Full Text
UDC 517.941.9:517.432.1
ON CERTAIN SINGULAR BOUNDARY VALUE PROBLEMS FOR THE BESSEL EQUATION AND THEIR APPLICATIONS IN MATHEMATICAL PHYSICS
E. A. Yushkova
1°. In this note we investigate the singular boundary value problem of finding a function satisfying the equation
\[ (xy')' + \left(\frac{\lambda}{x} - x\right)y = 0 \quad (0 < x < \gamma), \tag{1.1} \]
the boundary condition of the third kind
\[ y'(\gamma) + \beta y(\gamma) = 0 \quad (\beta > 0) \tag{1.2} \]
and the boundedness requirement as \(x \to 0\)
\[ y(0) < \infty . \tag{1.3} \]
Let us write the general solution of equation (1.1):
\[ y = a(\lambda) I_{\sqrt{-\lambda}}(x) + b(\lambda) K_{\sqrt{-\lambda}}(x) \tag{1.4} \]
(\(I_\nu(x)\) is the modified Bessel function of the first kind; \(K_\nu(x)\) is the Macdonald function), and investigate the character of the spectrum of eigenvalues of the problem under consideration.
Assuming that the eigenvalues \(\lambda\) are complex, and using condition (1.3), we obtain
\[ y = I_{\sqrt{-\lambda}}(x) \quad (\operatorname{Re}\sqrt{-\lambda} > 0). \tag{1.5} \]
In this case the boundary condition (1.2) gives the following equation for the eigenvalues \(\lambda\):
\[ I'_{\sqrt{-\lambda}}(\gamma) + \beta I_{\sqrt{-\lambda}}(\gamma) = 0. \tag{1.6} \]
Substituting now into the easily derived relation
\[ \int_0^\gamma I_{\sqrt{-\lambda_1}}(x)\, I_{\sqrt{-\lambda_2}}(x)\,\frac{dx}{x} = 0 \quad (\lambda_1 \ne \lambda_2), \tag{1.7} \]
\(\lambda_{1,2} = c \pm id\), we arrive at a contradiction, since the left-hand side of equality (1.7) is essentially positive.
Let us now suppose that \(\lambda\) is a negative real number. In this case the eigenfunctions are defined by formulas (1.5) and (1.6) with \(\sqrt{-\lambda} = \nu > 0\). Rewriting equation (1.6) in the form
\[ I_{\nu+1}(\gamma)+\frac{\nu}{\gamma} I_\nu(\gamma)+\beta I_\nu(\gamma)=0, \tag{1.8} \]
we see that the last equality is impossible, since the left-hand side of expression (1.8) consists of positive terms.
It is not difficult to verify that for positive values \(\lambda=\tau^2\) there exist real eigenfunctions
\[ y(x,\tau)=K_{i\tau}(x)\bigl[I'_{i\tau}(\gamma)+\beta I_{i\tau}(\gamma)\bigr]- \]
\[ - I_{i\tau}(x)\bigl[K'_{i\tau}(\gamma)+\beta K_{i\tau}(\gamma)\bigr], \tag{1.9} \]
satisfying both equation (1.1) and conditions (1.2), (1.3).
Thus, it has been established that the problem under consideration has a continuous spectrum of real positive eigenvalues.
\(2^\circ\). In the present section a formal derivation is given of the inversion theorem for an integral transform whose kernel is the eigenfunctions (1.9) of the boundary-value problem (1.1)—(1.3).
