Page 226 - Handbook Of Integral Equations

b y(t)
30. dt = f(x), 0 < k <1.
a |x – t| k
It is assumed that |a| + |b| < ∞. Solution:
1 1 d x f(t) dt 1 2 1 x Z(t)F(t)
y(x)= cot( πk) – cos ( πk) dt,
2π 2 dx a (x – t) 1–k π 2 2 a (x – t) 1–k
where
1+k 1–k d t dτ b f(s) ds
Z(t)=(t – a) 2 (b – t) 2 , F(t)= .
dt (t – τ) k Z(s)(s – τ) 1–k
a τ
•
Reference: F. D. Gakhov (1977).
a
y(t)
31. dt = f(x), 0 < k <1, 0 < a ≤ ∞.
2
2 k
0 |x – t |
Solution:
1 a 2–2k t k
2Γ(k) cos πk d t F(t) dt s f(s) ds
y(x)= – 2 x k–1 , F(t)= .
1+k 2 dx 1–k 1–k
π Γ x 2 2 0 2 2
2 (t – x ) 2 (t – s ) 2
•
Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975).
b y(t)
32. dt = f(x), 0 < k <1, λ >0.
a |x – t |
λ k
λ
1 . The transformation
◦
1–λ
λ
λ
z = x , τ = t , w(τ)= τ λ y(t)
leads to an equation of the form 3.8.30:
B

w(τ)
|z – τ| k dτ = F(z),
A
λ
λ
where A = a , B = b , F(z)= λf(z 1/λ ).
2 . Solution with a =0:
◦
λ(3–2k)–2 λ(k+1)–2

λ(k–1) d b t 2 dt t s 2 f(s) ds
y(x)= –Ax 2 ,
1–k
1–k
dx x (t – x ) 2 0 (t – s ) 2
λ
λ
λ
λ
λ 2 πk
1+ k
–2
A = cos Γ(k) Γ ,
2π 2 2
where Γ(k) is the gamma function.
1
y(t)
33. dt = f(x), 0 < k <1, λ >0, m >0.
0 |x – t |
λ
m k
The transformation
1–m
λ m
z = x , τ = t , w(τ)= τ m y(t)
leads to an equation of the form 3.8.30:
w(τ) 1/λ
1
|z – τ| k dτ = F(z), F(z)= mf(z ).
0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 205