Page 165 - Handbook Of Integral Equations

2.3-5. Kernels Containing Combinations of Hyperbolic Functions

x
k m
53. y(x) – A cosh (λx) sinh (µt)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) = sinh (µt).
x

54. y(x) – A + B cosh(λx)+ B(x – t)[λ sinh(λx) – A cosh(λx)] y(t) dt = f(x).
a
This is a special case of equation 2.9.32 with b = B and g(x)= A.
x

55. y(x) – A + B sinh(λx)+ B(x – t)[λ cosh(λx) – A sinh(λx)] y(t) dt = f(x).
a
This is a special case of equation 2.9.33 with b = B and g(x)= A.
x

m
k
56. y(x) – A tanh (λx) coth (µt)y(t) dt = f(x).
a
k
m
This is a special case of equation 2.9.2 with g(x)= A tanh (λx) and h(t) = coth (µt).
2.4. Equations Whose Kernels Contain Logarithmic
Functions
2.4-1. Kernels Containing Logarithmic Functions

x

1. y(x) – A ln(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A ln(λx) and h(t)=1.
Solution:
x (λx) Ax
y(x)= f(x)+ A ln(λx)e –A(x–t) At f(t) dt.
a (λt)
x
2. y(x) – A ln(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = ln(λt).
Solution:
x
(λx) Ax
y(x)= f(x)+ A ln(λt)e –A(x–t) At f(t) dt.
a (λt)
x
3. y(x)+ A (ln x – ln t)y(t) dt = f(x).
a
This is a special case of equation 2.9.5 with g(x)= A ln x.
Solution: x
1

y(x)= f(x)+ u (x)u (t) – u (x)u (t) f(t) dt,
1
2
2
1
W
a
where the primes denote differentiation with respect to the argument specified in the paren-
theses; and u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear
–1
homogeneous ordinary differential equation u xx +Ax u = 0, with u 1 (x) and u 2 (x) expressed
in terms of Bessel functions or modified Bessel functions, depending on the sign of A:
√ √ √ √
1
W = , u 1 (x)= xJ 1 2 Ax , u 2 (x)= xY 1 2 Ax for A >0,
π
√ √ √ √
1
W = – , u 1 (x)= xI 1 2 –Ax , u 2 (x)= xK 1 2 –Ax for A <0.
2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 144