Page 221 - Handbook Of Integral Equations

3.1-3. Kernels Containing Integer Powers of x and t or Rational Functions

b
3
11. x – t y(t) dt = f(x).

a
Let us remove the modulus in the integrand:
x b

3
3
(x – t) y(t) dt + (t – x) y(t) dt = f(x). (1)
a x
Differentiating (1) twice yields
x b

6 (x – t)y(t) dt +6 (t – x)y(t) dt = f (x).
xx
a x
This equation can be rewritten in the form 3.1.2:
b

1

|x – t| y(t) dt = f (x). (2)
6 xx
a
Therefore the solution of the integral equation is given by
y(x)= 1 y (x). (3)
12 xxxx
The right-hand side f(x) of the equation must satisfy certain conditions. To obtain these
conditions, one must substitute solution (3) into (1) with x = a and x = b and into (2) with
x = a and x = b, and then integrate the four resulting relations by parts.
b

x – t y(t) dt = f(x).
12. 3 3
a
3
This is a special case of equation 3.8.3 with g(x)= x .
b

2
13. xt – t y(t) dt = f(x) 0 ≤ a < b < ∞.
3
a
2
The substitution w(t)= t y(t) leads to an equation of the form 3.1.2:
b

|x – t|w(t) dt = f(x).
a
b

14. 2 3
x t – t y(t) dt = f(x).
a
The substitution w(t)= |t| y(t) leads to an equation of the form 3.1.8:
b

2 2

x – t w(t) dt = f(x).
a
a

15. 3 3 β >0.
x – βt y(t) dt = f(x),
0
3
3
This is a special case of equation 3.8.4 with g(x)= x and β = λ .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 200