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MATHEMATICAL PHYSICS
I. G. PORTNOV
EXACT SOLUTION OF THE FREEZING PROBLEM WITH AN ARBITRARY CHANGE OF TEMPERATURE AT A FIXED BOUNDARY
(Presented by Academician I. I. Artobolevskii, 22 XII 1961)
The freezing problem \((^1)\) reduces to the solution of the system of equations
\[ \frac{\partial u_i}{\partial t} = \frac{a_i t_0}{x_0^2}\frac{\partial^2 u_i}{\partial x^2} \quad (i=1,\ 0<x<\xi;\quad i=2,\ \xi<x<\infty) \tag{1} \]
with the additional conditions
\[ u_1(0,t)=f(t)<0,\quad u_2(x,0)=g(x)>0,\quad \xi(0)=0 \tag{2} \]
and the conditions at the moving boundary of the phase transition \(\xi(t)\):
\[ u_1\big|_{\xi}=u_2\big|_{\xi}=0,\quad \frac{\partial u_1}{\partial x}\bigg|_{\xi} -\frac{k_2}{k_1}\frac{\partial u_2}{\partial x}\bigg|_{\xi} = \frac{\lambda \rho x_0^2}{k_1 t_0}\frac{d\xi}{dt}. \tag{3} \]
Here \(x,t\) are dimensionless coordinate and time; \(u_i(x,t)\) is the temperature; \(k_i,\ a_i\) are the coefficients of thermal conductivity and temperature conductivity; \(\lambda\) is the latent heat of phase transition; \(\rho\) is the density; the indices \(1,2\) refer respectively to the solid and liquid phases; \(x_0\) and \(t_0\) are arbitrarily chosen scales for measuring length and time.
Let us first consider the case when \(u_2\equiv 0\). We make the change of variables
\[ t=\tau,\quad x=y\xi(\tau). \tag{4} \]
In the new variables equation (1) has the form (we omit the index \(i=1\) and put \(a_1t_0=x_0^2\))
\[ \xi^2(\tau)\frac{\partial u}{\partial \tau} - \xi(\tau)\xi'(\tau)y\frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial y^2}. \tag{5} \]
We apply to equation (5) the two-sided Laplace transform \((^2)\) with respect to the variable \(y\), regarding \(\tau\) as a parameter. After passing to the transforms we obtain
\[ \xi^2(\tau)\frac{\partial v}{\partial \tau} + \xi(\tau)\xi'(\tau)p\frac{\partial v}{\partial p} = p^2v. \tag{6} \]
The general solution of equation (6), under the assumption that
\[ \xi(0)\xi'(0)\ne 0, \tag{7} \]
is given by the expression
\[ v(p;\tau)=e^{p^2\tau/\xi^2(\tau)}F\left(\frac{p}{\xi}\right), \tag{8} \]
where \(F(p/\xi)\) is an arbitrary function.*
We represent the function \(F\) in the form
\[ F(p/\xi)=F_1(p/\xi)+F_2(p/\xi). \tag{9} \]
* The special solution of equation (6), when condition (7) is not satisfied \((^3)\), is obtained from the general one by a limiting transition and corresponds to the case of compatible initial and boundary conditions.
and we shall assume that \(F_1\) and \(F_2\) are images of arbitrary one-sided functions \(\varphi_1\) and \(\varphi_2\)
\[ F_1(p/\xi)=\varphi_1(\xi y)\,U(\xi y), \qquad F_2(p/\xi)=\varphi_2(\xi y)\,U(-\xi y). \tag{10} \]
Applying the multiplication rule for images \((^2)\), we find
\[ u(y;\tau)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty} \exp\left[-\frac{\xi^2(y-\eta)^2}{4\tau}\right] \{\varphi_1(\xi\eta)U(\xi\eta)+\varphi_2(\xi\eta)U(-\xi\eta)\} \frac{\xi\,d\eta}{2\sqrt{\tau}} . \tag{11} \]
Thus the temperature field is determined by the Poisson integral, while the arbitrary functions \(\varphi_1(x)U(x)\), \(\varphi_2(x)U(-x)\) (\(U(x)\) is the unit function) give the initial temperature distribution for \(x>0\) and \(x<0\), respectively.
