Abstract Generated abstract
The paper constructs finite integral Hankel transforms from expansions in eigenfunctions of a Sturm-Liouville problem whose differential equation is Bessel’s equation on a finite interval with third-kind boundary conditions. It derives the eigenfunctions, characteristic equation, normalization, and inversion formula, showing that known finite Hankel transforms arise as special cases. The limiting interval with one endpoint at zero is analyzed separately for different orders, including the cases m less than one half and m equal to zero, yielding generalized kernels and eigenvalue equations. The transform is then applied to reduce boundary-value problems for equations separable in a cylindrical radial variable, incorporating radial boundary conditions into transformed equations.
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MATHEMATICS
M. V. ELISTRATOVA
ON FINITE INTEGRAL HANKEL TRANSFORMS
(Presented by Academician I. N. Vekua, 3 V 1961)
The known finite integral Fourier and Hankel transforms are introduced from expansions of a function \(f(x)\) in Fourier, Fourier–Bessel, and Bessel–Dini series. It is natural, in order to obtain new finite integral transforms, to consider more general orthogonal series—series in eigenfunctions.
Let us expand the function \(f(x)\) in a Fourier series
\[ f(x)=\sum_n \frac{\displaystyle \int_a^b x Q_n(\lambda,x) f(x)\,dx} {\displaystyle \int_a^b x Q_n^2(\lambda,x)\,dx} \,Q_n(\lambda,x) \tag{1} \]
in the eigenfunctions of the Sturm–Liouville problem
\[ (xu')' + \left[\lambda^2 x-\frac{m^2}{x}\right]u(x)=0, \qquad a \leq x \leq b,\qquad h \geq 0,\qquad H \geq 0; \]
\[ u'-hu\big|_{x=a}=0;\qquad u'+Hu\big|_{x=b}=0. \tag{2} \]
It is not difficult to see that the eigenfunctions are equal to*
\[ Q_n(\lambda_n x)=\lambda_n V_m(\lambda_n x)+H W_m(\lambda_n x), \tag{3} \]
where the notation has been introduced
\[ V_m(\lambda_n x)=Y'_m(\lambda_n b)J_m(\lambda_n x) - J'_m(\lambda_n b)Y_m(\lambda_n x), \]
\[ W_m(\lambda_n x)=Y_m(\lambda_n b)J_m(\lambda_n x) - J_m(\lambda_n b)Y_m(\lambda_n x). \tag{4} \]
The eigenvalues \(\lambda_n\) are the positive roots of the transcendental equation
\[ \lambda_n^2 V'_m(\lambda_n a) + H\lambda_n W'_m(\lambda_n a) - h\lambda_n V_m(\lambda_n a) - hH W_m(\lambda_n a) =0. \tag{5} \]
Obviously (*),
\[ \int_a^b x Q_m^2(\lambda_n x)\,dx = \frac{x^2}{2} \left\{ \left(1-\frac{m^2}{\lambda_n^2 x^2}\right)Q_m^2(\lambda_n x) + Q_m'^2(\lambda_n x) \right\}\Bigg|_a^b, \]
\[ f(x)=2\sum_n \frac{\displaystyle \lambda_n^2\int_a^b x\,[\lambda_n V_m(\lambda_n x)+H W_m(\lambda x)]f(x)\,dx} {\displaystyle [b^2\lambda_n^2-m^2+b^2H^2]\lambda_n^2 V_m^2(\lambda_n b) - [a^2\lambda_n^2-m^2+a^2h^2]Q_m^2(\lambda_n a)} \,Q_m(\lambda_n x). \tag{6} \]
\[ \text{* The prime everywhere denotes differentiation with respect to } \lambda_n x. \]
Summation in (6) is carried out over all positive roots of equation (5). But
\[ V_m(\lambda_n b)=\frac{2}{\pi \lambda_n b};\qquad Q_m(\lambda_n a)= \frac{2}{\pi a}\, \frac{\lambda_n J'_m(\lambda_n b)+HJ_m(\lambda_n b)} {\lambda_n J'_m(\lambda_n a)-hJ_m(\lambda_n a)}, \]
and the series (6) can be represented in the form
\[ f(x)=\frac{\pi^2}{2}\sum_n \times \]
\[ \times \frac{ \lambda_n^2[\lambda_n J'_m(\lambda_n a)-hJ_m(\lambda_n a)]^2 \displaystyle \int_a^b x[\lambda_n V_m(\lambda_n x)+HW_m(\lambda_n x)]f(x)\,dx }{ \displaystyle \frac{b^2\lambda_n^2-m^2+b^2H^2}{b^2} [\lambda_n J'_m(\lambda_n a)-hJ_m(\lambda_n a)]^2 - \frac{a^2\lambda_n^2-m^2+a^2h^2}{a^2} [\lambda_n J'_m(\lambda_n b)+HJ_m(\lambda_n b)]^2 } \times \]
\[ \times Q_m(\lambda_n x). \tag{7} \]
Introduce the finite integral transform
\[ \bar f(\lambda_n)=\int_a^b x[\lambda_n V_m(\lambda_m x)+HW_m(\lambda_n x)]f(x)\,dx. \tag{8} \]
The inversion formula for it is obtained from expansion (7). Following Sneddon, we shall call the transform (8)—(7) a Hankel transform. For particular values of \(h\) and \(H\) we obtain the known finite Hankel transforms.
