Page 301 - Handbook Of Integral Equations

4.3-5. Kernels Containing Combination of Hyperbolic Functions

b
k m
50. y(x) – λ cosh (βx) sinh (µt)y(t) dt = f(x).
a
k
m
This is a special case of equation 4.9.1 with g(x) = cosh (βx) and h(t) = sinh (µt).
b
51. y(x) – λ [A sinh(αx) cosh(βt)+ B sinh(γx) cosh(δt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.18 with g 1 (x) = sinh(αx), h 1 (t)= A cosh(βt), g 2 (x)=
sinh(γx), and h 2 (t)= B cosh(δt).

b
k
m
52. y(x) – λ tanh (γx) coth (µt)y(t) dt = f(x).
a
k m
This is a special case of equation 4.9.1 with g(x) = tanh (γx) and h(t) = coth (µt).
b
53. y(x) – λ [A tanh(αx) coth(βt)+ B tanh(γx) coth(δt)]y(t) dt = f(x).
a
This is a special case of equation 4.9.18 with g 1 (x) = tanh(αx), h 1 (t)= A coth(βt), g 2 (x)=
tanh(γx), and h 2 (t)= B coth(δt).

4.4. Equations Whose Kernels Contain Logarithmic
Functions

4.4-1. Kernels Containing Logarithmic Functions

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

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

b
3. y(x) – λ (ln x – ln t)y(t) dt = f(x).
a
This is a special case of equation 4.9.3 with g(x)=ln x.

b ln(γx)
4. y(x) – λ y(t) dt = f(x).
a ln(γt)
1
This is a special case of equation 4.9.1 with g(x) = ln(γx) and h(t)= .
ln(γt)

b ln(γt)
5. y(x) – λ y(t) dt = f(x).
a ln(γx)
1
This is a special case of equation 4.9.1 with g(x)= and h(t) = ln(γt).
ln(γx)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 280