Abstract
Full Text
UDC 532.72
HYDROMECHANICS
N. N. KOCHINA
SOME SOLUTIONS OF AN INHOMOGENEOUS DIFFUSION EQUATION
(Presented by Academician L. I. Sedov on 11 X 1968)
We consider the following problem, close to those studied in works \((^{1-9})\): to find a bounded solution of the inhomogeneous diffusion equation
\[ \partial u / \partial t = a^2 \partial^2 u / \partial x^2 + F[u(\bar{x}, t)], \tag{1} \]
periodic, or such that the initial function \(u(x,0)\) over a previously unknown time interval \(T\) increases by a factor \(k\) \((k>1)\), with the boundary condition
\[ u(0,t)=0 \tag{2} \]
in the semi-infinite domain \(0<x<\infty\). Here the function \(F(u)\) has the form shown in Fig. 1, i.e. \(F[u(\bar{x},t)]=c\) \((c>0)\), if \(u(\bar{x},t)<u_*\), until \(u(\bar{x},t)=u_*\); if at \(t=T_1\), \(u(\bar{x},T_1)=u_*\), then for \(t>T_1\), \(F[u(\bar{x},t)]=-d\) \((d>0)\), if \(u(\bar{x},t)>u_{**}\) \((u_{**}<u_*)\), until \(u(\bar{x},t)=u_{**}\) at \(t=T\). In doing this we shall assume that
Fig. 1
\[ u(x,T)=k\omega(x) \quad (\omega(x)=u(x,0);\ k\geq 1), \tag{3} \]
so that it must be
\[ u_* = k u(\bar{x},0). \]
Thus, the formulation of the problem is analogous to that given in works \((^8,^9)\).
Below, for \(k\geq 1\), a solution of this problem is found for large values of \(T/T_1\) and small \(d/c\), under the assumption that the quantities \(c,d,T_1\), and \(T\) are related by the relation \((c+d)T_1=dT\). A solution of this problem is also obtained under the condition \(\bar{x}\gg 2a\sqrt{T}\), and, if \(k=1\), then \((c+d)T_1=dT\).
As an example of the problem described by equation (1) with conditions (2), (3), one may consider the following problem of filtration theory in a hydraulic formulation: let groundwater be situated in the region \(0\leq x<\infty\); let \(u(x,t)\) denote the groundwater level; at the section \(x=0\) there is a wall of a canal in which a constant, in particular zero, liquid level is maintained; at the section \(x=\bar{x}\) \((0<\bar{x}<\infty)\) the height of the groundwater mound is measured; as soon as the height of the mound reaches the value \(u_*\), uniform irrigation carried out with infiltration intensity (taking evaporation into account) \(mc\) is stopped, and only evaporation with intensity \(md\) occurs; when the height of the mound has decreased to the value \(u_{**}\), irrigation is again begun with the previous intensity; find the solution of the problem on the spreading of the groundwater mound. Here \(m\) is the active porosity of the soil, \(a^2=k\bar{u}/m\), \(k\) is the filtration coefficient, and \(\bar{u}\) is the mean depth of the flow.
Let, for \(0<t<T_1\), \(F[u(\bar{x},t)]=c\), and for \(T_1<t<T\), \(F[u(\bar{x},t)]=-d\); at \(t=T\) relation (3) is satisfied. We shall find the solution \(u(x,t)\) of equation (1) with boundary condition (2).
The function \(u(x,t)\), expressed in terms of the as yet unknown function \(\omega(x)\), is determined by the formulas
\[ \begin{aligned} u(x,t)&=v(c,x,t)+w(x,t) && (0\leq t\leq T_1),\\ u(x,t)&=v(c,x,t)-v(c,x,t-T_1)-v(d,x,t-T_1)+w(x,t) && (T_1\leq t\leq T), \end{aligned} \tag{4} \]
where
\[ v(c,x,t)=c\left[-\frac{x^2}{2a^2}+\left(t+\frac{x^2}{2a^2}\right)\Phi\left(\frac{x}{2a\sqrt{t}}\right)+\frac{x\sqrt{t}}{a\sqrt{\pi}}\exp\left(-\frac{x^2}{4a^2t}\right)\right], \]
\[ w(x,t)=\frac{\exp(-x^2/4a^2t)}{a\sqrt{\pi t}}\int_0^\infty \exp\left(-\frac{\xi^2}{4a^2t}\right)\operatorname{sh}\frac{x\xi}{2a^2t}\,\omega(\xi)\,d\xi, \tag{5} \]
\[ \Phi(y)=\frac{2}{\sqrt{\pi}}\int_0^y \exp(-S^2)\,dS. \]
Condition (3) gives the integral equation