Let us pass to the interval \([0,b]\). For fixed \(\lambda\), every solution of Bessel’s equation has a finite number of zeros on the interval \([a,b]\) as \(a\to0\). It follows from this \({}^{(2)}\) that the equation \((xu')'+[\lambda^2x-\frac{m^2}{x}]u(x)=0\) and the boundary condition \(u'+Hu\big|_{x=b}=0\) determine a discrete spectrum of eigenvalues on \([0,b]\); the corresponding eigenfunctions are square-integrable. For \(m\geq \tfrac12\), only \(J_m(\lambda_n x)\) is square-integrable in a neighborhood of zero, and the requirement of integrability replaces the boundary condition \(u'-hu\big|_{x=a}=0\). The eigenfunctions in this case are \(J_m(\lambda_n x)\), and the eigenvalues satisfy the equation \(\lambda_n J'_m(\lambda_n b)+HJ_m(\lambda_n b)=0\). This case corresponds to the second Hankel transform, and for \(H\to\infty\) to the first.
If \(m<\tfrac12\), \(J_m(\lambda_n x)\) and \(Y_m(\lambda_n x)\) are square-integrable, and the requirement of integrability does not replace the boundary condition at the left endpoint. Let us see how the latter is transformed as \(a\to0\).
Expression (3) for the eigenfunctions can be rewritten in the form
\[ Q_m(\lambda_n x)= -\frac{1}{\sin \pi m}\{[\lambda_n J'_{-m}(\lambda_n b)+HJ_{-m}(\lambda_n b)]J_m(\lambda_n x)- \]
\[ -[\lambda_n J'_m(\lambda_n b)+HJ_m(\lambda_n b)]J_{-m}(\lambda_n x)\}. \]
For fixed \(\lambda\), \(m\ne0\), and \(x\to0\), the estimates
\[ J_m(\lambda x)=\frac{(\lambda x)^m}{2^m\Gamma(1+m)}+O(x^{m+2});\qquad J'_m(\lambda x)=m\,\frac{(\lambda x)^{m-1}}{2^m\Gamma(1+m)}+O(x^{m+1)}. \]
are valid. Therefore the characteristic equation, as \(a\to0\), takes the form
\[ \frac{\lambda_n^m}{2^m\Gamma(1+m)} [\lambda_n J'_{-m}(\lambda_n b)+HJ_{-m}(\lambda_n b)](ma^{m-1}-ha^m)+ \]
\[ +\frac{\lambda_n^{-m}}{2^{-m}\Gamma(1-m)} (ma^{-m-1}+ha^{-m})[\lambda_n J'_m(\lambda_n b)+ \]
\[ +HJ_m(\lambda_n b)]+O(a^{-m+1})+O(a^{-m+2})=0. \tag{9} \]
We shall vary \(h\) together with \(a\) according to the law
\[ h=\frac{C\cdot 2^{-m}\Gamma(1-m)ma^{m-1}-2^{m}\Gamma(1+m)ma^{-m-1}} {C\cdot 2^{-m}\Gamma(1-m)a^{m}+2^{m}\Gamma(1+m)a^{-m}} \]
(\(C\) is an arbitrary constant), which is possible, since the spectrum is completely determined by the boundary condition at the right end.
