Page 188 - Handbook Of Integral Equations

2.7-6. Kernels Containing Logarithmic and Trigonometric Functions

x
k
m
54. y(x) – A cos (λx)ln (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A cos (λx) and h(t)=ln (µt).
x

m
k
55. y(x) – A cos (λt)ln (µx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A ln (µx) and h(t) = cos (λt).
x

k
m
56. y(x) – A sin (λx)ln (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A sin (λx) and h(t)=ln (µt).
x

k
m
57. y(x) – A sin (λt)ln (µx)y(t) dt = f(x).
a
m k
This is a special case of equation 2.9.2 with g(x)= A ln (µx) and h(t) = sin (λt).
x
m
k
58. y(x) – A tan (λx)ln (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A tan (λx) and h(t)=ln (µt).
x
k
m
59. y(x) – A tan (λt)ln (µx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A ln (µx) and h(t) = tan (λt).
x
m
k
60. y(x) – A cot (λx)ln (µt)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A cot (λx) and h(t)=ln (µt).
x

k
m
61. y(x) – A cot (λt)ln (µx)y(t) dt = f(x).
a
m
k
This is a special case of equation 2.9.2 with g(x)= A ln (µx) and h(t) = cot (λt).
2.8. Equations Whose Kernels Contain Special
Functions
2.8-1. Kernels Containing Bessel Functions
x

1. y(x) – λ J 0 (x – t)y(t) dt = f(x).
0
Solution:
x
y(x)= f(x)+ R(x – t)f(t) dt,
0
where
√ λ 2 √ λ x √ J 1 (t)
2
2
2
R(x)= λ cos 1–λ x + √ sin 1–λ x + √ sin 1–λ (x–t) dt.
1–λ 2 1–λ 2 0 t
•
Reference: V. I. Smirnov (1974).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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