Page 157 - Handbook Of Integral Equations

x

8. y(x)+ A t cosh[λ(x – t)]y(t) dt = f(x).
a
This is a special case of equation 2.9.28 with g(t)= At.

x

k
m
9. y(x)+ A t cosh (λx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= –A cosh (λx) and h(t)= t .
x

k
m
10. y(x)+ A x cosh (λt)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= –Ax and h(t) = cosh (λt).
x

11. y(x) – A cosh(kx)+ B – AB(x – t) cosh(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A cosh(kx).
Solution:
x

y(x)= f(x)+ R(x, t)f(t) dt,
a
G(x) B 2 x B(x–s) A
R(x, t)=[A cosh(kx)+ B] + e G(s) ds, G(x)=exp sinh(kx) .
G(t) G(t) k
t
x

12. y(x)+ A cosh(kt)+ B + AB(x – t) cosh(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A cosh(kt).
Solution:
x
y(x)= f(x)+ R(x, t)f(t) dt,
a
G(t) B 2 x B(t–s) A
R(x, t)= –[A cosh(kt)+ B] + e G(s) ds, G(x)=exp sinh(kx) .
G(x) G(x) t k
∞ √

13. y(x)+ A cosh λ t – x y(t) dt = f(x).
x
√
This is a special case of equation 2.9.62 with K(x)= A cosh λ –x .

2.3-2. Kernels Containing Hyperbolic Sine

x
14. y(x) – A sinh(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A sinh(λx) and h(t)=1.
Solution:
x

y(x)= f(x)+ A sinh(λx)exp A cosh(λx) – cosh(λt) f(t) dt.
a λ

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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