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On Certain Hankel Transformations
A. I. Tseitlin
In a number of problems of mathematical physics, for example in axisymmetric problems of the theory of elasticity (the plane problem, bending and vibrations of plates and shells, etc.), one encounters differential equations containing the operation
\[ l_1^2(y)=\left(\frac{d^2}{dx^2}+\frac{1}{x}\frac{d}{dx}\right)^2 y . \tag{1} \]
For solving these equations on the half-axis \([0,\infty)\), the classical integral Hankel transform and operational calculus for the operator \(B=\frac{d}{dx}x\frac{d}{dx}\) [1] are applicable. In the present note we shall construct integral transforms that make it possible to solve such equations on the half-interval \([a,\infty)\), \(a>0\), when the methods mentioned cannot be applied. The kernels of these transforms consist of cylindrical functions, and therefore, following Sneddon [2], we shall refer them to Hankel transforms.
Consider the self-adjoint operation
\[ l_2^2(z)=\left(\frac{d^2}{dx^2}+\frac{1}{4x^2}\right)^2 z \quad (a\leq x<\infty), \tag{2} \]
which, under the substitution \(z=x^{1/2}y\), is reduced to operation (1) on \(y\) with weight \(x^{1/2}\); moreover, without loss of generality one may put \(a=1\). Denote by \(L\) the self-adjoint linear singular operator generated by operation (2) and defined on the set of all functions \(z(x)\in L_2(1,\infty)\) satisfying certain boundary conditions at the regular end \(x=1\). Here it is assumed that the functions \(z(x)\), together with their derivatives up to the third order inclusive, are absolutely continuous, and that \(z^{\mathrm{IV}}(x)\in L_2(1,\infty)\). We shall be interested in self-adjoint boundary conditions for the function \(y(x)=x^{-1/2}z(x)\) of the following type:
\[ \theta_1 y'(1)+y''(1)=0, \]
\[ \theta_2 y(1)+y'(1)-y''(1)-y'''(1)=0 \quad (\operatorname{Im}\theta_1=\operatorname{Im}\theta_2=0). \tag{3} \]
Such boundary conditions occur, in particular, in the plane problem of the theory of elasticity and in problems on plates. Under the boundary conditions (3) the operator \(L\) has a simple spectrum and, consequently, on the basis of the theorem on expansion in eigenfunctions of a self-adjoint operator [3], for all \(f(x)\in L_2(1,\infty)\) the formula
\[ f(x)=\int_{-\infty}^{\infty} F(\lambda) z(\lambda,x)\,d\sigma(\lambda) \tag{4} \]
holds.
where \(z(\lambda,x)\) is the solution of the equation, bounded at infinity,
\[ l_2^2(z)-\lambda z \equiv l_2(z)\pm \lambda^{1/2}z=0\quad (\operatorname{Im}\lambda=0), \tag{5} \]
satisfying the boundary conditions at \(x=1\); \(\sigma(\lambda)\) is the spectral distribution function. The integral (4) converges in the mean square. The function \(F(\lambda)\) belongs to \(L_{2,\sigma}\) and is determined by the formula
\[ F(\lambda)=\int_1^\infty f(x)z(\lambda,x)\,dx, \tag{6} \]
where the integral converges in \(L_{2,\sigma}\). On the basis of formulas (4) and (6), integral transformations may be introduced; for this it is only necessary to determine the unknown function \(\sigma(\lambda)\). The usual methods for constructing the distribution function, for example the use of the Stieltjes inversion formula or passage to the limit from a regular interval to a singular one, in the case under consideration lead to known difficulties. Therefore it will be simpler to proceed in another way, connected with the use of generalized functions, whose theory has now been developed with all necessary rigor.