Consider the problem of finding the function \(F(\tau)\) from an integral relation of the form
\[ f(x)=\int_0^\infty F(\tau)y(x,\tau)\,d\tau \qquad (0<x<\gamma). \tag{2.1} \]
Using a method related to that applied in [1], we compose the following equation for a certain function \(W(x,t)\):
\[ x\frac{\partial}{\partial x}\left(x\frac{\partial W}{\partial x}\right)-x^2W-\frac{\partial W}{\partial t}=0 \tag{2.2} \]
\[ (t>0,\quad 0<x<\gamma) \]
and give its solution satisfying the conditions
\[ \left[\frac{\partial W}{\partial x}+\beta W\right]_{x=\gamma}=0,\qquad W|_{x=0}<\infty,\qquad W|_{t=0}=f(x). \tag{2.3} \]
With the aid of the Laplace transform
\[ \overline W(x,p)=\int_0^\infty W(x,t)\exp(-pt)\,dt \tag{2.4} \]
after some calculations we obtain
\[ \overline W(x,p)=\int_0^x \left[K_{\sqrt p}(x)I_{\sqrt p}(\xi)-\right. \]
\[ \left.-\Phi(\gamma,p)I_{\sqrt p}(x)I_{\sqrt p}(\xi)\right]\frac{f(\xi)}{\xi}\,d\xi+ \]
\[ +\int_x^\gamma \left[I_{\sqrt p}(x)K_{\sqrt p}(\xi)-\Phi(\gamma,p)I_{\sqrt p}(\xi)I_{\sqrt p}(x)\right]\frac{f(\xi)}{\xi}\,d\xi, \tag{2.5} \]
where
\[ \Phi(\gamma,p)= \frac{K'_{\sqrt p}(\gamma)+\beta K_{\sqrt p}(\gamma)} {I'_{\sqrt p}(\gamma)+\beta I_{\sqrt p}(\gamma)}. \tag{2.6} \]
Applying the Riemann—Mellin inversion formula, we shall have
\[ W(x,t)=\int_0^\gamma\left(\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty} \frac{G(x,\xi,p)}{I'_{\sqrt p}(\gamma)+\beta I_{\sqrt p}(\gamma)}\exp pt\,dp\right)\frac{f(\xi)}{\xi}\,d\xi, \tag{2.7} \]
where
\[ G(x,\xi,p)= \begin{cases} I_{\sqrt p}(\xi)\psi(x,p), & \xi\leq x,\\ I_{\sqrt p}(x)\psi(\xi,p), & \xi\geq x, \end{cases} \tag{2.8} \]
\[ \psi(x,p)=K_{\sqrt p}(x)[I'_{\sqrt p}(\gamma)+\beta I_{\sqrt p}(\gamma)]- \]
\[ -I_{\sqrt p}(x)[K'_{\sqrt p}(\gamma)+\beta K_{\sqrt p}(\gamma)]. \tag{2.9} \]
The solution found in (2.7)—(2.9) can be transformed into real form by integration along a cut drawn along the negative part of the real axis of the plane of the complex variable \(p\).
Omitting the corresponding transformations, we give the final result
\[ W(x,t)=\int_0^\gamma \frac{f(\xi)}{\xi}\times \]
\[ \times\left\{\frac{2}{\pi}\int_0^\infty \frac{\tau y(x,\tau)y(\xi,\tau)\operatorname{sh}\pi\tau\exp(-\tau^2 t)} {[I'_{i\tau}(\gamma)+\beta I_{i\tau}(\gamma)] [I'_{-i\tau}(\gamma)+\beta I_{-i\tau}(\gamma)]}\,d\tau\right\}d\xi . \tag{2.10} \]
Changing in equality (2.10) the order of integration and passing to the limit as \(t\to 0\), we obtain the expansion
\[ f(x)=\frac{2}{\pi^2}\int_0^\infty \frac{\tau\operatorname{sh}\pi\tau\, y(x,\tau)\,d\tau} {|I'_{i\tau}(\gamma)+\beta I_{i\tau}(\gamma)|^2} \int_0^\gamma y(\xi,\tau)\frac{f(\xi)}{\xi}\,d\xi, \tag{2.11} \]
which also gives the required formula
\[ F(\tau)=\frac{2\tau\operatorname{sh}\pi\tau} {\pi^2|I'_{i\tau}(\gamma)+\beta I_{i\tau}(\gamma)|^2} \int_0^\gamma y(x,\tau)\frac{f(x)}{x}\,dx. \tag{2.12} \]
Let us note that the quantity \(F(\tau)\) may be regarded as an integral transform of the given function \(f(x)\) with respect to the eigenfunctions of the form (1.9), and relation (2.1) as the corresponding inversion formula.
As \(\gamma\to\infty\), relation (2.11) passes into the well-known Kontorovich—Lebedev formula [2—4]
\[ f(x)=\frac{2}{\pi^2}\int_0^\infty K_{i\tau}(x)\tau\operatorname{sh}\pi\tau\,d\tau \int_0^\infty f(\xi)K_{i\tau}(\xi)\frac{d\xi}{\xi}. \tag{2.13} \]
In the case of the boundary condition of the first kind, the expansion found, as \(\beta\to\infty\), passes into the formula
\[ f(x)=\frac{2}{\pi^{2}}\int_{0}^{\infty}\varphi(x,\tau)\frac{\tau\operatorname{sh}\pi\tau}{|I_{i\tau}(\gamma)|^{2}}\,d\tau \int_{0}^{\gamma}\varphi(\xi,\tau)\frac{f(\xi)}{\xi}\,d\xi, \tag{2.14} \]
\[ \varphi(x,\tau)=K_{i\tau}(x)I_{i\tau}(\gamma)-I_{i\tau}(x)K_{i\tau}(\gamma), \]
obtained in [5] by a somewhat different method.
