UDC 532.72

HYDROMECHANICS

N. N. KOCHINA

ON THE SOLUTION OF A DIFFUSION PROBLEM WITH A NONLINEAR BOUNDARY CONDITION

(Presented by Academician L. I. Sedov, 9 VII 1966)

The problem of self-oscillations arising under certain conditions in electrochemical systems with a falling characteristic is reduced to finding a periodic solution of the diffusion equation \(\partial c/\partial t = D\,\partial^2 c/\partial x^2\) for the concentration \(c\) of the discharging substance in the semi-infinite region \(0 \le x < \infty\) with the nonlinear boundary condition \(\dfrac{dc(0,t)}{dx} = \chi_i[c(0,t)]\) \((i=1\) for \(0<t<T_1,\ i=2\) for \(T_1<t<T)\), where \(D\) is the diffusion coefficient, \(\chi(c)\) is an S-shaped function, \(c(0,0)=c(0,T)=c_-\), \(c(0,T_1)=c_+\), \(c_-=\min c(0,t)\), \(c_+=\max c(0,t)\), and the value \(c_-\) is attained by the function \(c(0,t)\) only at \(t=0\) and \(t=T\), while the value \(c_+\) only at \(t=T_1\), the period \(T\) and the quantity \(T_1\) being unknown in advance \((^1,^2)\). Below this problem is reduced to the solution of a nonlinear integral equation, and it is shown how to find this solution in some cases.

We shall seek the concentration in the form \(c(x,t)=qx+r+u(x,t)\), where the quantities \(q\) and \(r\) are unknown in advance and \(\lim_{x\to\infty} u(x,t)=0\); then it is necessary to find a periodic solution of the equation

\[ \partial u/\partial t = D\,\partial^2 u/\partial x^2 \tag{1} \]

with the nonlinear boundary condition

\[ \partial u(0,t)/\partial x=\Psi_i[u(0,t)] \quad \bigl(\Psi_i[u(0,t)]=\chi_i[c(0,t)]-q,\quad i=1,2, \]
\[ u(0,0)=u(0,T)=u_-,\quad u(0,T_1)=u_+,\quad \Psi'_1(u_+)=\Psi'_2(u_-)=\infty\bigr), \tag{2} \]

where \(\Psi(u)\) is an S-shaped function (Fig. 1, solid curves).

Assuming at first \(\Psi_i[u(0,t)]\) and \(u(x,0)=\Phi(x)\) to be known functions of \(t\) and \(x\), respectively, we write the solution of (1)—(2) as

\[ u(x,t)= \frac{1}{\sqrt{\pi Dt}} \int_0^\infty \Phi(\alpha)K(x,\alpha,Dt)\,d\alpha - \sqrt{\frac{D}{\pi}} \int_0^t \frac{\Psi[u(0,\tau)]\exp[-x^2/4D(t-\tau)]\,d\tau} {\sqrt{t-\tau}} \tag{3} \]

\[ \bigl(K(x,\alpha,Dt)=\exp[-x^2/4Dt-\alpha^2/4Dt]\operatorname{ch}x\alpha/2Dt\bigr). \]

Putting \(t=T\) in formula (3) and taking into account that \(u(x,T)=u(x,0)=\Phi(x)\), we reduce the problem to the solution of a linear integral equation for the function \(\Phi(x)\), under the assumption that \(\Psi_i[u(0,t)]\), \(T_1\), and \(T\) are known:

\[ \Phi(x)= \frac{1}{\sqrt{\pi Dt}} \int_0^\infty \Phi(\alpha)K(x,\alpha,DT)\,d\alpha +\Omega(x) \tag{4} \]

\[ \left( \Omega(x)= -\sqrt{\frac{D}{\pi}} \left\{ \int_{T_1}^{T} \frac{\Psi_2[u(0,\sigma)]\exp[-x^2/4D(T-\sigma)]\,d\sigma} {\sqrt{T-\sigma}} + \int_{0}^{T_1} \frac{\Psi_1[u(0,\sigma)]\exp[-x^2/4D(T-\sigma)]\,d\sigma} {\sqrt{T-\sigma}} \right\} \right). \]

