Abstract Generated abstract
The paper studies a boundary value problem for the two-dimensional heat equation in a half-plane with zero initial data and a general constant-coefficient differential boundary condition on the boundary. Using heat-potential representations, the boundary trace is reduced first to an integro-differential equation and then, through fractional inverse heat operators, to integral equations for the unknown boundary value. The analysis shows that the problem is not always solvable, and for the principal boundary operator gives a necessary and sufficient condition: the roots of the associated algebraic polynomial must lie between the branches of the hyperbola \(y^2 - x^2 = 1\) in the complex plane. Under this condition, the general problem is reduced to a Volterra-type integral equation with an estimable kernel, allowing solution by successive approximations.
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MATHEMATICAL PHYSICS
E. I. KIM
ON THE CONDITIONS FOR SOLVABILITY OF A BOUNDARY-VALUE PROBLEM FOR THE HEAT-CONDUCTION EQUATION
(Presented by Academician I. M. Vinogradov on 4 V 1961)
- Find a solution of the heat-conduction equation
\[ \frac{\partial u}{\partial t} = a^2\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) \tag{1} \]
in the domain \(D\) \((0<x<\infty,\ -\infty<y<+\infty,\ 0<t<T_0)\), satisfying the initial condition
\[ u(x,y,0)=0 \tag{2} \]
and the boundary condition
\[ \left. \sum_{k=0}^{m}\sum_{j=0}^{k} a_{kj}\frac{\partial^k u}{\partial x^j\,\partial y^{k-j}} \right|_{x=0} = f(y,t), \tag{3} \]
where \(a_{kj}\) are constants, and \(f(y,t)\) is a known function which, together with its derivative with respect to \(y\), satisfies the inequality:
\[ |f(y,t)|,\ |f'_y(y,t)| \leq M e^{\delta^2 y^2}. \tag{4} \]
We shall seek a solution \(u(x,y,t)\), continuous together with its derivatives up to order \(m\) with respect to \(x\) and \(y\) in \(\overline D\), and such that
\[ \left| \frac{\partial^k u}{\partial x^j\,\partial y^{k-j}} \right| \leq M_1 e^{\delta^2 r^2} \quad (j=0,1,\ldots,k;\ k=0,1,\ldots,m), \tag{5} \]
where \(r=\sqrt{x^2+y^2}\); \(M,\ M_1,\ \delta^2\) are certain constants, and \(T_0\) is also a constant satisfying the inequality
\[ T_0<\frac{1}{4a^2\delta^2}. \tag{6} \]
In the present work we shall show that the posed problem does not always have a solution, in contrast to the one-dimensional problem \((^1)\), and we shall indicate conditions under which our problem has a solution.
- It is obvious that if the posed problem has a solution, then it can always be expressed by the formula*
\[ u(x,y,t) = \int_0^t d\tau \int_{-\infty}^{+\infty} \psi(\eta,\tau) \frac{x}{4a^2\pi(t-\tau)^2} \exp\left[ -\frac{x^2+(y-\eta)^2}{4a^2(t-\tau)} \right]d\eta, \tag{7} \]
where
\[ \psi(\eta,\tau)=u(+0,\eta,\tau). \tag{8} \]
From the continuity of the derivatives of the function \(u(x,y,t)\) up to order \(m\) with respect to \(x,y\) in \(\overline D\) there follows the existence and continuity of the derivatives
* We assume that \(m\) is an even number. If \(m\) is an odd number, then instead of (7) we take the simple-layer potential.
\[
\frac{\partial^{m-k/2}\psi}{\partial y^{m-k}\,\partial t^{k/2}}\quad (k=0,2,4,\ldots,m),
\]
where
\[
\left|F^j\left[\psi_\eta^{(k-j)}\right]\right|\le M_2 e^{\delta_2\eta^2}
\quad
\left(j=0,1,\ldots,k;\quad k=0,1,\ldots,\frac m2\right);
\tag{9}
\]
\[
F^j\left[\psi_\eta^{(k-j)}\right]_{\tau=0}=0
\quad
\left(j=0,1,\ldots,k;\quad k=0,1,\ldots,\frac m2\right),
\tag{10}
\]
where
\[
F=\frac{\partial}{\partial t}-a^2\frac{\partial^2}{\partial\eta^2},\qquad
F\left[F^{k-1}\right]=F^k,\qquad F^0=1,
\tag{11}
\]
and conversely, if the function \(\psi(\eta,\tau)\) satisfies inequalities (9) and conditions (10), then, under condition (6), the function \(u(x,y,t)\) defined by formula (7) satisfies equation (1), the initial condition (8), and conditions (5). Therefore, in (7) we choose the function \(\psi(\eta,\tau)\) from the class of functions satisfying inequalities (9) and conditions (10) so that the function \(u(x,y,t)\) satisfies the boundary condition (3).
First we consider the simplest problem, i.e., the problem with boundary condition
\[
\sum_{k=0}^{m} a_{m,k}\left.\frac{\partial^m u}{\partial x^k\partial y^{m-k}}\right|_{x=0}=\varphi(y,t).
