Page 237 - Handbook Of Integral Equations

a

13. k sinh(λx) – t y(t) dt = f(x).

0
This is a special case of equation 3.8.5 with g(x)= k sinh(λx).
a

14. x – k sinh(λt) y(t) dt = f(x).

0
This is a special case of equation 3.8.6 with g(x)= k sinh(λt).
3.3-3. Kernels Containing Hyperbolic Tangent

b

15. tanh(λx) – tanh(λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x) = tanh(λx).
Solution:
2

y(x)= 1 d cosh (λx)f (x) .
x
2λ dx
The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2 of
◦
equation 3.8.3).
a

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

0
This is a special case of equation 3.8.4 with g(x) = tanh(βx) and λ = µ/β.
b
k k
17. | tanh x – tanh t| y(t) dt = f(x), 0 < k <1.
0
k
This is a special case of equation 3.8.3 with g(x) = tanh x.
Solution:
1 d 2 k–1

y(x)= cosh x coth xf (x) .
x
2k dx
The right-hand side f(x) must satisfy certain conditions. As follows from item 3 of equation
◦
3.8.3, the admissible general form of the right-hand side is given by

f(x)= F(x)+ Ax + B, A = –F (b), B = 1 bF (b) – F(0) – F(b) ,

x 2 x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
b
y(t)
18. dt = f(x), 0 < k <1.
a |tanh(λx) – tanh(λt)| k
This is a special case of equation 3.8.7 with g(x) = tanh(λx)+ β, where β is an arbitrary
number.
a

19. k tanh(λx) – t y(t) dt = f(x).

0
This is a special case of equation 3.8.5 with g(x)= k tanh(λx).
a

20. x – k tanh(λt) y(t) dt = f(x).

0
This is a special case of equation 3.8.6 with g(x)= k tanh(λt).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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