Page 256 - Handbook Of Integral Equations

b y(t)
7. dt = f(x), 0 < k <1.
a |g(x) – g(t)| k

Let g ≠ 0. The transformation
x
1
z = g(x), τ = g(t), w(τ)= y(t)
g (t)

t
leads to an equation of the form 3.1.30:
w(τ)
B
|z – τ| k dτ = F(z), A = g(a), B = g(b),
A
where F = F(z) is the function which is obtained from z = g(x) and F = f(x) by eliminating x.
1 y(t)
8. dt = f(x), 0 < k <1.
0 |g(x) – h(t)| k
Let g(0) = 0, g(1) = 1, g >0; h(0)=0, h(1) = 1, and h >0.

t
x
The transformation
1
z = g(x), τ = h(t), w(τ)= y(t)

h (t)
t
leads to an equation of the form 3.1.29:
1
w(τ)

|z – τ| k dτ = F(z),
0
where F = F(z) is the function which is obtained from z = g(x) and F = f(x) by eliminating x.
b
9. y(t)ln |g(x) – g(t)| dt = f(x).
a

Let g ≠ 0. The transformation
x
1
z = g(x), τ = g(t), w(τ)= y(t)
g (t)

t
leads to Carleman’s equation 3.4.2:
B

ln |z – τ|w(τ) dτ = F(z), A = g(a), B = g(b),
A
where F = F(z) is the function which is obtained from z = g(x) and F = f(x) by eliminating x.
1

10. y(t)ln |g(x) – h(t)| dt = f(x).
0
Let g(0) = 0, g(1) = 1, g >0; h(0)=0, h(1) = 1, and h >0.

t
x
The transformation
1
z = g(x), τ = h(t), w(τ)= y(t)
h (t)

t
leads to an equation of the form 3.4.2:
1

ln |z – τ|w(τ) dτ = F(z),
0
where F = F(z) is the function which is obtained from z = g(x) and F = f(x) by eliminating x.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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