\tag{12}
\]
Differentiating (7) successively and taking (10) into account, we obtain
\[
\left.\frac{\partial^{2n-1+k}u}{\partial x^{2n-1}\partial y^k}\right|_{x=0}
=-\frac{1}{a^{2n}}\int_0^t d\tau\int_{-\infty}^{+\infty}
F^n\left[\psi_\eta^{(k)}\right]g(y-\eta,t-\tau)\,d\eta;
\tag{13}
\]
\[
\left.\frac{\partial^{2n+k}u}{\partial x^{2n}\partial y^k}\right|_{x=0}
=-\frac{1}{a^{2n}}F^n\left[\psi_\eta^{(k)}\right],
\tag{14}
\]
where
\[
g(y-\eta,t-\tau)=\frac{1}{2\pi a(t-\tau)}
\exp\left[-\frac{(y-\eta)^2}{4a^2(t-\tau)}\right].
\tag{15}
\]
Substituting the found limiting values (13) and (14) into (12), we obtain the following integro-differential equation:
\[
\sum_{k=0}^{m/2}\frac{a_{m,2k}}{a^{2k}}F^k\left[\psi_y^{(m-2k)}\right]
-
\sum_{k=1}^{m/2}\frac{a_{m,2k-1}}{a^{2k}}
\int_0^t d\tau\int_{-\infty}^{+\infty}
F^k\left[\psi_\eta^{(m-2k+1)}\right]g(y-\eta,t-\tau)\,d\eta
=\varphi(y,t).
\tag{16}
\]
- To simplify the last equation we introduce the operators
\[ F^{-1}[\psi]=\int_0^t d\tau\int_{-\infty}^{+\infty} \psi(\eta,\tau)G(y-\eta,t-\tau)\,d\eta,\qquad F^{-k}=F^{-1}\left[F^{-k+1}\right], \tag{17} \]
where
\[ G(y-\eta,t-\tau)=\frac{1}{2a\sqrt{\pi(t-\tau)}} \exp\left[-\frac{(y-\eta)^2}{4a^2(t-\tau)}\right]. \]
By direct calculation we verify that the operators \(F^{-1}\) and \(F\) are mutually inverse under condition (10).
In passing, let us introduce one more operator:
\[
F^{-1/2}[\psi]=\int_0^t d\tau\int_{-\infty}^{+\infty}
\psi(\eta,\tau)g(y-\eta,t-\tau)\,d\eta.
\tag{18}
\]
It is easy to see that
\[ F^{-1/2}\left[F^{-1/2}[\psi]\right]=F^{-1}[\psi]. \tag{19} \]
Applying these operators successively, we obtain the following formulas useful to us:
\[ F^{-k}[\psi]=\int_{0}^{t}d\tau\int_{-\infty}^{+\infty} \psi(\eta,\tau)\frac{(t-\tau)^{k-1}}{(k-1)!}\, G(y-\eta,t-\tau)\,d\eta, \]
\[ F^{-k-1/2}[\psi]=\int_{0}^{t}d\tau\int_{-\infty}^{+\infty} \psi(\eta,\tau)\frac{(t-\tau)^{k-1/2}}{\Gamma(k+1/2)}\, G(y-\eta,t-\tau)\,d\eta. \]
- Apply the operator \(F^{-m/2}\) to equation (16). Then we obtain the following equivalent equation
\[ a_{m,m}\psi(y,t)+\sum_{k=1}^{m}(-1)^k a_{m,m-k}a^k \int_{0}^{t}d\tau\int_{-\infty}^{+\infty}\psi_{\eta}^{(k)}(\eta,\tau)\times \]
\[ \times \frac{(t-\tau)^{k/2-1}}{\Gamma(k/2)}\, G(y-\eta,t-\tau)\,d\eta = a^m F^{-m/2}[\varphi]=\varphi_1(y,t). \tag{20} \]
This equation has been well studied by us \((^2)\).
In order that equation (20) have a solution in the class of ordinary functions satisfying (9) and (10), it is necessary and sufficient that the roots of the algebraic equation
\[ \sum_{k=0}^{m} a_{m,k}z^k=0 \tag{21} \]
lie between the branches of the hyperbola
\[ y^2-x^2=1 \tag{22} \]
in the complex \(z\)-plane.
