Page 99 - Handbook Of Integral Equations

x

β γ
96. A tanh (λx)+ B cos (µt)+ C y(t) dt = f(x).
a
β γ
This is a special case of equation 1.9.6 with g(x)= A tanh (λx) and h(t)= B cos (µt)+ C.
x
β γ
97. A tanh (λx)+ B sin (µt)+ C y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= A tanh (λx) and h(t)= B sin (µt)+ C.

1.7-6. Kernels Containing Logarithmic and Trigonometric Functions

x

β γ
98. A cos (λx)+ B ln (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A cos (λx) and h(t)= B ln (µt)+ C.
x

β γ
99. A cos (λt)+ B ln (µx)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= B ln (µx)+ C and h(t)= A cos (λt).
x

β γ
100. A sin (λx)+ B ln (µt)+ C y(t) dt = f(x).
a
γ
β
This is a special case of equation 1.9.6 with g(x)= A sin (λx) and h(t)= B ln (µt)+ C.
x

β γ
101. A sin (λt)+ B ln (µx)+ C y(t) dt = f(x).
a
β
γ
This is a special case of equation 1.9.6 with g(x)= B ln (µx) and h(t)= A sin (λt)+ C.
1.8. Equations Whose Kernels Contain Special
Functions
1.8-1. Kernels Containing Bessel Functions

x

1. J 0 [λ(x – t)]y(t) dt = f(x).
a
Solution:
1 d 2 2 2 x
y(x)= + λ (x – t) J 1 [λ(x – t)] f(t) dt.
λ dx 2 a
Example. In the special case λ = 1 and f(x)= A sin x, the solution has the form y(x)= AJ 0 (x).
x

2. [J 0 (λx) – J 0 (λt)]y(t) dt = f(x).
a
d f (x)

x
Solution: y(x)= – .
dx λJ 1 (λx)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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