The homogeneous equation (4) has the solution \(\Phi(x) \equiv C\); the nonhomogeneous equation has a solution only if \(\int_0^\infty \Omega(x)\,dx=0\). Changing, in this condition, where the notation (4) has been introduced, the order of integration, we reduce it to the form

\[ \int_{T_1}^{T} \Psi_2[u_2(0,\sigma)]\,d\sigma + \int_0^{T_1} \Psi_1[u_1(0,\sigma)]\,d\sigma =0. \tag{5} \]

Using equality (5), by the method of successive approximations we find the solution \(\Phi(x)\) of equation (4) in the form of a series uniformly convergent by Abel’s test (3). Substituting this series into formula (3) and performing termwise integration, we obtain the functions \(u_1(x,t)\) \((0\le t\le T_1)\) and \(u_2(x,t)\) \((T_1\le t\le T)\) in the form of uniformly convergent series

\[ u_1(x,t) = -\sqrt{\frac{D}{\pi}} \left\{ \int_0^t \exp\left[-\frac{x^2}{4D(t-\sigma)}\right] \frac{\Psi_1[u_1(0,\sigma)]\,d\sigma}{\sqrt{t-\sigma}} + S(x,t) \right\}, \]

\[ \begin{aligned} u_2(x,t) &= -\sqrt{\frac{D}{\pi}} \left\{ \int_{T_1}^{t} \exp\left[-\frac{x^2}{4D(t-\sigma)}\right] \frac{\Psi_2[u_2(0,\sigma)]\,d\sigma}{\sqrt{t-\sigma}} \right.\\ &\qquad\left. + \int_0^{T_1} \exp\left[-\frac{x^2}{4D(t-\sigma)}\right] \frac{\Psi_1[u_1(0,\sigma)]\,d\sigma}{\sqrt{t-\sigma}} + S(x,t) \right\}, \end{aligned} \tag{6} \]

\[ \begin{aligned} S(x,t) = \sum_{n=1}^{\infty} \left\{ \int_{T_1}^{T} \exp\left[-\frac{x^2}{4D(nT+t-\sigma)}\right] \frac{\Psi_2[u_2(0,\sigma)]\,d\sigma}{\sqrt{nT+t-\sigma}} \right.\\ \left. + \int_0^{T_1} \exp\left[-\frac{x^2}{4D(nT+t-\sigma)}\right] \frac{\Psi_1[u_1(0,\sigma)]\,d\sigma}{\sqrt{nT+t-\sigma}} \right\}. \end{aligned} \]

Putting \(x=0\) in formulas (6), we reduce the problem under consideration to finding a solution of a nonlinear integral equation for determining, under fulfillment of condition (5), the functions \(u_i(0,t)\) \((i=1,2)\) and the quantities \(T_1,T,q\), and \(r\).

Suppose that \(T_1=T/2\), \(\Psi_2[u_2(0,T/2+\tau)]\equiv-\Psi_1[u_1(0,\tau)]\). From solution (6) it is easy to see that then \(u_2(0,T/2+\tau)=-u_1(0,\tau)\), \(\Psi_2(u)=-\Psi_1(-u)\). Returning to the originally formulated problem for the concentration \(c(x,t)\), we find that in this case (which we shall call symmetric), denoting \(u_1(0,t)=u(t)\), \(\Psi_1(u_1)=\Psi(u)\), \(r=(c_+ + c_-)/2\), \(\Psi_i(v)=\chi_i[(c_+ + c_-)/2+v]-q\) \((i=1,2)\).