Let \(q_1,q_2,\ldots,q_\nu\) be the roots of equation (21) with multiplicities \(n_1,n_2,\ldots,n_\nu\), and let
\[ \frac{x^m}{\displaystyle\sum_{k=0}^{m}a_{m,k}x^k} = 1+\sum_{k=1}^{\nu}\sum_{j=1}^{n_k}\beta_{k,j}\, \frac{1}{(x-q_k)^j}. \tag{23} \]
If \(q_1,q_2,\ldots,q_\nu\) lie between the branches of the hyperbola (22), then the solution of equation (20) is written in the form
\[ \psi(y,t)=\varphi_1(y,t)+ \sum_{k=1}^{\nu}\sum_{j=1}^{n_k}\beta_{k,j} \int_{0}^{t}d\tau\int_{-\infty}^{+\infty} w_{q_k}^{(j-1)}(y-\eta,t-\tau;q_k)\, \varphi'_{1\eta}(\eta,\tau)\,d\eta, \tag{24} \]
where
\[ w(y,t;q_k)=-\frac{1}{2\pi t}e^{-y^2/4a^2t} - \]
\[ -\frac{a q_k}{2\sqrt{\pi t}\,(1+q_k^2)^{1/2}} \frac{\partial}{\partial y} \left[ e^{-y^2/4a^2(1+q_k^2)t} \operatorname{erfc}\left( -\frac{q_k}{\sqrt{1+q_k^2}}\frac{y}{2a\sqrt{t}} \right) \right]. \tag{25} \]
- Now consider the problem with boundary condition (3). To this end, we rewrite it in the following form:
\[ \left. \sum_{k=0}^{m} a_{m,k}\frac{\partial^{m}u}{\partial x^{k}\partial y^{m-k}} \right|_{x=0} = f(x,y)- \left. \sum_{k=0}^{m-1}\sum_{j=0}^{k} a_{k,j}\frac{\partial^{k}u}{\partial x^{j}\partial y^{k-j}} \right|_{x=0} = \varphi_1(y,t). \tag{26} \]
On the basis of (13) and (14),
\[ \begin{aligned} \varphi_1(y,t) &=a^{m}\int_{0}^{t}dt_1\int_{-\infty}^{+\infty} f(y_1,t_1)\frac{(t-t_1)^{m/2-1}}{\Gamma(m/2)} G(y-y_1,t-t_1)\,dy_1 \\ &\quad -\sum_{k=0}^{m-1}\sum_{j=0}^{k} a_{k,j}a^{m-j} \int_{0}^{t}dt_1\int_{-\infty}^{+\infty} \psi_{y_1}^{(k-j)}(y_1,t_1) \frac{(t-t_1)^{m/2-j/2-1}}{\Gamma(m/2-j/2)} G(y-y_1,t-t_1)\,dy_1 . \end{aligned} \tag{27} \]
Substituting (27) into (24), we again obtain the integro-differential equation
\[ \psi(y,t)=f_1(y,t)- \sum_{n=0}^{m-1}\sum_{i=0}^{n}a_{n,i} \int_{0}^{t}d\tau\int_{-\infty}^{+\infty} \psi_{\eta}^{(n-i)}(\eta,\tau)K_i(y-\eta,t-\tau)\,d\eta, \tag{28} \]
where
\[ f_1(y,t)=\int_{0}^{t}d\tau\int_{-\infty}^{+\infty} f(\eta,\tau)K_0(y-\eta,t-\tau)\,d\eta, \]
\[ \begin{aligned} K_i(y-\eta,t-\tau) &= \frac{a^{m-i}(t-\tau)^{(m-i)/2-1}}{\Gamma((m-i)/2)} G(y-\eta,t-\tau) \\ &\quad -a^{m-i}\sum_{k=1}^{\nu}\sum_{j=1}^{n_k}\beta_{kj}\, \frac{\partial^{j}}{\partial\eta\,\partial q_k^{\,j-1}} \int_{0}^{t}dt_1\int_{-\infty}^{+\infty} \frac{(t_1-\tau)^{(m-i)/2-1}}{\Gamma((m-i)/2)} \\ &\qquad\qquad\qquad\qquad\qquad\qquad \times \omega(y-y_1,t-t_1;q_k) G(y_1-\eta,t_1-\tau)\,dy_1 . \end{aligned} \tag{29} \]
Transferring the derivatives from the unknown function in (28) onto \(K_i\), we obtain the final integral equation
\[ \psi(y,t)=f_1(y,t)+ \int_{0}^{t}d\tau\int_{-\infty}^{+\infty} K(y-\eta,t-\tau)\psi(\eta,\tau)\,d\eta, \tag{30} \]
where
\[ K(y-\eta,t-\tau)= \sum_{n=1}^{m-1}\sum_{i=0}^{n}(-1)^{n+i}a_{n,i} \frac{\partial^{\,n-i}}{\partial\eta^{\,n-i}} K_i(y-\eta,t-\tau). \]
A detailed investigation shows that
\[ \left|K(y-\eta,t-\tau)\right| \leq \frac{M}{t-\tau}e^{-\delta_1^{2}(y-\eta)^2/(t-\tau)}. \tag{31} \]
Consequently, equation (30) can be integrated by the method of successive approximations.
Kharkov Polytechnic Institute
named after V. I. Lenin
Received
4 V 1961
REFERENCES CITED
\(^{1}\) A. N. Tikhonov, Matem. sborn., 26 (68), 1 (1950).
\(^{2}\) E. I. Kim, DAN, 125, No. 4 (1959).
\(^{4}\) S. L. Sobolev, Equations of Mathematical Physics, 1947.