To determine the unknown functions \(\varphi_1,\varphi_2\) and \(\xi\), we shall use conditions (2)—(3). Setting \(\vartheta=2\sqrt{\tau}\) in (11), we obtain
\[ u(0,\vartheta)=f(\vartheta)= \frac{1}{\sqrt{\pi}}\int_{0}^{\infty} e^{-\alpha^2}\varphi_1(\alpha\vartheta)\,d\alpha +\frac{1}{\sqrt{\pi}}\int_{-\infty}^{0} e^{-\alpha^2}\varphi_2(\alpha\vartheta)\,d\alpha; \tag{12} \]
\[ 0=\int_{-\xi/\vartheta}^{\infty} e^{-\beta^2}\varphi_1(\xi+\vartheta\beta)\,d\beta +\int_{-\infty}^{-\xi/\vartheta} e^{-\beta^2}\varphi_2(\xi+\vartheta\beta)\,d\beta; \tag{13} \]
\[ \frac{\sqrt{\pi}\lambda\rho a}{k}\frac{d\xi}{d\vartheta}= \int_{-\xi/\vartheta}^{\infty} e^{-\beta^2}\varphi_1(\xi+\vartheta\beta)\,\beta\,d\beta +\int_{-\infty}^{-\xi/\vartheta} e^{-\beta^2}\varphi_2(\xi+\vartheta\beta)\,\beta\,d\beta . \tag{14} \]
Regarding the prescribed function \(f(\vartheta)\) as an analytic function of \(\vartheta\),
\[ f(\vartheta)=\sum_{k=0}^{\infty} f_k \vartheta^k, \tag{15} \]
we shall seek the solution of the problem in the form
\[ \varphi_i(\omega)\sum_{k=0}^{\infty}\varphi_{ik}\omega^k \quad (i=1,2), \qquad \xi(\vartheta)=\sum_{k=1}^{\infty}\xi_k\vartheta^k . \tag{16} \]
Using asymptotic estimates of the corresponding integrals, it can be shown that the power series for \(\varphi_i(\omega)\) have infinite radii of convergence. This entails uniform convergence of the indicated series and their boundedness for all \(\omega\), which, in turn, leads to uniform convergence in \(\vartheta\) of the integrals entering (12)—(14), and makes it possible to differentiate these integrals with respect to the parameter \(\vartheta\) any number of times. From the boundedness of the right-hand side of (14) follows the existence and boundedness of \(d\xi/d\vartheta\). Differentiating both sides of (14) with respect to \(\vartheta\), we obtain that \(d^2\xi/d\vartheta^2\) is also bounded. Continuing to differentiate both sides of (14) with respect to \(\vartheta\), we find that \(\xi(\vartheta)\) is an analytic function of \(\vartheta\).
Thus, the series (16) give an exact solution of the problem under consideration. The coefficients \(\varphi_{in}\), \(\xi_{n+1}\) for \(n\geqslant0\) are determined by a system of equations whose right-hand sides depend on the known coefficients \(f_k\) and \(\varphi_{ik},\xi_{k+1}\) with index \(k<n\):
\[ [P_{1n}+(-1)^n P_{2n}]\varphi_{1n}+(h_1-h_2)\xi_{n+1} = \]
\[ =(-1)^n\frac{2f_n P_{2n}}{P_n} -\frac{1}{n!}(A_{1n}-A_{2n}+B_{1n}-B_{2n}), \]
\[ [Q_{1n}+(-1)^n Q_{2n}]\varphi_{1n} -\bigl[(n+1)H+(h_1-h_2)\xi_1\bigr]\xi_{n+1} = \]
\[ =(-1)^n\frac{2f_n Q_{2n}}{\Phi_n} -\frac{1}{n!}(C_{1n}-C_{2n}-D_{1n}+D_{2n}), \tag{17} \]
\[ \varphi_{2n}=(-1)^n\left(\frac{2f_n}{\Phi_n}-\varphi_{1n}\right) \qquad (H=\sqrt{\pi}\lambda\rho a/k). \]
Here
\[ \begin{gathered} A_{in}=\int_{-\xi_1}^{\infty} e^{-\beta^2} R_{in}\,d\beta,\qquad B_{in}=\sum_{k=0}^{n-1}(S_{ik})_0^{(n-k-1)}-n!\,h_i\xi_{n+1},\\ C_{in}=\int_{-\xi_1}^{\infty} e^{-\beta^2} R_{in}\,\beta\,d\beta, \end{gathered} \tag{18} \]
