Abstract Generated abstract
The paper studies mixed initial-boundary value problems on a semi-infinite interval in which the solution satisfies a hyperbolic equation on a finite subinterval and a parabolic equation on the remaining half-line, with possible finite jumps of the function and derivative at the interface. Using the Laplace transform, the authors derive an integral representation of the solution and invert it by analyzing poles and a branch cut of the transformed problem. The associated spectral problem is shown to have a mixed spectrum, consisting of a continuous positive real part and a discrete set of complex conjugate eigenvalues, and an expansion formula for a given function in the corresponding eigenfunctions is obtained. The method is also indicated for other boundary and interface conditions and for equations with piecewise constant coefficients.
Full Text
UDC 517.946
MATHEMATICAL PHYSICS
I. T. Lozanovskaya, Ya. S. Uflyand
ON A CLASS OF PROBLEMS OF MATHEMATICAL PHYSICS WITH A MIXED SPECTRUM OF EIGENVALUES
(Presented by Academician B. P. Konstantinov, March 4, 1965)
In this work we investigate certain mixed problems for a semi-infinite interval in the one-dimensional case, when the sought function on a finite interval \((0<x<l)\) satisfies an equation of hyperbolic type, and on the remaining part of the interval—an equation of parabolic type. It is assumed here that at the point \(x=l\) the function and its derivative may have finite discontinuities.*
\(1^\circ\). The simplest problem of the class under consideration consists in finding a function \(u(x,t)\) satisfying the equations
\[ \frac{\partial^2 u}{\partial x^2}=\alpha\frac{\partial^2 u}{\partial t^2},\quad 0<x<l;\qquad \frac{\partial^2 u}{\partial x^2}=\beta\frac{\partial u}{\partial t},\quad l<x<\infty, \tag{1} \]
and the conditions
\[ u\big|_{x=0}=0,\quad u\big|_{x=l-0}=\mu u\big|_{x=l+0},\quad \partial u/\partial x\big|_{x=l-0}=\nu\,\partial u/\partial x\big|_{x=l+0}, \]
\[ u(\infty,t)<\infty, \tag{2} \]
\[ u\big|_{t=0}=f(x),\quad 0<x<\infty;\qquad \partial u/\partial t\big|_{t=0}=0,\quad 0<x<l. \]
Applying the Laplace transform
\[ \bar u(x)=\int_0^\infty u(x,t)e^{-pt}\,dt, \tag{3} \]
we arrive at the equations
\[ \bar u''-\alpha p^2\bar u=-\alpha p f(x),\quad 0<x<l;\qquad \bar u''-\beta p\bar u=-\beta f(x),\quad l<x<\infty \tag{4} \]
and the boundary conditions
\[ \bar u(0)=0,\quad \bar u(l-0)=\mu\bar u(l+0),\quad \bar u'(l-0)=\nu\bar u'(l+0),\quad \bar u(\infty)<\infty. \tag{5} \]
The solution of this boundary-value problem can be expressed in the form**
\[ \bar u=\frac{1}{D(p)}\int_0^\infty f(\xi)\Phi(x,\xi,p)\,d\xi; \tag{6} \]
\[ \Phi= \begin{cases} \sqrt{\alpha}\,[\delta\sqrt p\,\operatorname{ch}\sqrt{\alpha}\,p(l-x)+\operatorname{sh}\sqrt{\alpha}\,p(l-x)]\operatorname{sh}\sqrt{\alpha}\,p\xi, & 0<\xi<x<l,\\[6pt] \mu\sqrt{\dfrac{\beta}{p}}\,\operatorname{sh}\sqrt{\alpha}\,px\,e^{\sqrt{\beta p}(l-\xi)}, & 0<x<l<\xi<\infty,\\[8pt] \dfrac{\alpha\sqrt p}{\nu\sqrt\beta}\,\operatorname{sh}\sqrt{\alpha}\,p\xi\,e^{\sqrt{\beta p}(l-x)}, & 0<\xi<l<x<\infty,\\[8pt] \sqrt{\dfrac{\beta}{p}}\,[\operatorname{sh}\sqrt{\alpha}\,l\,\operatorname{ch}\sqrt{\beta p}(l-x)-\delta\,\operatorname{ch}\sqrt{\alpha}\,pl\,\operatorname{sh}\sqrt{\beta p}(l-x)]e^{\sqrt{\beta p}(l-\xi)}, & l<x<\xi<\infty, \end{cases} \tag{7} \]
* Problems of this kind arise, in particular, in the study of transient processes in piecewise-inhomogeneous media; see, for example, works \((^{1,5})\), where the flow of electric fluid in a channel is considered, taking into account the conductivity of its walls, as well as note \((^4)\), devoted to the calculation of a composite electric line.