\[ \varphi(x)=\psi(x)+\lambda\int_0^\infty K(x,\eta)\varphi(\eta)\,d\eta \tag{6} \]
for finding the function \(\varphi(x)\). Here it is assumed that
\[ \varphi(x)=\omega(x),\qquad \psi(x)=f(x)\qquad (k=1), \]
\[ \varphi(x)=-g(x)+\omega(x),\qquad g(x)=D\left[1+\sum_{m=1}^{\infty}\frac{1}{k^m}\Phi\left(\frac{x}{2a\sqrt{mT}}\right)\right], \tag{7} \]
\[ \psi(x)=-D+f(x)/k\qquad (k>1), \]
where
\[ f(x)=\gamma x^2[1-\Phi(\alpha x)]-\zeta x^2[1-\Phi(\beta x)]+\mu\Phi(\alpha x)+ \chi\Phi(\beta x)-\delta x e^{-\alpha^2x^2}+\varepsilon x e^{-\beta^2x^2}, \tag{8} \]
\[ \left(\gamma=\frac{c+d}{2a^2},\ \zeta=\frac{c}{2a^2},\ \alpha=\frac{1}{2a\sqrt{T-T_1}},\ \beta=\frac{1}{2a\sqrt{T}},\right. \]
\[ \left.\mu=(c+d)(T_1-T),\ \chi=cT,\ \delta=\frac{(c+d)\sqrt{T-T_1}}{a\sqrt{\pi}},\right. \]
\[ \left.\varepsilon=\frac{c\sqrt{T}}{a\sqrt{\pi}},\right. \]
\[ D=[cT_1+d(T_1-T)]/k,\quad f(\infty)=Dk\right); \]
\[ \lambda=\frac{1}{ka\sqrt{\pi T}},\qquad K(x,\eta,T)=\exp\left[-\frac{x^2+\eta^2}{4a^2T}\right]\operatorname{sh}\frac{x\eta}{2a^2T}. \tag{9} \]
Consider the homogeneous integral equation corresponding to (6) for the case \(k=1\)
\[ \varphi(x)=\frac{1}{a\sqrt{\pi T}}\int_0^\infty \exp\left[-\frac{x^2+\eta^2}{4a^2T}\right]\operatorname{sh}\frac{x\eta}{2a^2T}\,\varphi(\eta)\,d\eta. \tag{10} \]
As is known, for (10) the relation
\[ \int_0^\infty \eta\exp\left(-\frac{\eta^2}{4a^2T}\right)\operatorname{sh}\frac{x\eta}{2a^2T}\,d\eta = a\sqrt{\pi T}\,x\exp\frac{x^2}{4a^2T} \tag{11} \]
holds.
In consequence of (11), it is clear that the integral equation (10), for any \(B\), has the solution \(\varphi(x)=Bx\). Formulas (4), (5), and (7) show that, for boundedness of the solution \(u(x,t)\) of the problem under consideration, the condition \(B=0\) must be satisfied.
Introducing the notation
\[ R(x,\eta,T)=\sum_{m=1}^{\infty}\frac{1}{\sqrt{m k^{m-1}}}\exp\left[-\frac{x^2+\eta^2}{4a^2 mT}\right]\operatorname{sh}\frac{x\eta}{2a^2mT}, \tag{12} \]
one can see that, if in formula (8) \(\psi(x)\to0\) as \(x\to\infty\), the solution \(\varphi(x)\) of integral equation (6) is represented in the form
\[ \varphi(x)=\psi(x)+\frac{1}{ka\sqrt{\pi T}}\int_{0}^{\infty}R(x,\eta,T)\psi(\eta)\,d\eta . \tag{13} \]
It follows from formulas (7) and (8) that for \(k=1\), \(\psi(\infty)=D\), and for \(k\ne1\), \(\psi(\infty)=0\). For large \(m\) the general term of the series (12) has the form
\[ \frac{1}{2a^2T}\frac{x\eta\psi(\eta)}{k^{m-1}m^{3/2}}+\cdots . \]
In the case \(k=1\), solution (13), by virtue of (8), exists only when \(D=0\), which gives the condition
\[ cT_1=d(T-T_1). \tag{14} \]
Thus, we have obtained the following result: the problem sought has a solution defined by formula (13), for \(k>1\), for arbitrary values of the constants \(c,d,T_1\), and \(T\); for \(k=1\), the solution of the problem exists only for certain values of \(c,d,T_1\), and \(T\), namely those connected by equation (14).
The graphs of the function \(f(x)\) for various values of the parameters \(c,d,T_1\), and \(T\), connected by condition (14), are presented in Fig. 2. Here the solid curve corresponds to the values \(c=1\), \(d=1/99\), \(T_1=1\), and \(T=100\), and the dashed curve to the values \(c=1\), \(d=1\), \(T_1=1\), \(T=2\) \((a=1)\).
The constants \(T_1\) and \(T\) are connected with the quantities \(u_*\) and \(u_{**}\) by the relations
\[ u(\bar{x},T_1)=u_*,\qquad ku(\bar{x},0)=u_{**}. \tag{15} \]
In the case \(k=1\), the period of self-oscillation \(T\) is determined from equation (14), which must therefore be consistent with the equation \(ku(\bar{x},0)=u_{**}\).