From (4) and (6) we have:
\[ s(\lambda)\int_1^\infty z(\lambda,x)z(\lambda_1,x)\,dx=\delta(\lambda-\lambda_1), \tag{7} \]
\[ \int_{-\infty}^{\infty} z(\lambda,x)z(\lambda,x_1)s(\lambda)\,d\lambda =\delta(x-x_1)+T(x), \tag{8} \]
where \(T(x)\) and \(s(\lambda)=\sigma'(\lambda)\) are certain generalized functions, with \(T(x)\) having no essential points for \(x>1\); \(\delta(\lambda-\lambda_1)\) is the delta function. Relation (7) can be used to determine the function \(s(\lambda)\). Let us represent the solution of equation (5), bounded at infinity, for \(\lambda>0\) in the form
\[ z(\xi,x)=\sqrt{x}\,[J_0(\xi x)+B(\xi)Y_0(\xi x)+C(\xi)K_0(\xi x)],\quad \xi=|\lambda^{1/4}|, \tag{9} \]
where \(J_0(\xi x)\), \(Y_0(\xi x)\) are Bessel functions of the first and second kinds; \(K_0(\xi x)\) is the Macdonald function; \(B(\xi)\), \(C(\xi)\) are arbitrary functions determined from the boundary conditions at \(x=1\). Using (9), we rewrite (7) in the form
\[ s(\lambda)\left[ D(\xi,\xi_1)\int_0^\infty xJ_0(\xi x)J_0(\xi_1 x)\,dx +\frac{B(\xi)B(\xi_1)}{J_0(\xi)J_0(\xi_1)} \right. \]
\[ \left. {}\times \int_1^\infty xW_0(\xi,x)W_0(\xi_1,x)\,dx +K(\lambda,\lambda_1) -D(\xi,\xi_1)\times \int_0^1 xJ_0(\xi x)J_0(\xi_1 x)\,dx \right] =\delta(\lambda-\lambda_1). \tag{10} \]
Here
\[ D(\xi,\xi_1)=1-\frac{B(\xi)B(\xi_1)Y_0(\xi)Y_0(\xi_1)} {J_0(\xi)J_0(\xi_1)}; \]
\(K(\lambda,\lambda_1)\) is an expression containing integrals in whose integrands enter the Macdonald functions or products of Bessel functions of the first and second kinds;
\[ W_0(\xi,x)=J_0(\xi)Y_0(\xi x)-Y_0(\xi)J_0(\xi x). \]
is the kernel of Weber’s integral transform (see, for example, [1]), defined by the formulas
\[ F(\xi)=\int_1^\infty x f(x) W_0(\xi,x)\,dx;\qquad f(x)=\int_0^\infty \frac{\xi W_0(\xi,x)F(\xi)\,d\xi}{J_0^2(\xi)+Y_0^2(\xi)}. \tag{11} \]
From (11) we have
\[ \xi \int_1^\infty x W_0(\xi,x) W_0(\xi_1,x)\,dx = [J_0^2(\xi)+Y_0^2(\xi)]\delta(\xi-\xi_1). \tag{12} \]
Moreover,
\[ \xi \int_0^\infty x J_0(\xi x)J_0(\xi_1 x)\,dx=\delta(\xi-\xi_1). \tag{13} \]
Since \(K(\lambda,\lambda_1)\) does not contain singular generalized functions, it follows from (10), (11), and (12) that
\[ K(\lambda,\lambda_1)-D(\xi,\xi_1)\int_0^1 xJ_0(\xi x)J_0(\xi_1 x)\,dx \equiv 0. \tag{14} \]
Now, integrating the right- and left-hand sides of (10) over all \(\lambda\), we find
\[ s(\xi)=\frac{1}{4\xi^2[1+B^2(\xi)]}. \tag{15} \]
In the case \(\lambda<0\), the solution of equation (5) has the form
\[ z(\xi,x)=\sqrt{x}\,[A_1(\xi)u_0(\xi x)+B_1(\xi)v_0(\xi x)+C_1(\xi)f_0(\xi x)+D_1(\xi)g_0(\xi x)], \tag{16} \]
where
\[ u_0(x)=\operatorname{Re}J_0(x\sqrt{i});\qquad v_0(x)=\operatorname{Im}J_0(x\sqrt{i});\qquad f_0(x)=\operatorname{Re}H_0^{(1)}(x\sqrt{i}), \]
\[ g_0(x)=\operatorname{Im}H_0^{(1)}(x\sqrt{i}) \]
—functions analogous to Thomson functions. As \(x\to\infty\), the following asymptotic formulas hold [4, 5]:
\[ u_0(x)\sim (2\pi x)^{-1/2}\exp\left(\frac{x}{\sqrt{2}}\right) \cos\left(\frac{x}{\sqrt{2}}-\frac{\pi}{8}\right); \]
\[ v_0(x)\sim -(2\pi x)^{-1/2}\exp\left(\frac{x}{\sqrt{2}}\right) \sin\left(\frac{x}{\sqrt{2}}-\frac{\pi}{8}\right); \]
\[ f_0(x)\sim \sqrt{\frac{2}{\pi x}}\exp\left(-\frac{x}{\sqrt{2}}\right) \sin\left(\frac{x}{\sqrt{2}}+\frac{\pi}{8}\right); \tag{17} \]
\[ g_0(x)\sim -\sqrt{\frac{2}{\pi x}}\exp\left(-\frac{x}{\sqrt{2}}\right) \cos\left(\frac{x}{\sqrt{2}}+\frac{\pi}{8}\right), \]
therefore in (16) one must set \(A_1=B_1=0\), and
\[ z(\xi,x)=C_1(\xi)\sqrt{x}\,f_0(\xi x)+D_1(\xi)\sqrt{x}\,g_0(\xi x). \tag{18} \]
Satisfying the boundary conditions at \(x=1\) and setting the characteristic determinant equal to zero, we obtain an equation for determining the discrete eigenvalues. Let us consider the three cases of boundary conditions most important for plates:
- \(\theta_1=\theta_2=\infty\) (clamped edge);
- \(\theta_1=\nu,\quad \theta_2=\infty\); \(\nu\) is Poisson’s ratio (supported edge);
- \(\theta_1=\nu,\quad \theta_2=0\) (free edge).
