Abstract Generated abstract
This paper presents a method for solving a mixed boundary value problem for a coupled system of two Fourier type equations arising in thermodiffusion. Using Fourier and Laplace transforms, the solution with prescribed initial data and provisional boundary values is represented through integrals involving theta functions and unknown boundary functions. The boundary conditions are then integrated in time to avoid singular derivatives at the endpoints, leading to a system of Volterra integral equations of the first kind for the unknown boundary functions. The paper shows that this system can be reduced to generalized Abel equations and then to Volterra equations of the second kind with continuous kernels, after which the solution follows by standard integral equation methods.
Full Text
Doklady of the Academy of Sciences of the USSR
1958. Vol. 120, No. 2
MATHEMATICS
A. P. PRUDNIKOV
SOLUTION OF A MIXED BOUNDARY-VALUE PROBLEM IN THE THEORY OF THERMODIFFUSION
(Presented by Academician A. A. Dorodnitsyn, December 3, 1957)
In the present article a method is proposed for solving a mixed boundary-value problem for a system of two equations of Fourier type, occurring in the theory of thermodiffusion (1). The solution is represented in the form of a sum of integrals containing arbitrary functions of the integration parameter. The latter are determined from the boundary conditions by solving a system of Volterra integral equations.
1°. Consider the system of equations
\[ \frac{\partial u_1}{\partial t} = a_1 \frac{\partial^2 u_1}{\partial x^2} + b_1 \frac{\partial u_2}{\partial t}, \qquad \frac{\partial u_2}{\partial t} = a_2 \frac{\partial^2 u_2}{\partial x^2} + b_2 \frac{\partial^2 u_1}{\partial x^2}, \tag{1} \]
\[ (a_1+b_1b_2+a_2)^2 \ne 4a_1a_2 \]
with initial conditions
\[ u_i(x,0)=f_i(x), \qquad 0<x<l, \tag{2} \]
and boundary conditions
\[ \begin{aligned} a_{11}u_1(0,t)+a_{12}u_2(0,t)+a_{13}\frac{\partial u_1(0,t)}{\partial x} +a_{14}\frac{\partial u_2(0,t)}{\partial x}&=\chi_1(t),\\ a_{21}u_1(0,t)+a_{22}u_2(0,t)+a_{23}\frac{\partial u_1(0,t)}{\partial x} +a_{24}\frac{\partial u_2(0,t)}{\partial x}&=\chi_2(t),\\[4pt] a_{31}u_1(l,t)+a_{32}u_2(l,t)+a_{33}\frac{\partial u_1(l,t)}{\partial x} +a_{34}\frac{\partial u_2(l,t)}{\partial x}&=\chi_3(t),\\ a_{41}u_1(l,t)+a_{42}u_2(l,t)+a_{43}\frac{\partial u_1(l,t)}{\partial x} +a_{44}\frac{\partial u_2(l,t)}{\partial x}&=\chi_4(t), \end{aligned} \tag{3} \]
where \(a_i, b_i, a_{kl}\) are constant coefficients; \(f_i(x), \chi_k(t)\) \((i=1,2;\ k,l=1,2,3,4)\) are arbitrary bounded integrable functions of their arguments. We shall regard the unknown functions
\[ u_i(0,t)=\varphi_i(t), \qquad u_i(l,t)=\psi_i(t) \tag{4} \]
as given; their determination will be considered later.
By the combined application of the Fourier and Laplace transforms, the solution of the problem (1), (2), (4) can be obtained in the form
\[ u_i(x,t)=\sum_{s,j}^{1,2}\left(A_{sj}^i V_{sj}+B_{sj}^i W_{sj}\right), \tag{5} \]
where \(A_{sj}^{i},\ B_{sj}^{i}\) are constant coefficients;
\[ V_{sj} = -\frac{1}{l}\int_{0}^{t} \frac{\partial}{\partial x} \vartheta_{3}\left[ \frac{x}{2l},\,\frac{\mu_s(t-\tau)}{l^{2}} \right]\varphi_j(\tau)\,d\tau + \frac{1}{l}\int_{0}^{t} \frac{\partial}{\partial x} \vartheta_{3}\left[ \frac{l-x}{2l},\,\frac{\mu_s(t-\tau)}{l^{2}} \right]\psi_j(\tau)\,d\tau, \]
\[ W_{sj} = \frac{1}{2l}\int_{0}^{l} \left\{ \vartheta_{3}\left[ \frac{x-\xi}{2l},\,\frac{\mu_s t}{l^{2}} \right] - \vartheta_{3}\left[ \frac{x+\xi}{2l},\,\frac{\mu_s t}{l^{2}} \right] \right\} f_j(\xi)\,d\xi, \]
\[ \mu_s = \frac{1}{2} \left[ a_1+b_1b_2+a_2 + (-1)^{s+1} \sqrt{(a_1+b_1b_2+a_2)^2-4a_1a_2} \right], \]
\[ \vartheta_{3}(x,t) = \frac{1}{\sqrt{\pi t}} \sum_{n=-\infty}^{\infty} \exp\{-(x+n)^2/t\}. \]
\(2^\circ\). Let us proceed to determine the unknown functions \(\varphi_i(t)\), \(\psi_i(t)\) from the boundary conditions (3). Differentiating (5) with respect to \(x\) and replacing the second derivative with respect to \(x\) of the theta function \(\vartheta_3\) by its derivative with respect to \(t\), we obtain
\[ \frac{\partial u_i(x,t)}{\partial x} = \sum_{s,j=1}^{2} \left(A_{sj}^{i}P_{sj}+B_{sj}^{i}Q_{sj}\right), \tag{6} \]
where
\[ P_{sj} = -\frac{1}{\mu_s l} \int_{0}^{t} \frac{\partial}{\partial(t-\tau)} \vartheta_{3}\left[ \frac{x}{2l},\,\frac{\mu_s(t-\tau)}{l^{2}} \right]\varphi_j(\tau)\,d\tau + \]
\[ + \frac{1}{\mu_s l} \int_{0}^{t} \frac{\partial}{\partial(t-\tau)} \vartheta_{3}\left[ \frac{l-x}{2l},\,\frac{\mu_s(t-\tau)}{l^{2}} \right]\psi_j(\tau)\,d\tau, \tag{7} \]
\[ Q_{sj} = \frac{1}{2l}\int_{0}^{l} \frac{\partial}{\partial x} \left\{ \vartheta_{3}\left[ \frac{x-\xi}{2l},\,\frac{\mu_s t}{l^{2}} \right] - \vartheta_{3}\left[ \frac{x+\xi}{2l},\,\frac{\mu_s t}{l^{2}} \right] \right\} f_j(\xi)\,d\xi. \]
Taking into account the above representation of the function \(\vartheta_3\), it is not difficult to see that the derivative \(\dfrac{\partial}{\partial(t-\tau)}\vartheta_3\), which exists for all \(x\ne 0\) in the first and for all \(x\ne l\) in the second of the integrals on the right-hand side of equality (7), tends to infinity as \(t^{-3/2}\) when \(t\to 0\), if \(t\to \tau\) with \(x=0\) in the first and with \(x=l\) in the second of the mentioned integrals.