Finally, as \(\beta\to 0\), (2.11) passes into the formula
\[ f(x)=\frac{2}{\pi^{2}}\int_{0}^{\infty} \frac{\tau\operatorname{sh}\pi\tau\chi(x,\tau)}{|I'_{i\tau}(\gamma)|^{2}}\,d\tau \int_{0}^{\gamma}\chi(\xi,\tau)\frac{f(\xi)}{\xi}\,d\xi, \tag{2.15} \]
in which the eigenfunctions have the form
\[ \chi(x,\tau)=K_{i\tau}(x)I'_{i\tau}(\gamma)-I_{i\tau}(x)K'_{i\tau}(\gamma) \tag{2.16} \]
and correspond to the boundary condition of the second kind.
3°. The expansion formula obtained in item 2° may be applied to the solution of a number of boundary-value problems of mathematical physics, in particular, for wedge-shaped regions.
As one such possibility, let us consider a mixed boundary-value problem of the theory of heat conduction for a wedge of finite cross-section \((0<r<a,\ 0<\theta<\alpha,\ -\infty<z<\infty,\ r,\theta,z\) are cylindrical coordinates).
Suppose that the face \(\theta=0\) is thermally insulated, that on the portion \(\theta=\alpha\) a heat flux of prescribed density \(q(r,z)\) is supplied, and that the surface \(r=a\) freely radiates heat into the surrounding medium, whose temperature is assumed to be zero. The problem posed reduces to finding a function \(U(r,\theta,z)\) satisfying Laplace’s equation
\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial U}{\partial r}\right) +\frac{1}{r^{2}}\frac{\partial^{2}U}{\partial\theta^{2}} +\frac{\partial^{2}U}{\partial z^{2}}=0 \tag{3.1} \]
and the boundary conditions
\[ \left(\frac{\partial U}{\partial r}+\beta U\right)_{r=a}=0,\qquad \left.\frac{\partial U}{\partial\theta}\right|_{\theta=0}=0, \]
\[ \left.\frac{\partial U}{\partial\theta}\right|_{\theta=\alpha} =f(r,z)=\frac{rq(r,z)}{k}. \tag{3.2} \]
(\(k\) and \(\beta\) are, respectively, the coefficients of thermal conductivity and heat exchange.)
If the solution of the posed problem is sought in the form of a Fourier integral
\[ U(r,\theta,z)=\int_{-\infty}^{\infty}V(r,\theta,\mu)\exp i\mu z\,d\mu \tag{3.3} \]
and separation of variables is carried out,
\[ V(r,\theta,\mu)=R(r)T(\theta) \tag{3.4} \]
in the equation
\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial V}{\partial r}\right) +\frac{1}{r^{2}}\frac{\partial^{2}V}{\partial\theta^{2}} -\mu^{2}V=0, \tag{3.5} \]
then we arrive at the singular problem (1.1)—(1.3) considered in § 1, where
\[ x=\mu r,\qquad y(x,\tau)=R\left(\frac{x}{\mu}\right),\qquad \gamma=\mu a. \tag{3.6} \]
Putting further
\[ V=\int_{0}^{\infty}\frac{F(\tau)\operatorname{ch}\theta\tau}{\tau\operatorname{sh}a\tau}\,y(x,\tau)\,d\tau, \tag{3.7} \]
we satisfy the second of equalities (3.2), and the last boundary condition leads to the relation
\[ \left.\frac{\partial V}{\partial\theta}\right|_{\theta=a} = \int_{0}^{\infty}F(\tau)y(x,\tau)\,d\tau. \tag{3.8'} \]
Since, by the Fourier inversion theorem, the quantity
\[ \left.\frac{\partial V}{\partial\theta}\right|_{\theta=a} = \frac{1}{2\pi}\int_{-\infty}^{\infty} f\left(\frac{x}{\mu},z\right)\exp(-i\mu z)\,dz \tag{3.9} \]
is a known function of the variable \(x\), the sought function \(F(\tau)\) can be found by formula (2.12), after which the solution of the problem is given by expressions (3.3) and (3.6).
The method set forth is also applicable to the case of a wedge of finite length. In this case the integral in formula (3.3) is replaced by a Fourier series corresponding to the boundary conditions for \(z=\mathrm{const}\).
References
- Titchmarsh E. C. Eigenfunction expansions associated with second-order differential equations, 1, IL, 1960.
- Kontorovich M. I., Lebedev N. N. ZhETF, 8, 10—11, 1192, 1938.
- Lebedev N. N. PMM, 13, 5, 465, 1949.
- Ditkin V. A., Prudnikov A. P. Integral transforms and operational calculus. Fizmatgiz, Moscow, 1961.
- Naylor D. Journal of Math. and Mech., 12, 3, 375, 1963.