Abstract
Full Text
UDC 532.72
HYDROMECHANICS
N. N. KOCHINA
ON ONE SOLUTION OF THE ONE-DIMENSIONAL DIFFUSION EQUATION IN A BOUNDED DOMAIN
(Presented by Academician L. I. Sedov, May 17, 1968)
A qualitative study of self-oscillations arising under certain conditions in oil production or in some electrolytic systems was given in papers \((^{1-3})\). In papers \((^{3-10})\), periodic solutions of the one-dimensional diffusion equation in a semi-infinite domain of variation of the spatial variable \(x\), connected with this problem, were obtained. Below a periodic solution is found for a certain diffusion problem in a bounded domain.
As is known \((^{11})\), the diffusion equation
\[ \partial c / \partial t = D\,\partial^{2}c / \partial x^{2} \tag{1} \]
has periodic solutions of the form
\[ Ae^{\pm \sqrt{\omega/2D}\,x}\cos\left(\mp \sqrt{\omega/2D}\,x+\omega t\right),\qquad Ae^{\pm \sqrt{\omega/2D}\,x}\sin\left(\mp \sqrt{\omega/2D}\,x+\omega t\right). \tag{2} \]
For the semi-infinite interval \(0<x<\infty\), a periodic solution of equation (1) was obtained in paper \((^{4})\).
Let us now consider the following problem: to find a periodic solution of equation (1) for the finite interval \(0<x<l\), satisfying the boundary condition
\[ c(l,t)=c^{0}. \tag{3} \]
We shall seek the function \(c(x,t)\) in the form
\[ c(x,t)=c_{0}+(c^{0}-c_{0})x/l+u(z,\tau), \tag{4} \]
where \(u(z,\tau)\) is a periodic solution of the equation
\[ \partial u/\partial \tau=\partial^{2}u/\partial z^{2} \qquad (\tau=Dt/l^{2},\ z=x/l-1) \tag{5} \]
with period \(T\), satisfying the conditions
\[ u(0,\tau)=0,\qquad \int_{0}^{T}u(z,\tau)\,d\tau=0. \tag{6} \]
Using formulas (2), (5), and (6), it is easy to verify that the desired solution can be represented in the form
\[ u(z,\tau)=\sum_{k=1}^{\infty}u_{k}(z,\tau),\qquad u_{k}(z,\tau)=A_{k}\{e^{\rho_{k}z}\cos(\omega_{k}\tau+\rho_{k}z)- \tag{7} \]
\[ -e^{-\rho_{k}z}\cos(\omega_{k}\tau-\rho_{k}z)\} +B_{k}\{e^{\rho_{k}z}\sin(\omega_{k}\tau+\rho_{k}z)-e^{-\rho_{k}z}\sin(\omega_{k}\tau-\rho_{k}z)\}, \]
where
\[ \rho_{k}=\sqrt{\pi k/T},\qquad \omega_{k}=2\pi k/T, \tag{8} \]
and \(A_{k}\) and \(B_{k}\) denote constants which must be determined from the boundary condition.
We shall now seek a periodic solution of equation (1) for the finite interval \(0<x<l\) with the boundary condition
\[ \partial c(0,t)/\partial x=\Omega[c(0,t)], \tag{9} \]
where \(\Omega\) is an \(S\)-shaped function \((^{2})\), at \(x=0\), and with condition (3) at \(x=l\).
Using formulas (4) and (5), we reduce this problem to finding a solution of the diffusion equation (5) in the form (7), where the constants \(A_k\) and \(B_k\) are found from the boundary condition
\[ \partial u(-1,\tau)/\partial z = F[u(-1,\tau)] \quad (F(u)=l\Omega(c_0+u)-c^0+c_0). \tag{10} \]
Introducing the notation
\[ \begin{aligned} \alpha_k(z)&=2(A_k\cos \rho_k z\,\operatorname{sh}\rho_k z+B_k\sin \rho_k z\,\operatorname{ch}\rho_k z),\\ \beta_k(z)&=2(-A_k\sin \rho_k z\,\operatorname{ch}\rho_k z+B_k\cos \rho_k z\,\operatorname{sh}\rho_k z), \end{aligned} \tag{11} \]
we write equation (10) in the following form:
\[ \sum_{k=1}^{\infty}\left[\alpha'_k(1)\cos\omega_k\tau+\beta'_k(1)\sin\omega_k\tau\right] = \]
\[ =F\left\{-\sum_{k=1}^{\infty}\left[\alpha_k(1)\cos\omega_k\tau+\beta_k(1)\sin\omega_k\tau\right]\right\}. \tag{12} \]
Equation (12), taking into account the notation (11), gives an infinite system of nonlinear equations for finding the constants \(A_k\) and \(B_k\) \((k=1,2,\ldots)\).
