Page 303 - Handbook Of Integral Equations

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

b
3. y(x) – λ cos[β(x – t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.12 with g(x) = cos(βx) and h(t) = sin(βt).
Solution:

y(x)= f(x)+ λ A 1 cos(βx)+ A 2 sin(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.12.

b

4. y(x) – λ cos[β(x + t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.13 with g(x) = cos(βx) and h(t) = sin(βt).
Solution:

y(x)= f(x)+ λ A 1 cos(βx)+ A 2 sin(βx) ,
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.13.

∞
5. y(x) – λ cos(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) cos(xt) dt, (1)
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) cos(xt) dt, (2)
π 0
where f = f(x) is any continuous function absolutely integrable on the interval [0, ∞).
In particular, from (1) and (2) with f(x)= e –ax we obtain

2 a 2
–ax
y + (x)= e + for λ =+ ,
2
π a + x 2 π

2 a 2
–ax
y – (x)= e – for λ = – ,
2
π a + x 2 π
where a is any positive number.
•
Reference: M. L. Krasnov, A. I. Kisilev, and G. I. Makarenko (1971).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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