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UDC 517.948.3
ON A CLASS OF DUAL INTEGRAL EQUATIONS
A. I. TSEITLIN
The solution of boundary-value problems of mathematical physics with mixed boundary conditions by the method of integral transforms (or by expansion in eigenfunctions) is connected with determining the expansion coefficients from dual integral equations or dual series. In applications one especially often encounters dual equations whose kernels are cylindrical functions and, as a special case, trigonometric functions; in most works on the theory of dual integral equations the kernel is also considered to be just such a kernel. The first dual equations with a kernel in the form of a Bessel function were solved by Titchmarsh [1] and Busbridge [2]. Subsequently, dual equations with a Bessel kernel were studied in detail in works by various authors and were successfully applied to the solution of boundary-value problems of electrostatics, elasticity theory, and hydrodynamics. Shrivastav [3] considered dual integral equations with the kernel of the Weber transform, which is a certain combination of Bessel and Neumann functions.
In this note a formal solution is given for one class of dual integral equations generated by integral transforms constructed on the basis of expansions in eigenfunctions of the Bessel differential operator. The kernels of these equations may be cylindrical functions or various combinations of them, as well as functions related to cylindrical functions.
Consider dual integral equations of the following form:
\[ \int_{0}^{\infty} \xi^{2m} f(\xi) W_{\nu}(\xi,\eta)\,\xi\,d\xi = g_{1}(\eta) \qquad (0 \leq a < \eta < c), \]
\[ \int_{0}^{\infty} f(\xi) W_{\nu}(\xi,\eta)\,\xi\,d\xi = g_{2}(\eta) \qquad (c < \eta < \infty), \tag{1} \]
where \(m=1,2,3,\ldots\); \(f(\xi)\) is an unknown function; \(g_{1}(\eta)\) and \(g_{2}(\eta)\) are given functions. It is assumed that the kernel \(W_{\nu}(\xi,\eta)\) of the dual equations (1) is the inversion kernel of some integral transform which is an expansion in eigenfunctions of the Bessel differential operator on the interval \((a,\infty)\), i.e., that for any function \(\varphi(x)\in L_{2}(a,\infty)\) the inversion formulas
\[ \Phi(\xi)=\int_{a}^{\infty} x\varphi(x) Z_{\nu}(\xi,x)\,dx, \qquad \varphi(x)=\int_{0}^{\infty} \xi\Phi(\xi) W_{\nu}(\xi,x)\,d\xi, \tag{2} \]
hold, where
\[ Z_{\nu}(\xi,x)=A(\xi)J_{\nu}(\xi x)+B(\xi)N_{\nu}(\xi x) \]
is some combination of cylindrical functions of the first and second kind. It is also assumed that \(g_{1}(\eta)\) and \(g_{2}(\eta)\) are sufficiently regular so that the integrals considered below make sense. It is convenient to transform the inversion formulas (2) to the following equivalent form, using generalized functions:
\[ \xi_{1}\int_{a}^{\infty} x Z_{\nu}(\xi,x) W_{\nu}(\xi_{1},x)\,dx = \delta(\xi_{1}-\xi), \tag{3} \]
\[ x_{1}\int_{0}^{\infty} \xi Z_{\nu}(\xi,x_{1}) W_{\nu}(\xi,x)\,d\xi = \delta(x_{1}-x), \tag{4} \]
where \(\delta\) is the delta function.
We symmetrize the dual equations (1), setting \(f_1(\xi)=\xi^m f(\xi)\), and bring them to the form
\[ \begin{aligned} &\int_0^\infty \xi^{m+1} f_1(\xi) W_\nu(\xi,\eta)\,d\xi = g_1(\eta) \qquad (a<\eta<c),\\ &\int_0^\infty \xi^{-m+1} f_1(\xi) W_\nu(\xi,\eta)\,d\xi = g_2(\eta) \qquad (c<\eta<\infty). \end{aligned} \tag{5} \]
To solve the dual equations (5) one can construct integral operators that transform them into a single equation posed on the whole interval \((a,\infty)\) and admitting a solution by means of the inversion formulas (2). The kernels of these integral operators have the form
\[ K(x,\eta)=\int_0^\infty \xi^{-m+1} W_{\nu+m}(\xi,x) Z_\nu(\xi,\eta)\,d\xi, \tag{6} \]
\[ H(x;\eta)=\int_0^\infty \xi^{m+1} W_{\nu+m}(\xi,x) Z_\nu(\xi,\eta)\,d\xi, \tag{7} \]
where the integrals in (6) and (7) converge in the sense of generalized functions. Indeed, multiplying the left- and right-hand sides of the first equation (5) by \(\eta K(x,\eta)\) and integrating with respect to \(\eta\) from \(a\) to \(\infty\), after changing the order of integration and using relation (3), we obtain
\[ \int_0^\infty f_1(\xi) W_{\nu+m}(\xi,x)\,\xi\,d\xi = \int_a^\infty g_1(\eta)K(x,\eta)\,\eta\,d\eta. \tag{8} \]
An analogous result is obtained after multiplying the left- and right-hand sides of the second equation (5) by \(\eta H(x,\eta)\) and integrating with respect to \(\eta\) from \(a\) to \(\infty\):
\[ \int_0^\infty f_1(\xi) W_{\nu+m}(\xi,x)\,\xi\,d\xi = \int_a^\infty g_2(\eta)H(x,\eta)\,\eta\,d\eta. \tag{9} \]
The functions \(g_1(\eta)\) and \(g_2(\eta)\) are not defined on the whole interval \((a,\infty)\); however, as will be seen below, this difficulty is easily avoided.