If \(F[u(-1,\tau)]=\chi(\tau)\), where \(\chi(\tau)\) is a known periodic function of \(\tau\) with period \(T\), satisfying, by virtue of (6), the condition
\[ \int_0^T \chi(\tau)\,d\tau=0, \tag{13} \]
then, expanding \(\chi(\tau)\) in a Fourier series, one can find the coefficients \(A_k\) and \(B_k\) from (12) and (11).
Let us consider in more detail the case where \(F(u)\) is a piecewise-constant function
\[ F(u)=Q_1 \quad \text{for } 0<\tau<T_1;\qquad F(u)=Q_2 \quad \text{for } T_1<\tau<T. \tag{14} \]
It follows from (6) that in this case the constants \(Q_1,Q_2,T_1\), and \(T\) must be related by
\[ Q_1T_1+Q_2(T-T_1)=0. \tag{15} \]
Formulas (12), (11), and (13) give the following expressions for the coefficients \(A_k\) and \(B_k\):
\[ A_k=\frac{(Q_1-Q_2)}{\pi k\rho_k}\sin\frac{\pi kT_1}{T} \left[ (\mu_k-\nu_k)\cos\frac{\pi kT_1}{T} -(\mu_k+\nu_k)\sin\frac{\pi kT_1}{T} \right], \]
\[ B_k=\frac{(Q_1-Q_2)}{\pi k\rho_k}\sin\frac{\pi kT_1}{T} \left[ (\mu_k+\nu_k)\cos\frac{\pi kT_1}{T} +(\mu_k-\nu_k)\sin\frac{\pi kT_1}{T} \right], \tag{16} \]
\[ \mu_k=\operatorname{ch}\rho_k\cos\rho_k/(\operatorname{ch}2\rho_k+\cos2\rho_k),\qquad \nu_k=\operatorname{ch}\rho_k\sin\rho_k/(\operatorname{ch}2\rho_k+\cos2\rho_k). \]
We now give the solution of the desired problem in another form. Suppose first that the boundary condition at \(z=-1\) has the form
\[ \partial u(-1,\tau)/\partial z=\chi(\tau), \tag{17} \]
where the periodic function \(\chi(\tau)\), for which equality (13) is satisfied, is known. Consider equation (1). The source function \(G(x,t,\xi,\tau)\) for the interval \((0<x<l)\), satisfying the conditions \(\partial G(0,t,\xi,\tau)/\partial x=G(l,t,\xi,\tau)=0\), has the form \((^{11,3})\):
\[ G(x,t,\xi,\tau)=\frac{1}{2\sqrt{\pi D(t-\tau)}}\sum_{k=-\infty}^{\infty}(-1)^k\times \]
\[ \times\left[ \exp\left(-\frac{(x-\xi-2kl)^2}{4D(t-\tau)}\right) + \exp\left(-\frac{(x+\xi-2kl)^2}{4D(t-\tau)}\right) \right]. \]
It can also be represented in the following way:
\[ G(x,t,\xi,\tau)=\frac{2}{l}\sum_{k=0}^{\infty} \exp\left[-\frac{(2k+1)^2\pi D}{4l^2}(t-\tau)\right]\times \]
\[ \times \cos\frac{(2k+1)\pi\xi}{2l}\cos\frac{(2k+1)\pi x}{2l}. \]
Using the source function, we write the solution of equation (5) for \(x=0\) with boundary conditions (6) and (17)
\[ u(0,\tau)=-\int_{-\infty}^{\tau} K(\tau-\sigma)\chi(\sigma)\,d\sigma \quad \left(K(\sigma)=\left[1+2\sum_{n=1}^{\infty}(-1)^n e^{-n^2\sigma}\right]/\sqrt{\pi\sigma}\right). \tag{18} \]
Let now, analogously to (6),
\[ \chi(\tau)=\psi_i(\tau)\quad \text{for } \alpha_i+kT<\tau<\beta_i+kT \]
\[ (i=1,2;\ k=0,\pm 1,\pm 2,\ldots) \tag{19} \]
\[ \alpha_1=pT/2,\quad \alpha_2=\beta_1=T-pT/2,\quad \beta_2=T+pT/2,\quad 0<p<1. \]
It is easy to see that
\[ u_i(0,\tau)=-\int_{\alpha_i}^{\tau}K(\tau-\sigma)\psi_i(\sigma)\,d\sigma+S_i(\tau) \quad (\alpha_i\leq \tau\leq \beta_i), \tag{20} \]
\[ S_i(\tau)=-\sum_{k=0}^{-\infty} \left\{ \int_{\beta_i-T+kT}^{\alpha_i+kT} K(\tau-\sigma)\varphi_i(\sigma)\,d\sigma + \int_{\alpha_i-T+kT}^{\beta_i-T+kT} K(\tau-\sigma)\psi_i(\sigma)\,d\sigma \right\} \]
\[ (\varphi_1=\psi_2,\ \varphi_2=\psi_1). \]
The uniform convergence of the series \(S_i(\tau)\) follows from the fulfillment of condition (13) and from the fact that \(K(\sigma)\) is a monotonically decreasing function of its argument.
