Page 219 - Handbook Of Integral Equations

Substitute y(x) of (3) into (4). Integration by parts yields f (1) = f(1)+f(0) and f (1)–f (0) =

x
x
x
2f(1)+2f(0). Hence, we obtain the desired constraints for f(x):
f (1) = f(0) + f(1), f (0) + f (1) = 0. (5)

x x x
Conditions (5) make it possible to find the admissible general form of the right-hand side
of the integral equation:
f(x)= F(x)+ Ax + B,

A =– 1 F (1) + F (0) , B = 1 F (1) – F(1) – F(0) ,

2 x x 2 x
where F(x) is an arbitrary bounded twice differentiable function with bounded first derivative.
b
2. |x – t| y(t) dt = f(x), 0 ≤ a < b < ∞.
a
This is a special case of equation 3.8.3 with g(x)= x.
Solution:
1

y(x)= f (x).
2 xx
The right-hand side f(x) of the integral equation must satisfy certain relations. The
general form of f(x) is as follows:
f(x)= F(x)+ Ax + B,

A =– 1 F (a)+ F (b) , B = 1 aF (a)+ bF (b)– F(a)– F(b) ,

2 x x 2 x x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
a
3. |λx – t| y(t) dt = f(x), λ >0.
0
Here 0 ≤ x ≤ a and 0 ≤ t ≤ a.
◦
1 . Let us remove the modulus in the integrand:
λx a

(λx – t)y(t) dt + (t – λx)y(t) dt = f(x). (1)
0 λx
Differentiating (1) with respect to x, we find that
λx a

λ y(t) dt – λ y(t) dt = f (x). (2)
x
0 λx
2
Differentiating (2) yields 2λ y(λx)= f (x). Hence, we obtain the solution

xx
1
x

y(x)= f xx . (3)
2λ 2 λ
◦
2 . Let us demonstrate that the right-hand side f(x) of the integral equation must satisfy
certain relations. By setting x = 0 in (1) and (2), we obtain two corollaries
a a

ty(t) dt = f(0), λ y(t) dt =–f (0), (4)
x
0 0
Substitute y(x) from (3) into (4). Integrating by parts yields the desired constraints for f(x):

(a/λ)f (a/λ)= f(0) + f(a/λ), f (0) + f (a/λ)=0. (5)
x
x
x
Conditions (5) make it possible to establish the admissible general form of the right-hand
side of the integral equation:
f(x)= F(z)+ Az + B, z = λx;

A =– 1 F (a)+ F (0) , B = 1 aF (a)– F(a)– F(0) ,
2 z z 2 z
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 198