** For \(x,\xi<l\) and \(x,\xi>l\), \(\Phi(\xi,x,p)=\Phi(x,\xi,p)\).
where
\[ D(p)=\delta \sqrt{p}\,\operatorname{ch}\sqrt{\alpha}\,pl+\operatorname{sh}\sqrt{\alpha}\,pl,\qquad \operatorname{Re}\sqrt{p}>0,\qquad \delta=\frac{\mu}{\nu}\sqrt{\frac{\alpha}{\beta}} . \tag{8} \]
By the inversion formula we obtain the solution of the posed problem in the form of the complex integral
\[ u=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\bar u e^{pt}\,dp . \tag{9} \]
Since the singular points of the function \(\bar u\) are the simple poles \(p_n\)—the roots of the equation \(D(p)=0\)*, and also \(p=0\) (a branch point), the solution (9) is composed of the sum of residues and the integrals along the banks of the cut \(p=-\lambda\) \((\lambda>0)\):
\[ u(x,t)=2\operatorname{Re}\sum_{n=1}^{\infty} \frac{e^{p_n t}}{D'(p_n)} \int_{0}^{\infty} f(\xi)\Phi(x,\xi,p_n)\,d\xi- \]
\[ -\frac{1}{\pi}\operatorname{Im}\int_{0}^{\infty} \frac{e^{-\lambda t}}{D(\lambda e^{i\pi})}\,d\lambda \int_{0}^{\infty} f(\xi)\Phi(x,\xi,\lambda e^{i\pi})\,d\xi \qquad (\operatorname{Im}p_n>0). \tag{10} \]
\(2^\circ\). To the mixed problem of mathematical physics considered there corresponds the spectral problem**
\[ y''+\psi(x,\lambda)y=0;\qquad \psi(x,\lambda)= \begin{cases} -\alpha\lambda^2, & 0<x<l,\\ \beta\lambda, & l<x<\infty; \end{cases} \tag{11} \]
\[ y(0)=0,\qquad y(l-0)=\mu y(l+0),\qquad y'(l-0)=\nu y'(l+0),\qquad y(\infty)<\infty, \]
which has a mixed spectrum of eigenvalues, consisting of points of the real axis \(\lambda>0\) and the roots \(\lambda_n=-p_n\) of the equation
\[ \omega(\lambda)=\delta\sqrt{-\lambda}\,\operatorname{ch}\sqrt{\alpha}\lambda l-\operatorname{sh}\sqrt{\alpha}\lambda l,\qquad \operatorname{Re}\sqrt{-\lambda}>0 . \tag{12} \]
With the aid of the solution found in (10), one can indicate a formula for expanding a given function \(f(x)\) in the eigenfunctions of the boundary-value problem (11). If the eigenfunctions are defined by the formulas
\[ y_n(x) \begin{cases} \displaystyle \mu\,\frac{\operatorname{sh}\sqrt{\alpha}\lambda_n x} {\operatorname{sh}\sqrt{\alpha}\lambda_n l}, & 0<x<l,\\[1.2ex] \displaystyle e^{\sqrt{-\beta\lambda_n}(l-x)}, & l<x<\infty; \end{cases} \tag{13} \]
\[ y(x,\lambda)= \begin{cases} \mu\,\operatorname{sh}\sqrt{\alpha}\lambda x, & 0<x<l,\\ \operatorname{sh}\sqrt{\alpha}\lambda l\cos\sqrt{\beta\lambda}(l-x) -\delta\sqrt{\lambda}\operatorname{ch}\sqrt{\alpha}\lambda l\sin\sqrt{\beta\lambda}(l-x), & l<x<\infty, \end{cases} \tag{14} \]
and carry out in (10) the limiting passage as \(t\to 0\), then, after some transformations, we obtain the desired expansion formula***
\[ f(x)=2\operatorname{Re}\sum_{n=1}^{\infty} \frac{\operatorname{sh}\sqrt{\alpha}\lambda_n l}{\omega'(\lambda_n)} y_n(x)\int_{0}^{\infty} f(\xi)y_n(\xi)r_n(\xi)\,d\xi+ \]
\[ \text{* It can be proved that this equation has an infinite set of complex-conjugate simple roots, for which } \operatorname{Re}\sqrt{p}>0,\text{ and has no real roots.} \]
\[ \text{** Problems of this type apparently have not been considered in the literature.} \]
\[ \text{*** Establishing the class of functions for which formula (15) is valid will be the subject of a separate investigation.} \]
\[ + \frac{1}{\pi}\int_0^\infty \frac{y(x,\lambda)}{|\omega(\lambda)|^2}\,d\lambda \int_0^\infty f(\xi)y(\xi,\lambda)r(\xi,\lambda)\,d\xi \qquad (\operatorname{Im}\lambda_n>0), \tag{15} \]
where
\[ r_n(x)=\frac{\delta}{\mu^2}\sqrt{-\alpha\lambda_n}, \quad 0<x<l; \qquad r_n(x)=\sqrt{-\frac{\beta}{\lambda_n}}, \quad l<x<\infty; \tag{16} \]
\[ r(x,\lambda)=-\frac{\delta}{\mu^2}\sqrt{\alpha\lambda}, \quad 0<x<l; \qquad r(x,\lambda)=\sqrt{\frac{\beta}{\lambda}}, \quad l<x<\infty. \tag{17} \]
3°. In an analogous way, the corresponding expansions may be found for boundary conditions of the second or third kind at \(x=0\), as well as for more complicated conditions at the point \(x=l\) (for example, those containing derivatives with respect to time).
By the method set forth, one can also establish the mixed character of the spectrum for boundary-value problems connected with the equation
\[ \frac{\partial^2 u}{\partial x^2} = \alpha \frac{\partial^2 u}{\partial t^2} + \beta \frac{\partial u}{\partial t} + \gamma u, \qquad 0<x<\infty, \]
where \(\alpha,\beta,\gamma\) are piecewise constant coefficients. Exceptions may be some limiting cases, for example \(\beta=\gamma\equiv 0\), when the spectrum is discrete, or \(\alpha=\gamma\equiv 0\) (continuous spectrum). A continuous spectrum is also obtained in the case when, in (1), the equation of hyperbolic type takes place for \(-\infty<x<0\) (see (2)).
Physical-Technical Institute named after A. F. Ioffe
Academy of Sciences of the USSR
Received
24 II 1965
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