Page 247 - Handbook Of Integral Equations

π/2

λ
k
30. sin t cos m ty(ξ) dt = f(x), ξ = x sin t.
0
1 . Let λ > –1, m > –1, and k > 0. The transformation
◦
2 λ–1 k
2
z = x k , ζ = z sin t, w(ζ)= ζ 2 y ζ 2
leads to an equation of the form 1.1.43:
z
m–1 λ+m k
(z – ζ) 2 w(ζ) dζ = F(z), F(z)=2z 2 f z 2 .
0
◦
2 . Solution with –1< m <1:
π/2
2k π(1 – m) k–λ–1 d λ+1 λ+1 m
y(x)= sin x k x k sin t tan tf(ξ) dt ,
π 2 dx 0
k
where ξ = x sin t.

3.5-7. A Singular Equation

2π t – x
31. cot y(t) dt = f(x), 0 ≤ x ≤ 2π.
0 2
Here the integral is understood in the sense of the Cauchy principal value and the right-hand
2π
side is assumed to satisfy the condition f(t) dt =0.
0
Solution:
1 2π t – x
y(x)= – cot f(t) dt + C,
4π 2 0 2
where C is an arbitrary constant.
2π
It follows from the solution that y(t) dt =2πC.
0
The equation and its solution form a Hilbert transform pair (in the asymmetric form).
•
Reference: F. D. Gakhov (1977).

3.6. Equations Whose Kernels Contain Combinations of
Elementary Functions

3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions

b

1. ln cosh(λx) – cosh(λt) y(t) dt = f(x).

a
This is a special case of equation 1.8.9 with g(x) = cosh(λx).
b

2. ln sinh(λx) – sinh(λt) y(t) dt = f(x).

a
This is a special case of equation 1.8.9 with g(x) = sinh(λx).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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