The series with general term
\[ v_n(\sigma)=\Psi[u(\sigma)]\left\{-[(n-1/2)T+\tau-\sigma]^{-1/2} + [nT+\tau-\sigma]^{-1/2}\right\} \]
converges uniformly in the interval \(0\le\sigma\le T/2\), and the functions \(v_n\) are integrable; consequently, (6), where \(x=0\) has been put, reduces to the equation for finding the function \(u(t)\)

\[ u(t) = -\sqrt{\frac{D}{\pi}} \left\{ \int_0^t \frac{\Psi[u(\sigma)]\,d\sigma}{\sqrt{t-\sigma}} + \int_0^{T/2} \frac{\Psi[u(\sigma)]}{\sqrt{T}}\, Q\left(\frac{t-\sigma}{T}\right)d\sigma \right\} \qquad \left(0\le t\le \frac{T}{2}\right) \tag{7} \]

\[ \left( Q(z)= \sum_{n=1}^{\infty} \left\{ -\frac{1}{\sqrt{\,n-\tfrac12+z\,}} + \frac{1}{\sqrt{\,n+z\,}} \right\} \right). \]

Putting \(U=u/u_+\), \(\tau=t/T\), \(\lambda=\mu\sqrt{DT}/u_+\) and assuming that condition (2) is written in the form \(\partial u(0,t)/\partial x=\mu F(U)\), we rewrite (7) in the form

\[ U(\tau) = -\frac{\lambda}{\sqrt{\pi}} \left\{ \int_0^\tau \frac{F[U(\sigma)]\,d\sigma}{\sqrt{\tau-\sigma}} + \int_0^{1/2} F[U(\sigma)]\,Q(\tau-\sigma)\,d\sigma \right\}. \tag{8} \]

We shall consider the space \(C\) of continuous functions. Let
\(F(U)=F_0(U)+\varepsilon \xi(U)\), where \(F_0(U)\) is a function for which the solution \(\lambda \overline U(\tau)\) of equation (8) is known, \(\varepsilon>0\) is a small parameter, and \(\xi(U)\) is a function satisfying the Lipschitz condition

\[ \| \xi(U'')-\xi(U')\|<L\|U''-U'\|. \tag{9} \]

Denoting by \(BU\) the operator

\[ BU=\frac{1}{\sqrt{\pi}}\left\{ \int_0^\tau \frac{\xi[U(\sigma)]\,d\sigma}{\sqrt{\tau-\sigma}} + \int_0^{1/2} \xi[U(\sigma)]Q(\tau-\sigma)\,d\sigma \right\}, \tag{10} \]

we reduce equation (8) to the form

\[ U=\lambda \overline U-\varepsilon \lambda BU, \tag{11} \]

where (9) shows that the operator \(BU\) satisfies the Lipschitz condition

\[ \|BU''-BU'\|<a\|U''-U'\|. \tag{12} \]

We shall solve equation (11) by the method of successive approximations, putting

\[ U_{n+1}=\lambda_{n+1}\overline U-\varepsilon \lambda_n BU_n. \tag{13} \]

We shall consider the functions \(U_n(\tau)\) \((0\le \tau \le 1/2)\). As is easy to see,
\(\overline U(0)=-\overline U(1/2)\), \(BU_n(0)=-BU_n(1/2)\), and consequently, from (13),
\(U_{n+1}(0)=-U_{n+1}(1/2)\). The value \(\lambda_{n+1}\) is determined from the condition \(U_{n+1}(1/2)=1\):

\[ \lambda_{n+1}=\bigl[1+\varepsilon \lambda_n BU_n(1/2)\bigr]/\overline U(1/2). \tag{14} \]

From (13) it is clear that all the functions \(U_n(\tau)\) are continuous. Since for \(\tau=1/2\) the value \(BU_n\) is finite, the functions \(\overline U(\tau)\), for which a solution of (1)—(2) has already been found (in papers \((^1,^2,^4)\)), assume the value \(\overline U(1/2)\) only at \(\tau=1/2\), and in a neighborhood of \(\tau=0\) the functions \(\overline U(\tau)\) and \(BU_n(\tau)\) have respectively the form
\(\overline U(\tau)=a/\sqrt{\tau}+\ldots\), \(BU_n(\tau)=b/\sqrt{\tau}+\ldots\), then, for sufficiently small \(\varepsilon\), the value 1 is attained by the function \(U_{n+1}(\tau)\) only at \(\tau=1/2\).