Accordingly we obtain the three equations:
\[ \begin{gathered} \Delta_1(\xi)=f_0(\xi)g'_0(\xi)-f'_0(\xi)g_0(\xi)=0,\\ \Delta_2(\xi)=f_0(\xi)f^{(M)}_0(\xi)-g_0(\xi)g^{(M)}_0(\xi)=0,\\ \Delta_3(\xi)=f'_0(\xi)g^{(M)}_0(\xi)+g'_0(\xi)f^{(M)}_0(\xi)=0, \end{gathered} \tag{19} \]
where
\[ f^{(M)}_0(\xi)=-f_0(\xi)-(1-\nu)\frac{g'_0(\xi)}{\xi}; \]
\[ g^{(M)}_0(\xi)=g_0(\xi)-(1-\nu)\frac{f'_0(\xi)}{\xi} \]
are the functions introduced and tabulated by B. G. Korenev [4]. The graphs of the functions \(\Delta_1(\xi)\), \(\Delta_2(\xi)\), \(\Delta_3(\xi)\) are presented in the figure. These functions, as we see, have no zeros, and, consequently, the operator \(L\) in the three cases of boundary conditions considered has no discrete eigenvalues \(\lambda<0\). \(\lambda=0\) is also not an eigenvalue, since in this case
\[ z(x)=\sqrt{x}\,[A_2\ln x+B_2x^2\ln x+C_2x^2+D_2], \tag{20} \]
and the condition at infinity leads to the trivial solution. Thus, the spectrum of the operator \(L\) is continuous and fills the entire half-axis \(\lambda>0\). The function \(s(\lambda)\) for \(\lambda>0\) is determined by formula (15), while for \(\lambda<0\) it is equal to zero. The inversion formulas can now be written in the form
\[ f(x)=\int_0^\infty \Phi(\xi)R(\xi,x)\,d\xi, \]
\[ \Phi(\xi)=\int_1^\infty f(x)R(\xi,x)\,dx, \tag{21} \]
where
\[ R(\xi,x)=\frac{J_0(\xi x)+B(\xi)Y_0(\xi x)+C(\xi)K_0(\xi x)} {2\xi[1+B^2(\xi)]^{1/2}} \]
is the kernel of the integral transform obtained. The coefficients \(B(\xi)\) and \(C(\xi)\) will be: for boundary conditions 1)
\[ B(\xi)=-\frac{K_0(\xi)J'_0(\xi)-K'_0(\xi)J_0(\xi)} {K'_0(\xi)Y_0(\xi)-K_0(\xi)Y'_0(\xi)},\qquad C(\xi)=-\frac{J_0(\xi)+B(\xi)Y_0(\xi)}{K_0(\xi)}; \tag{22} \]
for boundary conditions 2)
\[ B(\xi)=-\frac{ 2J_0(\xi)K_0(\xi)+\dfrac{1-\nu}{\xi}\,[J'_0(\xi)K_0(\xi)-J_0(\xi)K'_0(\xi)] }{ 2Y_0(\xi)K_0(\xi)+\dfrac{1-\nu}{\xi}\,[Y'_0(\xi)K_0(\xi)-Y_0(\xi)K'_0(\xi)] }. \tag{23} \]
\[ C(\xi)=-\frac{J_0(\xi)+B(\xi)Y_0(\xi)}{K_0(\xi)}; \tag{23} \]
under boundary conditions 3)
\[ B(\xi)=-\frac{ 2\frac{1-\nu}{\xi}J_0'(\xi)K_0'(\xi)+K_0'(\xi)J_0(\xi)-K_0(\xi)J_0'(\xi) }{ 2\frac{1-\nu}{\xi}Y_0'(\xi)K_0'(\xi)+K_0'(\xi)Y_0(\xi)-K_0(\xi)Y_0'(\xi) }, \]
\[ C(\xi)=\frac{J_0'(\xi)+B(\xi)Y_0'(\xi)}{K_0'(\xi)}. \tag{24} \]
When the integral transform (21) is used, the coefficients \(B(\xi)\) and \(C(\xi)\) are chosen depending on the boundary conditions of the problem.
In conclusion, let us note that the use of generalized functions makes it possible to extend the integral transforms considered, in addition to \(L_2(1,\infty)\), to other classes of functions, analogously to what occurs in the case of the Fourier transform [6].
References
- V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus. Fizmatgiz, 1961.
- I. Sneddon, Fourier Transforms. IL, 1955.
- M. A. Naimark, Linear Differential Operators. Gostekhizdat, 1954.
- B. G. Korenev, Some Problems of the Theory of Elasticity and Heat Conduction Solvable in Bessel Functions. Fizmatgiz, 1960.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Part 1. IL, 1949.
- I. M. Gel'fand and G. E. Shilov, Generalized Functions and Operations on Them. Fizmatgiz, 1958.
Received by the editors
April 26, 1965
Central Scientific Research Institute of Building Structures
named after V. A. Kucherenko