Page 662 - Handbook Of Integral Equations

13.2-5. The Characteristic Equation With Hilbert Kernel

Consider the characteristic equation with Hilbert kernel

b(x) 2π ξ – x
a(x)ϕ(x) – cot ϕ(ξ) dξ = f(x). (43)
2π 2
0
Just as the characteristic integral equation with Cauchy kernel is related to the Riemann boundary
value problem, so the characteristic equation (43) with Hilbert kernel can be analytically reduced to
a Hilbert problem in a straightforward manner. In turn, the Hilbert problem can be reduced to the
Riemann problem (see Subsection 12.3-12), and hence the solution of Eq. (43) can be constructed
in a closed form.
For ν > 0, the homogeneous equation (43) (f(x) ≡ 0) has 2ν linearly independent solutions, and
the nonhomogeneous problem is unconditionally solvable and linearly depends on 2ν real constants.
For ν < 0, the homogeneous equation is unsolvable, and the nonhomogeneous equation is
solvable only under –2ν real solvability conditions.
Taking into account the fact that any complex parameter contains two real parameters, and
a complex solvability condition is equivalent to two real conditions, we see that, for ν ≠ 0, the
qualitative results of investigating the characteristic equation with Hilbert kernel completely agree
with the corresponding results for the characteristic equation with Cauchy kernel.

13.2-6. The Tricomi Equation

The singular integral Tricomi equation has the form

1
1 1
ϕ(x) – λ – ϕ(ξ) dξ = f(x), 0 ≤ x ≤ 1. (44)
0 ξ – x x + ξ – 2xξ
The kernel of this equation consists of two terms. The first term is the Cauchy kernel. The second
term is continuous if at least one of the variables x and ξ varies strictly inside the interval [0, 1];
however, for x = ξ = 0 and for x = ξ = 1, this kernel becomes infinite and is nonintegrable in the
square {0 ≤ x ≤ 1, 0 ≤ ξ ≤ 1}.
By using the function

1
1 1 1
Φ(z)= – ϕ(ξ) dξ,
2πi 0 ξ – z z + ξ – 2zξ
which is piecewise analytic in the upper and the lower half-plane, we can reduce Eq. (44) to the
Riemann problem with boundary condition on the real axis. The solution of the Tricomi equation
has the form

α
1 1 ξ (1 – x) α 1 1
C(1 – x) β
y(x)= f(x)+ – f(ξ) dξ + ,
α
1+ λ π 0 x (1 – ξ) α ξ – x x + ξ – 2xξ x 1+β
2 2
2 βπ
α = arctan(λπ)(–1< α < 1), tan = λπ (–2< β < 0),
π 2
where C is an arbitrary constant.
•
References for Section 13.2: P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov (1977), F. G. Tricomi (1985),
N. I. Muskhelishvili (1992).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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