From (10) it is seen that the operator \(BU\) is bounded \((\|BU\|\le B)\); from (14) it follows that the constants \(\lambda_n\) are bounded \((\lambda_n<\overline\lambda)\), if \(\varepsilon B/\overline U(1/2)<1\). Using also (13), (14), (12), and denoting

\[ \nu=\max\{2a\varepsilon\overline\lambda,\;2\varepsilon B,\; a\varepsilon\overline\lambda/\overline U(1/2),\; \varepsilon B/\overline U(1/2)\}, \tag{15} \]

we obtain the estimates

\[ \|U_{n+1}-U_n\|\le \nu\{\|U_n-U_{n-1}\|+|\lambda_n-\lambda_{n-1}|\}, \]

\[ |\lambda_{n+1}-\lambda|\le \nu\{\|U_n-U_{n-1}\|+|\lambda_n-\lambda_{n-1}|\}, \]

whence the inequalities follow

\[ \|U_{n+1}-U_n\|\le \xi(2\nu)^n M,\qquad |\lambda_{n+1}-\lambda|\le \xi(2\nu)^n M, \]

\[ \|U_{n+k}-U_n\|\le \xi(2\nu)^n M/(1-2\nu), \]

\[ |\lambda_{n+k}-\lambda_n|\le \xi(2\nu)^n M/(1-2\nu), \]

\[ \left(\xi=\frac{1}{4\nu},\quad M=\|U_2-U_1\|+|\lambda_2-\lambda_1|\right). \]

Thus, if \(2\nu<1\), i.e., according to (15), for sufficiently small \(\varepsilon\), the sequences \(U_n\) and \(\lambda_n\) are fundamental and, by virtue of the completeness of the space \(C\) and of the space of numerical sequences, each of these sequences converges to a limit \(U_*(\tau)\) and \(\lambda_*\), respectively; moreover, the function \(U_*(\tau)\) is continuous.

Let us show that the function \(U_*(\tau)\) and the constant \(\lambda_*\) satisfy equation (11). Using (11), (13), and (10), we have

\[ \|U_*-\lambda_*\overline U+\varepsilon \lambda_* BU_*\| \le \|U_*-U_{n+1}\|+\|\overline U\|\,|\lambda_{n+1}-\lambda_*|+ \]

\[ +a\varepsilon\lambda_*\|U_*-U_n\|+\varepsilon B|\lambda_*-\lambda_n|. \tag{16} \]

Since every term on the right-hand side of (16) tends to zero as \(n\) increases,
\(\|U_*-\lambda_*\overline U+\varepsilon \lambda_* BU_*\|=0\), whence it follows that \(U_*\) and \(\lambda_*\) satisfy (11).

We shall prove that \(\|U_*\|=1\). By means of the relations \(\|U_n\|=1\) and \(U_n(1/2)=1\), we find
\[ \|U_*\|\leq \|U_n\|+\|U_*-U_n\|\leq 1,\qquad U_*(1/2)=U_n(1/2)+U_*(1/2)-U_n(1/2)=1+U_*(1/2)-U_n(1/2), \]
hence \(U_*(1/2)=1,\ \|U_*\|=1\). It is easy to show that if \(\tau\ne 0\), then \(U_*(\tau)>-1\), and if \(\tau\ne 1/2\), then \(U_*(\tau)<1\).

