Page 305 - Handbook Of Integral Equations

4.5-2. Kernels Containing Sine

b
15. y(x) – λ sin(βx)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = sin(βx) and h(t)=1.
b
16. y(x) – λ sin(βt)y(t) dt = f(x).
a
This is a special case of equation 4.9.1 with g(x) = 1 and h(t) = sin(βt).

b
17. y(x) – λ sin[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.16 with g(x) = sin(βx) and h(t) = cos(βt).
Solution:

y(x)= f(x)+ λ A 1 sin(βx)+ A 2 cos(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.16.

b
18. y(x) – λ sin[β(x + t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.15 with g(x) = sin(βx) and h(t) = cos(βt).
Solution:

y(x)= f(x)+ λ A 1 sin(βx)+ A 2 cos(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.15.

∞
19. y(x) – λ sin(xt)y(t) dt =0.
0

Characteristic values: λ = ± 2/π. For the characteristic values, the integral equation has
infinitely many linearly independent eigenfunctions.

Eigenfunctions for λ =+ 2/π have the form

2 ∞
y + (x)= f(x)+ f(t) sin(xt) dt,
π 0
where f = f(x) is any continuous function absolutely integrable on the interval [0, ∞).

Eigenfunctions for λ = – 2/π have the form

2 ∞
y – (x)= f(x) – f(t) sin(xt) dt,
π 0
where f = f(x) is any continuous function absolutely integrable on the interval [0, ∞).
•
Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

∞
20. y(x) – λ sin(xt)y(t) dt = f(x).
0
Solution:
f(x) λ ∞
y(x)= π + π sin(xt)f(t) dt,
1 – λ 2 1 – λ 2
2 2 0

where λ ≠ ± 2/π.
•
References: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971), F. D. Gakhov and Yu. I. Cherskii (1978).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 284