Page 295 - Handbook Of Integral Equations

b

4. y(x) – λ cosh[β(x + t)]y(t) dt = f(x).
a
This is a special case of equation 4.9.12 with g(x) = cosh(βx) and h(t) = sinh(βt).
Solution:

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

5. y(x) – λ A k cosh[β k (x – t)] y(t) dt = f(x), n =1, 2, ...
a
k=1
This is a special case of equation 4.9.20.
b cosh(βx)
6. y(x) – λ y(t) dt = f(x).
a cosh(βt)
1
This is a special case of equation 4.9.1 with g(x) = cosh(βx) and h(t)= .
cosh(βt)
b cosh(βt)
7. y(x) – λ y(t) dt = f(x).
a cosh(βx)
1
This is a special case of equation 4.9.1 with g(x)= and h(t) = cosh(βt).
cosh(βx)
b

k m
8. y(x) – λ cosh (βx) cosh (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 4.9.1 with g(x) = cosh (βx) and h(t) = cosh (µt).
b
m
k
9. y(x) – λ t cosh (βx)y(t) dt = f(x).
a
m k
This is a special case of equation 4.9.1 with g(x) = cosh (βx) and h(t)= t .

b
m
k
10. y(x) – λ x cosh (βt)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x)= x and h(t) = cosh (βt).

b
11. y(x) – λ [A + B(x – t) cosh(βx)]y(t) dt = f(x).
a
This is a special case of equation 4.9.10 with h(x) = cosh(βx).
b
12. y(x) – λ [A + B(x – t) cosh(βt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.8 with h(t) = cosh(βt).

∞ y(t) dt
13. y(x)+ λ = f(x).
–∞ cosh[b(x – t)]
Solution with b > π|λ|:
∞
2λb sinh[2k(x – t)] b πλ
y(x)= f(x) – √ f(t) dt, k = arccos .
2 2
2
b – π λ –∞ sinh[2b(x – t)] π b
•
Reference: F. D. Gakhov and Yu. I. Cherskii (1978).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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