Consider the kernels (6) and (7). The integral standing on the right-hand side of (6) belongs to the discontinuous integrals, various special cases of which have been considered in the literature on Bessel functions [4, 5]. These integrals are characterized by the fact that they are equal to zero for \(\eta>x\). Consequently, equation (8) is completely determined on the interval \((a,c)\), for which it is sufficient to prescribe the function \(g_1(\eta)\) in the range \(a<\eta<c\).
On the other hand, replacing in (4) \(x_1\) by \(\eta\) and \(\nu\) by \(\nu+m\), applying to both sides the operator
\[ l=\left(\frac{d}{\eta\,d\eta}\right)^m \]
and using the known formula
\[ \left(\frac{d}{\eta\,d\eta}\right)^m \left[\eta^\nu Z_\nu(\eta)\right] = \eta^{\nu-m} Z_{\nu-m}(\eta), \]
we obtain
\[ \int_0^\infty \xi^{m+1} W_{\nu+m}(\xi,x) Z_\nu(\xi,\eta)\,d\xi = \eta^{m-1} l[\delta(\eta-x)]. \tag{10} \]
Here \(l[\delta(\eta-x)]\) is a generalized function acting according to the formula [6]
\[ (f(\eta),\,l[\delta(\eta-x)])=l[f(x)]. \tag{11} \]
Consequently, now (8) and (9) can be written in the form
\[ \int_0^\infty f_1(\xi) W_{\nu+m}(\xi,x)\,\xi\,d\xi = \begin{cases} \displaystyle \int_a^x g_1(\eta)K(x,\eta)\,\eta\,d\eta, & (a<x<c),\\[1.2ex] \displaystyle x^m\,l[g_2(x)], & (c<x<\infty). \end{cases} \tag{12} \]
Using the inversion formula (2), we obtain the solution of the dual equations
\[ f_1(\xi) = \int_a^c Z_{\nu+m}(\xi,x)\,x\,dx \int_a^x g_1(\eta)K(x,\eta)\,\eta\,d\eta + \int_c^\infty x^{m+1} Z_{\nu+m}(\xi,x)\,l[g_2(x)]\,dx. \tag{13} \]
Let, for example, \(a=0\), \(-\dfrac{3}{2}<\nu<-\dfrac{1}{2}\), and \(Z_\nu(\xi,x)=N_\nu(\xi x)\), where \(N_\nu(\xi x)\) is a cylindrical function of the second kind. Then, as is known, \(W_\nu(\xi,x)=\mathbf H_\nu(\xi x)\), where \(\mathbf H_\nu(x)\) is the Struve function, which is a solution of the differential equation
\[ x^2y''+xy'+(x^2-\nu^2)y= \frac{1}{\sqrt{\pi}}\, \frac{4}{\Gamma(\nu+1/2)} \left(\frac{x}{2}\right)^{\nu+1}. \]
The dual integral equations (5) in this case have the form
\[ \int_0^\infty \xi^{m+1} f_1(\xi)\mathbf H_\nu(\xi\eta)\,d\xi = g_1(\eta) \qquad (0<\eta<c), \]
\[ \int_0^\infty \xi^{-m+1} f_1(\xi)\mathbf H_\nu(\xi\eta)\,d\xi = g_2(\eta) \qquad (c<\eta<\infty). \tag{14} \]
According to (6),
\[ K(x,\eta)=\int_0^\infty \xi^{-m+1}\mathbf H_{\nu+m}(\xi x)N_\nu(\xi\eta)\,d\xi, \]
whence [5]
\[ K(x,\eta)= \begin{cases} \dfrac{2^{1-m}\eta^\nu}{x^{\nu+m}\Gamma(m)} (x^2-\eta^2)^{m-1}, & \text{for } x>\eta,\\[6pt] 0, & \text{for } x<\eta. \end{cases} \tag{15} \]
Consequently, the solution of the dual equations (14) is
\[ f_1(\xi)= \frac{2^{1-m}}{\Gamma(m)} \int_0^c x^{1-\nu-m}N_{\nu+m}(\xi x)\,dx \int_0^x (x^2-\eta^2)^{m-1}g_1(\eta)\eta\,d\eta + \]
\[ +\int_c^\infty x^{m+1}N_{\nu+m}(\xi x)\,l[g_2(x)]\,dx . \tag{16} \]
We note that the results obtained above can also be extended to the case of any non-integer \(m>0\); in (11) one must then use fractional differentiation.
References
- Titchmarsh E. Introduction to the Theory of Fourier Integrals. Fizmatgiz, 1948.
- Busbridge J. W. Dual integral equations, Proc. London Math. Soc. (2), 44, 1938.
- Srivastav R. P. A pair of dual integral equations involving Bessel functions of the first and the second kind. Proc. Edinburgh Math. Soc., 14, No. 2, 1964.
- Watson G. N. Theory of Bessel Functions, Part 1. IL, 1949.
- Gradshteyn I. S., Ryzhik I. M. Tables of Integrals, Sums, Series and Products. Fizmatgiz, 1962.
- Gelfand I. M., Shilov G. E. Generalized Functions and Operations on Them. Fizmatgiz, 1959.
Received by the editors
September 18, 1965
Central Scientific Research Institute
of Building Structures
named after V. A. Kucherenko