Page 138 - Handbook Of Integral Equations

Remark. For arbitrary f(x), expandable into power series, the formulas of item 2 can
◦
be used, in which one should set N = ∞. In this case, the radius of convergence of the
solution y(x) is equal to the radius of convergence of f(x).
3 . For logarithmic-polynomial right-hand side,
◦
N
n
f(x)=ln x A n x ,
n=0
the solution with logarithmic singularity at zero is given by


N N
A n n n
 A n D n λ
 ln x x + x for λ < λ 0 ,


1 – (λ/λ n ) [1 – (λ/λ n )]
 2

n=0 n=0
y(x)=
N N
 A n D n λ

 A n n n β
 ln x x + x + Cx for λ > λ 0 and λ ≠ λ n ,


 1 – (λ/λ n ) [1 – (λ/λ n )] 2
n=0 n=0
n k 2 n k

1 n (–1) n+1 π (–1)
λ n = , I(n)=(–1) ln 2 + , D n =(–1) + .
I(n) k 12 k 2
k=1 k=1
4 . For arbitrary f(x), the transformation
◦
1 2z
1 2τ
–z
–z
x = e , t = e , y(x)= e w(z), f(x)= e g(z)
2 2
leads to an integral equation with difference kernel of the form 2.9.51:
z
w(τ) dτ
w(z) – λ = g(z).
cosh(z – τ)
–∞
x x + b
42. y(x) – λ y(t) dt = f(x).
a t + b
This is a special case of equation 2.9.1 with g(x)= x + b.
Solution:
x x + b
y(x)= f(x)+ λ e λ(x–t) f(t) dt.
a t + b
2 x t
43. y(x)= y(t) dt.
2
(1 – λ )x 2 λx 1+ t
This equation is encountered in nuclear physics and describes deceleration of neutrons in
matter.
1 . Solution with λ =0:
◦
C
y(x)= ,
(1 + x) 2
where C is an arbitrary constant.
2 .For λ ≠ 0, the solution can be found in the series form
◦
∞
n
y(x)= A n x .
n=0
•
Reference: I. Sneddon (1951).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 117