Since the order of the expression \(ma^{m-1}-ha^m\) with respect to \(a\) is \(m-1\), in equation (9) one may omit the terms of orders \(-m+1\) and \(-m+2\), if \(-m+1>m-1\), and for \(m<1/2\) the eigenvalues satisfy the equation
\[ \lambda_n^{2m}\left[\lambda_n J'_{-m}(\lambda_n b)+HJ_{-m}(\lambda_n b)\right] +C\left[\lambda_n J'_m(\lambda_n b)+HJ_m(\lambda_n b)\right]=0. \tag{10} \]
The eigenfunctions are
\[ Q_m(\lambda_n x)=CJ_m(\lambda_n x)+\lambda_n^{2m}J_{-m}(\lambda_n x). \tag{11} \]
Thus, for \(m<1/2\) one may introduce the Hankel transform
\[ \bar f(\lambda_n)=\int_0^b x\left[CJ_m(\lambda_n x)+\lambda_n^{2m}J_{-m}(\lambda_n x)\right]f(x)\,dx, \]
where \(C\) is an arbitrary constant, and \(\lambda_n\) are the positive roots of the transcendental equation (10). This transform generalizes the second Hankel transform, obtained for an infinitely large value of \(C\).
Repeating analogous arguments for \(m=0\) and using the estimates
\[ Y_0(\lambda x)=\frac{2}{\pi}\left[\gamma+\ln\frac{\lambda}{2}\right]\left[1+O(x^2)\right];\qquad Y'_0(\lambda x)=\frac{2}{\pi\lambda x}+O(x|\ln x|), \]
we obtain an integral transform whose kernel is equal to
\[ x\left[\left(\frac{2}{\pi}\ln\lambda_n-C\right)J_0(\lambda_n x)-Y_0(\lambda_n x)\right], \tag{12} \]
and the eigenvalues satisfy the equation
\[ \left[\lambda_n Y'_0(\lambda_n b)+HY_0(\lambda_n b)\right] -\left[\lambda_n J'_0(\lambda_n b)+HJ_0(\lambda_n b)\right]\left(\frac{2}{\pi}\ln\lambda_n-C\right)=0. \tag{13} \]
Let us show that the finite integral transform introduced in (8)—(7) makes it possible to solve effectively boundary-value problems of the third kind for equations for which the corresponding homogeneous equations admit at least a partial separation of variables and represent, with respect to the separable variable, a Bessel equation of order \(m\) for hollow cylindrical bodies.
Suppose it is required to integrate the equation
\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)-\frac{m^2}{r^2}u +D_1(z,\theta,t)u=f(r,z,\theta,t) \tag{14} \]
under boundary conditions with respect to \(r\)
\[ \left.\frac{\partial u}{\partial r}-hu\right|_{r=r_0}=F_1(z,\theta,t);\qquad \left.\frac{\partial u}{\partial r}+Hu\right|_{r=R}=F_2(z,\theta,t);\qquad h\geqslant 0;\quad H\geqslant 0. \tag{15} \]
We pass to transforms by multiplying both sides of equation (14) by \(rQ_m(\lambda_n r)\) and integrating with respect to \(r\) from \(r_0\) to \(R\), where \(Q_m(\lambda_n r)\) are the eigenfunctions of problem (2).
We then obtain the equation
\[ D_1\bar{u}(\lambda_n)-\lambda_n^2\bar{u}(\lambda_n) = r_0Q_m(\lambda_n r_0)F_1(z,\theta,t) - RQ_m(\lambda_n R)F_2(z,\theta,t) + \int_{r_0}^{R} rQ_m(\lambda_n r)f\,dr, \tag{16} \]
which does not contain the variable \(r\) and includes the boundary conditions with respect to \(r\).
The desired solution is represented by the series in eigenfunctions
\[ u=\frac{\pi^2}{2}\sum_n \left\{ \left[ \lambda_n^2\,[\lambda_n J'_m(\lambda_n r_0)-hJ_m(\lambda_n r_0)]^2 \int_{r_0}^{R} r[\lambda_n V'_m(\lambda_n r)+HW'_m(\lambda_n r)]u\,dr \right] \left[ \frac{R^2\lambda_n^2-m^2+R^2H^2}{R^2} [\lambda_n J'_m(\lambda_n r_0)-hJ_m(\lambda_n r_0)]^2 - \frac{r_0^2\lambda_n^2-m^2+r_0^2h^2}{r_0^2} [\lambda_n J'_m(\lambda_n R)+HJ_m(\lambda_n R)]^{-1} Q_m(\lambda_n r) \right] \right\}. \tag{17} \]
The summation in (17) extends over all positive roots of the transcendental equation (5).
Kuibyshev Industrial Institute
named after V. V. Kuibyshev
Received
27 IV 1961
REFERENCES
¹ G. N. Watson, A Treatise on the Theory of Bessel Functions, Part 1, IL, 1949. ² B. M. Levitan, Expansion in Eigenfunctions, 1950.