\[ \begin{gathered} D_{in}=\sum_{k=0}^{n-1}(\gamma S_{ik})_0^{(n-k-1)}-n!\,h_i\xi_1\xi_{n+1},\qquad \Phi_k=\frac{2}{\sqrt{\pi}}\int_0^\infty e^{-\alpha^2}\alpha^k\,d\alpha,\qquad \gamma=\frac{\xi}{\vartheta}; \\ P_{in}=\int_{-\xi_1}^{\infty} e^{-\beta^2}(\xi_1+\beta)^n\,d\beta,\qquad Q_{in}=\int_{-\xi_1}^{\infty} e^{-\beta^2}(\xi_1+\beta)^n\beta\,d\beta, \end{gathered} \tag{19} \]
\[ \begin{gathered} R_{in}=\left(\frac{d^n\varphi_i}{d\vartheta^n}\right)_0-n!\,\varphi_{in}(\xi_1+\beta)^n,\qquad S_{ik}=\left.\gamma' e^{-\gamma^2}\frac{d^k\varphi_i}{d\vartheta^k}\right|_{\beta=-\gamma},\\ h_i=e^{-\xi_1^2}\varphi_{i0},\qquad \nu=(-1)^{i-1}\ (i=1,2),\qquad B_{i0}=-h_i\xi_1=\frac{1}{\xi_1}D_{i0}. \end{gathered} \]
Hence, for \(n=0\) we obtain the known solution corresponding to the case \(f(\vartheta)=f_0=\mathrm{const}\) (¹):
\[ \varphi_{10}=f_0\left(1-\frac{1}{\operatorname{erf}\xi_1}\right),\qquad \varphi_{20}=f_0\left(1+\frac{1}{\operatorname{erf}\xi_1}\right),\qquad -\lambda\rho\sqrt{\pi}\xi_1=\frac{k f_0}{a}\,\frac{e^{-\xi_1^2}}{\operatorname{erf}\xi_1}. \tag{20} \]
The coefficients of the series (16) representing the solution are given by formulas (20), and for \(n>0\) by the solutions of the system of linear algebraic equations (17). The uniqueness of the solution constructed is obvious. For \(n=1\) these coefficients have the form\(^*\)
\[ \xi_2=-\frac{\sqrt{\pi}e^{-\xi_1^2}f_1}{2H\xi_1(3+2\xi_1^2)},\qquad \varphi_{11}\sqrt{\pi}f_1\operatorname{erfc}\xi_1+\frac{4H}{\sqrt{\pi}}(1+\xi_1^2)\xi_2,\qquad \varphi_{21}=\varphi_{11}-2\sqrt{\pi}f_1. \tag{21} \]
Suppose that \(f(\vartheta)\), continued to the plane of a complex variable, is an entire function of exponential type. Then \(\varphi_i(\omega)\) will also be entire functions of exponential type and, consequently, \(F_i\) will be functions regular at infinity. Their Laurent expansions in a neighborhood of the infinitely distant point have the form
\[ F_i(\xi/p)=F_{i0}+F_{i1}\xi/p+F_{i2}(\xi/p)^2+\cdots \tag{22} \]
Using the expansions (22) and applying the integration rule (²) for \(i=1\) when \(\operatorname{Re}p>0\), and for \(i=2\) when \(\operatorname{Re}p<0\), we find
\[ u(y,\vartheta)=\frac12\sum_{k=0}^{\infty}\vartheta^k\{F_{1k} i^k\operatorname{erfc}(-\gamma y)+(-1)^{k+1}F_{2k}i^k\operatorname{erfc}(\gamma y)\}. \tag{23} \]
where
\[ i^n\operatorname{erfc}x=\int_x^\infty i^{\,n-1}\operatorname{erfc}\alpha\,d\alpha,\qquad i^0\operatorname{erfc}x=\operatorname{erfc}x=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-\alpha^2}\,d\alpha, \tag{24} \]
\[ 2n i^n\operatorname{erfc}x=i^{\,n-2}\operatorname{erfc}x-2x i^{\,n-1}\operatorname{erfc}x. \]
The solution in the form (23) is convenient for small values of the parameter \(\gamma=\xi/\vartheta\). By virtue of conditions (2), (3), \(u(0,\tau)=f(\tau)\), \(u(1,\tau)=0\). Consequently,
\[
-f(\tau)=\frac12\sum_{k=0}^{\infty}\vartheta^k\{F_{1k}[i^k\operatorname{erfc}(-\gamma)-i^k\operatorname{erfc}0]
\]
\[
+(-1)^{k+1}F_{2k}[i^k\operatorname{erfc}\gamma-i^k\operatorname{erfc}0]\}.