Fig. 2
Fig. 3
We now let, in formulas (4) and (5), the quantity \(T/T_1\) tend to infinity and \(d/c\) tend to zero in such a way that condition (14) is preserved. From (8), (12), and (13) it is clear that \(f(x)=\varphi(x)=\omega(x)=\psi(x)=0\), and formulas (4) and (5) pass into the following:
\[ \begin{aligned} u(x,t)&=v(c,x,t) &&(0\le t\le T_1),\\ u(x,t)&=v(c,x,t)-v(c,x,t-T_1) &&(T_1\le t\le T). \end{aligned} \tag{16} \]
Relations (16) give a solution of problem (1)—(3) for which
\[ \lim_{t\to\infty} u(x,t)=ku(x,0)=k\omega(x)=0. \]
Formulas (16) and (5) show that, for any \(\bar x\), \(\partial u(\bar x,t)/\partial t>0\) for \(0<t<T_1\), and \(\partial u(\bar x,t)/\partial t<0\) for \(T_1<t<T\).
The graph of the function \(u=u(\bar x,t)/c\), given by relations (16), for the case \(\bar x=31.622777,\ d=0,\ T_1=400,\ T=\infty,\ a=1\), is shown in Fig. 3.
Let now the quantity \(T/T_1\) be large, and \(d/c\) small, with condition (14) satisfied. Expanding the functions \(\psi(x)\) and \(R(x,\eta,T)\), given by formulas (7), (8), and (12), in series in powers of \(T^{-1/2}\) and \(T^{-1}\), respectively, and substituting these series into equation (13), we find an asymptotic representation of the function \(\varphi(x)\) for large values of \(T\). The first term of this expansion has the form
\[ \varphi(x)=\frac{\nu}{\sqrt{T}}\,x+\cdots, \]
where
\[ \nu=\frac{cT_1}{a\sqrt{\pi}}(-1+S),\qquad S=\sum_{n=0}^{\infty} \frac{(-1)^{n+1}1\cdot3\ldots(2n+1)\sigma(n+{}^3/2)} {2^{n+1}n!(n+2)}. \]
Here \(\sigma(s)\) denotes the series convergent for \(k\geq 1\),
\[ \sigma(s)=\sum_{m=1}^{\infty}\frac{1}{k^m m^s}, \]
which for \(k=1\) becomes the Riemann zeta function \(\zeta(s)\) \((^{11})\). It is easy to see that the series \(S\) entering the expression for the quantity \(\nu\) also converges.
Relations (4) and (5) show that, in the case \(\bar x\gg 2a\sqrt{T}\), and also for sufficiently large \(T/T_1\) and sufficiently small \(d/c\), if \(\bar x\leq 2a\sqrt{T}\), the conditions \(\partial u(\bar x,t)/\partial t>0\) for \(0<t<T_1\), \(\partial u(\bar x,t)/\partial t<0\) for \(T_1<t<T\) are also satisfied, and the quantities \(T_1\) and \(T\) can be found from relations (14) and (15).
Let us now consider problem (1)—(3) for \(k=1\) on the finite interval \(0<x<l\). The function \(u(x,0)=\varphi(x)\) is determined by formulas (12)—(13), where \(R(x,\eta,T)\) is replaced by
\[ R(x,\eta,T)+\sum_{s=1}^{\infty}\{R(x-2sl,\eta,T)+R(x+2sl,\eta,T)\}, \]
and
\[ \psi(x)=\sum_{n=-\infty}^{\infty}[2\Lambda(\alpha_n)-\Lambda(\beta_n)+\Lambda(\gamma_n)], \]
\[ \Lambda(\alpha)=\alpha^2\left[cM(\alpha/\sqrt{T})-(c+d)M(\alpha/\sqrt{T-T_1})\right], \]
\[ M(y)=\exp(-y^2)/\sqrt{\pi}y+[1+1/2y^2]\Phi(y)-1, \]
\[ \alpha_n=(x-2nl)/2a,\qquad \beta_n=\alpha_n+l/2a,\qquad \gamma_n=-\alpha_n+l/2a. \]
From the results obtained in work \((^9)\) there follows the existence of a solution of the problem under consideration for \(k=1\) in the finite domain \(0\leq x\leq l\).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
26 IX 1968
REFERENCES
- N. N. Kochina, PMM, 28, issue 4, 699 (1964).
- N. N. Kochina, DAN, 165, No. 5, 1015 (1965).
- N. N. Kochina, DAN, 174, No. 2, 305 (1967).
- N. N. Kochina, DAN, 179, No. 6, 1297 (1968).
- B. M. Entov, Izv. AN SSSR, Mechanics and Mechanical Engineering, No. 1, 31 (1964).
- A. Ya. Gokhshtein, DAN, 140, No. 5, 1114 (1961).
- Yu. I. Neimark, I. M. Kublanov, Avtomatika i Telemekh., 14, No. 1, 34 (1953).
- Yu. P. Gor'kov, Differential Equations, 2, No. 7, 943 (1966).
- Yu. S. Kolesov, DAN, 176, No. 6, 1240 (1967).
- I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, Moscow, 1962.
- E. K. Titchmarsh, The Riemann Zeta-Function, IL, 1947.