Page 241 - Handbook Of Integral Equations

3.4-3. An Equation Containing the Unknown Function of a Complicated Argument

1

15. A ln t + B)y(xt) dt = f(x).
0
The substitution ξ = xt leads to an equation of the form 1.9.3 with g(x)= –A ln x:
x

A ln ξ – A ln x + B y(ξ) dξ = xf(x).
0
3.5. Equations Whose Kernels Contain Trigonometric
Functions

3.5-1. Kernels Containing Cosine
∞

1. cos(xt)y(t) dt = f(x).
0
2 ∞
Solution: y(x)= cos(xt)f(t) dt.
π 0
Up to constant factors, the function f(x) and the solution y(t) are the Fourier cosine
transform pair.
•
References: H. Bateman and A. Erd´ elyi (vol. 1, 1954), V. A. Ditkin and A. P. Prudnikov (1965).
b

2. cos(λx) – cos(λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x) = cos(λx).
Solution:
1 d f (x)

x
y(x)= – .
2λ dx sin(λx)
The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2 of
◦
equation 3.8.3).
a

3. cos(βx) – cos(µt) y(t) dt = f(x), β >0, µ >0.

0
This is a special case of equation 3.8.4 with g(x) = cos(βx) and λ = µ/β.
b
k
4. cos x – cos t y(t) dt = f(x), 0 < k <1.
k
a
k
This is a special case of equation 3.8.3 with g(x) = cos x.
Solution:
1 d f (x)

x
y(x)= – .
2k dx sin x cos k–1 x
◦
The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2 of
equation 3.8.3).
b y(t)
5. dt = f(x), 0 < k <1.
a |cos(λx) – cos(λt)| k
This is a special case of equation 3.8.7 with g(x) = cos(λx)+ β, where β is an arbitrary
number.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 220