We shall prove uniqueness of the solution \(U_*,\lambda_*\) of equation (11). Suppose equation (11) has two solutions \(U_*,\lambda_*\) and \(U_{**},\lambda_{**}\). Then from (11) it follows that
\[ \|U_*-U_{**}\|\leq \alpha\varepsilon\lambda_*\|U_*-U_{**}\|+ \left(\|\overline U\|+\varepsilon\|BU_{**}\|\right)|\lambda_*-\lambda_{**}|, \tag{17} \]
\[ |\lambda_*-\lambda_{**}|\leq \alpha\varepsilon\lambda_*\lambda_{**}\|U_*-U_{**}\|. \tag{18} \]

Substituting (18) into (17), we find the inequality
\[ \|U_*-U_{**}\|\leq N\|U_*-U_{**}\|\bigl(N=\alpha\varepsilon\lambda_*[1+\lambda_{**}(\|\overline U\|+\varepsilon\|BU_{**}\|)]\bigr). \tag{19} \]
It is clear that, for sufficiently small \(\varepsilon\), \(\|U_*-U_{**}\|=0\), i.e. \(U_*=U_{**}\), and then from (11) \(\lambda_*=\lambda_{**}\).

Fig. 1

The solution of equation (1) with condition (2) in explicit form for rectangular oscillations \(\chi_i(c)=\chi_i=\mathrm{const}\) was found in papers \({}^{1,4}\), and for the case \(\chi_i(c)=\chi_i+hc\ (h>0)\), in paper \({}^{4}\).

In paper \({}^{2}\) the inverse problem was solved: the functions
\(\Psi_i[u_i(0,t)]\) are regarded as known as functions of \(t\):
\[ \Psi_1[u_1(0,t)]=\psi(t),\qquad \Psi_2[u_2(0,t)]=\varphi(t), \]
and the case in which \(\psi(t)\) is defined by the formula
\[ \psi(t)=e'+d'(-p/2+1-\tau)^q+ f'(\tau-p/2)^{1/2} \tag{20} \]
\[ (0<p<1,\ e'<0,\ d'<0,\ f'>0,\ 1/2<q<1). \]
The function \(\varphi(t)\) has an analogous form; moreover, certain conditions are imposed on the constants entering the expressions for \(\psi(t)\) and \(\varphi(t)\) \({}^{2}\).

Thus, in the present article, for the symmetric case, it has been shown how to find the solution of equation (1) with condition (2), where the function \(\chi(c)\) differs little from that occurring in these three problems.

Let us dwell in more detail on the case when the prescribed function \(F(U)\) in a neighborhood of \(U=1\) has the form \(F(U)=F(1)+B(1-U)^q+\cdots\), and the function \(\overline U(\tau)\) represents the solution of the inverse problem \(\mu F_0[U(\tau)]=\psi(t)\), where \(\psi(t)\) is given by formula (20), and, by symmetry, \(p=1/2\) \({}^{2}\). Near the value \(\tau=1/2\),
\[ \overline U=1+\overline C(\tau-1/2)+\cdots,\qquad \overline C=\sqrt{DT}\,[a_1e'+a_2(q)d'+a_3f']/\mu \quad {}^{2}. \]
Using the notation adopted by us, we find, as shown in \({}^{2}\), that
\[ \overline C=(d'/\mu B)^{1/q}. \]
Taking into account that \(F(1)=F_0(1)\), \(F(-1)=F_0(-1)\), it is easy to show that, from the prescribed values \(B\), \(F(1)\), and \(F(-1)\), one can uniquely find the constants \(e'/\mu\), \(d'/\mu\), and \(f'/\mu\). Putting \(\varepsilon=0\) in (11), from (14) we find \(\lambda\), i.e. the period \(T\) as a function of \(\mu\) and \(u_+\), and from (11) the solution of the problem under consideration in the first approximation, when \(F(U)=F_0(U)\) (the dotted curve in Fig. 1). Now the problem is reduced to determining the solution \(U\) of equation (11), where \(\varepsilon \zeta(U)=F(U)-F_0(U)\) is a known function, with \(\zeta'(1)=0\), so that condition (9) is satisfied.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
10 VI 1966

CITED LITERATURE

\({}^{1}\) A. Ya. Gokhshtein, DAN, 140, No. 5 (1961).
\({}^{2}\) N. N. Kochina, PMM, 28, issue 4 (1964).
\({}^{3}\) G. M. Fikhtengol’ts, Course of Differential and Integral Calculus, 2, 1948.
\({}^{4}\) N. N. Kochina, DAN, 165, No. 5 (1965).