To eliminate this difficulty, let us replace conditions (3) by others, which are obtained from (3) by integration with respect to \(t\). Integrating (6) with respect to \(t\), we find
\[ \int_{0}^{t} \frac{\partial u_i(x,\tau)}{\partial x}\,d\tau = \sum_{s,j=1}^{2} \left(A_{sj}^{i}P_{sj}^{*}+B_{sj}^{i}Q_{sj}^{*}\right), \tag{8} \]
where
\[ P_{sj}^{*} = \int_{0}^{t} P_{sj}(x,\tau)\,d\tau, \qquad Q_{sj}^{*} = \int_{0}^{t} Q_{sj}(x,\tau)\,d\tau. \]
Integrating (7) with respect to \(t\) and changing the order of integration by means of Dirichlet’s formula, we obtain
\[ P_{sj}^{*}(x,T)=-\frac{1}{\mu_s l}\int_0^T \varphi_j(\tau)\,d\tau \int_\tau^T \frac{\partial}{\partial(t-\tau)} \vartheta_3\left[\frac{x}{2l},\frac{\mu_s(t-\tau)}{l^2}\right]\,dt+ \]
\[ +\frac{1}{\mu_s l}\int_0^T \psi_j(\tau)\,d\tau \int_\tau^T \frac{\partial}{\partial(t-\tau)} \vartheta_3\left[\frac{l-x}{2l},\frac{\mu_s(t-\tau)}{l^2}\right]\,dt. \]
Hence it follows that for all \(0<x<l\) we have
\[ P_{sj}^{*}(x,T)=-\frac{1}{\mu_s l}\int_0^T \varphi_j(\tau)\, \vartheta_3\left[\frac{x}{2l},\frac{\mu_s(T-\tau)}{l^2}\right]\,d\tau+ \]
\[ +\frac{1}{\mu_s l}\int_0^T \psi_j(\tau)\, \vartheta_3\left[\frac{l-x}{2l},\frac{\mu_s(T-\tau)}{l^2}\right]\,d\tau. \tag{9} \]
By direct calculations, as was done in \((^2)\), it can be shown that formula (9) is valid for \(x=0,\ x=l\). Finding expressions (8) for these values of \(x\) and substituting them into the equations obtained after integrating (3) with respect to \(t\) leads to a system of Volterra integral equations of the first kind
\[ \sum_{\beta=1}^{4}\int_0^t K_{\alpha\beta}(t,\tau)\,\varphi_\beta(\tau)\,d\tau =g_\alpha(t)\qquad(\alpha=1,2,3,4), \tag{10} \]
where \(\varphi_3=\psi_1,\ \varphi_4=\psi_2\). The functions \(K_{\alpha\beta}(t,\tau)\) and \(g_\alpha(t)\) have a form that makes it possible to reduce system (10) to a system of generalized Abel integral equations \((^3)\), which can easily be reduced to a system of Volterra integral equations of the second kind
\[ \varphi_\alpha(t)+\sum_{\beta=1}^{4}\int_0^t \theta_{\alpha\beta}(t,\tau)\,\varphi_\beta(\tau)\,d\tau =\omega_\alpha(t)\qquad(\alpha=1,2,3,4), \tag{11} \]
where \(\theta_{\alpha\beta}(t,\tau)\) and \(\omega_\alpha(t)\) are continuous functions of the variables \(t\) and \(\tau\). The solution of the posed problem is obtained after determining, by a known method \((^4)\), the functions \(\varphi_\alpha(t)\) from the system of equations (11) and substituting them into equalities (5).
Computing Center
Academy of Sciences of the USSR
Received
2 XII 1957
REFERENCES
- A. V. Lykov, Heat and Mass Transfer in Drying Processes, Moscow—Leningrad, 1956.
- A. Datsev, Yearbook of Sofia University, 43, 113 (1946–1947).
- E. Goursat, A Course of Mathematical Analysis, 3, part 2, Moscow—Leningrad, 1934.
- G. M. Muntz, Integral Equations, Leningrad—Moscow, 1934.