\tag{25}
\]
\[ \text{———} \]
\(^*\) The following coefficients are not given because of lack of space. From this solution, by a limiting passage, exact solutions with compatible initial and boundary conditions are obtained. In particular, when \(f_0\to 0\) and \(\xi_k=0\) \((k\geq 3)\), one obtains a solution with a uniformly moving wave \(\xi=\beta t\).
Differentiating (24) with respect to \(y\), we obtain, for \(y=1\),
\[ \left. \frac{\partial u}{\partial y} \right|_{y=1} = \frac{1}{2}\gamma \left\{ \frac{2}{\sqrt{\pi}} e^{-\gamma^2}(F_{10}+F_{20}) + \sum_{k=1}^{\infty} \vartheta^k \left[ F_{1k} i^{k-1}\operatorname{erfc}(-\gamma) + (-1)^k F_{2k} i^{k-1}\operatorname{erfc}\gamma \right] \right\}. \tag{26} \]
From (25) and (26) it follows that, to within terms of order \(\gamma^2\vartheta\) and higher,
\[ \left. \frac{\partial u}{\partial y} \right|_{y=1} = - f(\tau). \tag{27} \]
Substituting (27) into the last of conditions (3) and integrating with respect to \(\tau\), we obtain a convenient approximate formula expressing the law of motion of the phase-transition surface for the case when terms of order \(\gamma^2\vartheta\) may be neglected:
\[ \xi(\tau) = \left[ -\frac{2k}{\lambda \rho a} \int_{0}^{\tau} f(\tau)\,d\tau \right]^{1/2}. \tag{28} \]
In the general case, when
\[ u_2(x,0)=g(x)=\sum_{k=0}^{\infty}\frac{1}{k!}g_k x^k \qquad (g(x)\ne 0,\ x>0), \tag{29} \]
the exact solution in series is constructed analogously. An approximate solution for small values of the parameter \(\gamma\) can be obtained by using conditions (29), (3) and a solution in a form analogous to (23):
\[ u_2(y,\vartheta) = \frac{1}{2}\sum_{k=0}^{\infty} \vartheta^k \left(\frac{a_2}{a_1}\right)^{k/2} \left[ H_{1k}i^k\operatorname{erfc} \left( -\gamma \sqrt{\frac{a_1}{a_2}}\,y \right) + (-1)^{k+1}H_{2k}i^k\operatorname{erfc} \left( \gamma \sqrt{\frac{a_1}{a_2}}\,y \right) \right]. \tag{30} \]
Considering (29) as a limiting condition as \(y\to\infty,\ \tau\to 0\), we obtain that \(H_{1k}=g_k\).
Taking into account the second of conditions (3), we find, to within terms of order \(\gamma^2\vartheta\) and higher,
\[ \left. \frac{\partial u_2}{\partial y} \right|_{y=1} = \frac{1}{2}\sum_{k=0}^{\infty} \vartheta^k \left(\frac{a_2}{a_1}\right)^{k/2} (-1)^k \frac{H_{2k}-(-1)^k g_k}{2^k \Pi(k/2)}. \tag{31} \]
Substituting (27) and (31) into the last of conditions (3) and integrating with respect to \(\tau\), we obtain
\[ \frac{\xi^2}{4\tau} = \gamma^2 = -\frac{k_1}{2\lambda\rho a_1} \left\{ \frac{1}{\tau}\int_{0}^{\tau} f(\tau)\,d\tau + \frac{k_2}{k_1} \sum_{n=0}^{\infty} (-1)^n \bigl(H_{2n}-(-1)^n g_n\bigr) \left(\frac{a_2}{a_1}\right)^{n/2} \frac{\tau^{n/2}}{(n+2)\Pi(n/2)} \right\}. \tag{32} \]
The approximate values \(\gamma_0^{(n)}\) and \(H_{2n}\) are determined from relation (32) and the identity \(u_2(1,\vartheta)\equiv 0\).
Received
21 III 1961
REFERENCES
- A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Moscow—Leningrad, 1951.
- B. van der Pol, H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transformation, Moscow, 1952.
- V. V. Stepanov, A Course of Differential Equations, Moscow—Leningrad, 1953.