLEM*

(Presented by Academician S. L. Sobolev, 16 X 1961)

A one-phase problem is considered which is a generalization of the classical Stefan problem; namely, one seeks \(y(t)\) and \(u(x,t)\) such that \(u\) and \(\partial u/\partial x\) are defined and continuous in the closure of the parabolic domain
\(D:\{0<x<y(t);\ t>0\}\), while \(u\) satisfies inside \(D\) the equation

\[ a^2\frac{\partial^2 u}{\partial x^2} + F\left(x,t,u,\frac{\partial u}{\partial x},y,\dot y\right) = \frac{\partial u}{\partial t} \qquad (a=\mathrm{const}) \tag{1} \]

and on the boundary of \(D\) the conditions

\[ \left.\frac{\partial u}{\partial x}\right|_{x=0} = \left.f(t,u)\right|_{x=0}; \qquad u(x,0)=\varphi(x); \tag{2} \]

\[ u(y(t),t)=\psi[y(t)]; \qquad \frac{dy}{dt} = \left.\Phi\left(t,u,\frac{\partial u}{\partial x},y\right)\right|_{x=y}; \qquad y(0)=l>0. \]

It is assumed that \(F\), \(f\), and \(\Phi\) have bounded partial derivatives with respect to all their arguments, and that \(\psi\) and \(\varphi\) have bounded derivatives respectively up to the second and up to the third order inclusive. Under these assumptions the following theorems are proved:

Theorem 1. Put

\[ w(t)=u(0,t); \qquad q(x,t)=\frac{\partial}{\partial x}u(x,t); \qquad v(t)=\frac{\partial}{\partial x}u(y(t),t); \qquad \dot y(s)=z(t). \tag{3} \]

Then, if \(u,y\) are a solution of problem (1), (2), then \(u,w,q,v,y,z\) are solutions of the system of integral equations

\[ \begin{aligned} u={}& -a^2\int_0^t f(\tau,w)G(x,0,t-\tau)\,d\tau +\int_0^l \varphi(\xi)G(x,\xi,t)\,d\xi \\ &+\int_0^t d\tau\int_0^{y(\tau)} F(\xi,\tau,\ldots)G(x,\xi,t-\tau)\,d\xi \\ &+a^2\int_0^t [v(\tau)+\psi(y(\tau))z(\tau)] G(x,y(\tau),t-\tau)\,d\tau \\ &-a^2\int_0^t \psi(y(\tau)) \frac{\partial}{\partial \xi}G(x,y(\tau),t-\tau)\,d\tau \equiv U(t,x\mid u,w,q,v,y,z); \end{aligned} \tag{4} \]

\[ w=\left.U\right|_{x=0}; \]

\[ \begin{aligned} q(x,t)={}& a^2\int_0^t f(\tau,w)\frac{\partial}{\partial \xi} g(x,0,t-\tau)\,d\tau +\int_0^l \varphi(\xi)g(x,\xi,t)\,d\xi \\ &-\int_0^t d\tau\int_0^{y(\tau)} F(\xi,\tau,\ldots)\frac{\partial}{\partial \xi} g(x,\xi,t-\tau)\,d\xi \\ &-a^2\int_0^t v(\tau)\frac{\partial}{\partial \xi} g(x,y(\tau),t-\tau)\,d\tau + \end{aligned} \]

* Reported at the Fourth All-Union Mathematical Congress.

\[ +\int_0^t \dot\psi(y(\tau))z(\tau)g(x,y(\tau),t-\tau)\,d\tau \equiv Q(t,x\mid u,w,q,v,y,z); \]

\[ v(t)=2Q\bigm|_{x=y(t)}, \qquad y(t)=l+\int_0^t z(\tau)\,d\tau; \qquad z(t)=\Phi(t,\psi(y),v,y). \]

Here \(g\) and \(G\) are the Green’s functions of the first and second boundary-value problems for the heat equation on the half-line \(x>0\), i.e.

\[ g(x,\xi,t)=E(x-\xi,a^2t)-E(x+\xi,a^2t); \]

\[ G(x,\xi,t)=E(x-\xi,a^2t)+E(x+\xi,a^2t); \qquad E(x,t)=\frac{e^{-x^2/4t}}{2\sqrt{\pi t}}. \]

Theorem 2. If \(u,w,q,v,z,y\) are solutions of system (4) satisfying a Lipschitz condition in all arguments, then \(u,y\) are solutions of the original problem (1), (2).

Theorem 3. The unique solution of system (4) satisfying the conditions of Theorem 2 can be constructed by Picard’s method of successive approximations, if the process is begun with arbitrary functions \(u_0,w_0,v_0,q_0,y_0,z_0\) having bounded partial derivatives in all their arguments and satisfying the conditions

\[ \dot\varphi(0)=f(0,w_0); \qquad \dot\varphi(l)=q_0(l,0)-v_0(0); \qquad \dot y_0(t)=z_0(t). \tag{5} \]

The iterative process converges on some small time interval \((0,T)\), where \(T\) depends on the estimates of \(f,F,\Phi,\psi,\varphi\), on the estimates of all the above-mentioned partial derivatives of \(f,\ldots,\varphi\), and on \(l=y(0)>0\), with \(T\to0\) as \(l\to0\).

Theorem \(3^*\). A solution of system (4) can be constructed for sufficiently small \(t>0\) by the finite-difference method.

The computational scheme in this case is analogous to that which was used earlier [1].

Theorem 4. The solution of system (4) is stable in the following sense: suppose that, along with \(a^2,l,F,f,\varphi,\psi,\Phi\), there are given \(a^{2*},l^*,F^*,f^*,\varphi^*,\psi^*,\Phi^*\) possessing the required differentiability properties. Let \(u^*,q^*,w^*,v^*,y^*,z^*\) be the solution of the system obtained from (4) by replacing \(a^2,l,\ldots,\Phi\) by \(a^{2*},l^*,\ldots,\Phi^*\). Then for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, on some small interval \((0,T)\) independent of \(\varepsilon\), the inequalities

\[ |u(x,t)-u^*(x,t)|<\varepsilon; \qquad |q(x,t)-q^*(x,t)|<\varepsilon \]

\[ \text{for } 0\le x\le \min(y(t),y^*(t)); \]

\[ |v(t)-v^*(t)|<\varepsilon; \qquad |y(t)-y^*(t)|<\varepsilon; \qquad |z(t)-z^*(t)|<\varepsilon \tag{6} \]

hold as soon as the moduli of the differences \(a^2\) and \(a^{2*}\); \(l\) and \(l^*\); \(F,f,\Phi\) and respectively \(F^*,f^*,\Phi^*\) and all their partial derivatives of first order; \(\psi,\psi^*\) and their derivatives up to second order; \(\varphi,\varphi^*\) and their derivatives up to third order inclusive do not exceed \(\delta\).

The results remain valid for the analogous problem with any finite number of phase-separation boundaries, provided that all phases coexist at the initial moment.

Computing Center
of the Latvian State University
named after Pēteris Stučka

Received
27 IX 1961

REFERENCES

1. L. I. Rubinshtein, Izv. Vyssh. uchebn. zaved., Mathematics, No. 4 (1958).