Consider again the case of a piecewise-constant function \(F(u)\) of the form (14). Using the following relation from (18),
\[ f(a,b)=\int_a^b K(y)\,dy = \frac{2}{\sqrt{\pi}} \left\{ \sqrt{b}-\sqrt{a} + 2\sum_{n=1}^{\infty}(-1)^n \left[ n\sqrt{\pi}\left(\Phi\left(\frac{n}{\sqrt{b}}\right)- \Phi\left(\frac{n}{\sqrt{a}}\right)\right) + \sqrt{b}e^{-n^2/b}-\sqrt{a}e^{-n^2/a} \right] \right\} \]
\[ \left(\Phi(z)=\frac{2}{\sqrt{\pi}}\int_0^z e^{-y^2}\,dy\right), \tag{21} \]
we write the solution \(u(0,\tau)\) in the following way:
\[ u_i(0,\tau)=Q_i f(\tau-\alpha_i,0)+S_i(\tau) \quad (\alpha_i\leq \tau\leq \beta_i), \]
\[ S_i(\tau)=\sum_{k=0}^{-\infty} \left\{ q_i f(\tau-\beta_i+T-kT,\ \tau-\alpha_i-kT) + Q_i f(\tau-\alpha_i+T-kT,\right. \]
\[ \left. \tau-\beta_i+T-kT) \right\} \quad (q_1=Q_2,\ q_2=Q_1). \tag{22} \]
Using condition (13), it is easy to see that \(du_1(0,\tau)/d\tau>0\), \(du_2(0,\tau)/d\tau<0\).
Let us return to the more general case, when the periodic function \(\chi(\tau)\), defined by formula (19), is given. From formulas (20) we find
\[ \frac{du_i(0,\tau)}{d\tau} = -\frac{1}{\sqrt{\pi}} \left\{ \frac{\psi_i(\alpha_i)}{\sqrt{\tau-\alpha_i}} + \int_{\alpha_i}^{\tau} \frac{\psi_i'(\sigma)\,d\sigma}{\sqrt{\tau-\sigma}} + \right. \]
\[ \left. + \int_{\alpha_i}^{\tau} \frac{\psi_i(\sigma)}{(\tau-\sigma)^{5/2}} \sum_{n=1}^{\infty} \left[(-1)^n e^{-n^2/(\tau-\sigma)}(2n^2-\tau+\sigma)\right]\,d\sigma \right\} + \frac{dS_i}{d\tau}, \]
\[ \frac{dS_i}{d\tau} = -\sum_{k=0}^{-\infty} \left\{ \int_{\beta_i-T+kT}^{\alpha_i+kT} K'(\tau-\sigma)\varphi_i(\sigma)\,d\sigma + \int_{\alpha_i-T+kT}^{\beta_i-T+kT} K'(\tau-\sigma)\psi_i(\sigma)\,d\sigma \right\} \]
\[ (\alpha_i<\tau<\beta_i). \tag{23} \]
It is easy to see that, by virtue of (13), \(dS_1/d\tau>0\). It can be shown that, if \(\psi_1(\tau)<0,\ \psi_2(\tau)>0,\ (1-p)T<2,\ pT<2\), then, in order for the inequalities \(du_1(0,\tau)/d\tau>0\) \((\alpha_1<\tau<\beta_1)\), \(du_2(0,\tau)/d\tau<0\) \((\alpha_2<\tau<\beta_2)\) to hold, it is sufficient that the relations \((i=1,2)\) hold.
\[ (-1)^i\left[\frac{\psi_i(\alpha_i)}{\sqrt{\tau-\alpha_i}}+\int_{\alpha_i}^{\tau}\frac{\psi'_i(\sigma)\,d\sigma}{\sqrt{\tau-\sigma}}-\int_{\alpha_i}^{\tau}\frac{e^{-1/(\tau-\sigma)}(2-\tau+\sigma)\psi_i(\sigma)\,d\sigma}{(\tau-\sigma)^{5/2}}\right]>0. \tag{24} \]
The function \(F\) entering condition (10) may then be regarded as having the form
\[ F(u,\partial u/\partial \tau)= \begin{cases} F_1(u) & \text{for } \partial u/\partial \tau>0,\quad F_1=\psi_1(\tau),\ u=u_1(0,\tau),\\ F_2(u) & \text{for } \partial u/\partial \tau<0,\quad F_2=\psi_2(\tau),\ u=u_2(0,\tau). \end{cases} \]
As an example, let us consider the case in which the functions \(\psi_i(\tau)\) in formula (19) are prescribed by the dependences
\[ \psi_i(\tau)=e_i+d_i(\beta_i-\tau)^{q_i}+f_i(\tau-\alpha_i)^{1/2};\qquad e_1<0,\ d_1<0,\ f_1>0, \tag{25} \]
\[ \frac{1}{2}<q_i<1;\quad e_2>0,\ d_2>0,\ f_2<0. \]
A periodic solution of equation (1) with conditions (3) and (17), where \(\chi(\tau)\) is given by relations (19) and (25), in the semi-infinite domain \(0<x<\infty\) was found in paper \((^6)\). The solution of the corresponding problem in the bounded domain \(0<z<1\) is given by formulas (20), (19), and (25). Moreover, if we introduce the notation
\[ I_1(x,q)=\int_x^{\infty} e^{-z}z^{1/2-q}\,dz,\qquad I_2(x,q)=\int_x^{\infty} e^{-z}z^{q-1/2}\,dz, \]
\[ I_3(x)=\int_x^{\infty} e^{-z}\sqrt{\frac{z}{x}-1}\,dz,\qquad I_4(x)=\int_x^{\infty}\frac{e^{-z}}{z}\sqrt{\frac{z}{x}-1}\,dz, \tag{26} \]
then inequalities (24), sufficient for having \(du_1(0,\tau)/d\tau>0\), \(du_2(0,\tau)/d\tau<0\), take the form \((p_1=1-p,\ p_2=p)\)
\[ (-1)^i\left[e_i\left(1-2e^{-1/p_iT}\right)-d_i\sqrt{p_iT}\left\{\frac{(p_iT)^{q_i-1/2}}{2q_i-1} +2I_1\left(\frac{1}{p_iT},q_i\right)-\right.\right. \]
\[ \left.\left. -I_2\left(\frac{1}{p_iT},q_i\right)\right\} +f_i\sqrt{p_iT}\left\{\frac{\pi}{2}-2I_3\left(\frac{1}{p_iT}\right)+I_4\left(\frac{1}{p_iT}\right)\right\}\right]>0. \tag{27} \]
From (13) and (25) there also follows a relation among the constants \(e_i,d_i,f_i,q_i,p\) and \(T\), coinciding with equality (4.4) of article \((^6)\):
\[ e_2pT+\frac{d_2}{q_2+1}(pT)^{q_2+1}+\frac{2}{3}f_2(pT)^{3/2} +e_1(1-p)T+\frac{d_1}{q_1+1}[(1-p)T]^{q_1+1} +\frac{2}{3}f_1[(1-p)T]^{3/2}=0. \tag{28} \]
From (26) it follows that \(\lim_{x\to\infty} I_j(x,q_i)=0\) \((j=1,2,3,4)\) and \(\lim_{x\to\infty}[(2q_i-1)x^{q_i-1/2}I_i(x,q_i)]=0\) \((i=1,2)\). Therefore, in the limit as \(T\to0\), inequalities (27) pass into inequalities (4.3) of article \((^6)\).
Thus, if there exists a solution of the problem in the semi-infinite domain, then, at least for sufficiently small values of the period \(T\), there also exists a solution of the corresponding problem in the finite domain. (Indeed, \(T\to0\) \((l\to\infty)\), since from formula (5) it follows that the period \(T\) is related to the actual period \(T_*\) of the self-oscillation by the dependence \(T=DT_*/l^2\).)
Putting in formulas (20) \(\psi_1(\sigma)=F_1[u_1(0,\sigma)]\), \(\psi_2(\sigma)=F_2[u_2(0,\sigma)]\), we obtain a nonlinear integral equation for finding the function \(u(0,\sigma)\) \((\alpha_1\leq\sigma\leq\beta_2)\).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
8 V 1968
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