Page 220 - Handbook Of Integral Equations

a

4. |x – λt| y(t) dt = f(x), λ >0.
0
Here 0 ≤ x ≤ a and 0 ≤ t ≤ a.
Solution:
1

y(x)= λf (λx).
2 xx
The right-hand side f(x) of the integral equation must satisfy the relations

aλf (aλ)= f(0) + f(aλ), f (0) + f (aλ)=0.
x
x
x
Hence, it follows the general form of the right-hand side:

f(x)= F(x)+ Ax + B, A = – 1 F (λa)+ F (0) , B = 1 aλF (aλ) – F(λa) – F(0) ,

2 x x 2 x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
3.1-2. Kernels Quadratic in the Arguments x and t
a
2
5. Ax + Bx – t y(t) dt = f(x), A >0, B >0.

0
2
This is a special case of equation 3.8.5 with g(x)= Ax + Bx .
a

6. x – At – Bt y(t) dt = f(x), A >0, B >0.
2
0
2
This is a special case of equation 3.8.6 with g(x)= At + Bt .
b

7. xt – t y(t) dt = f(x) 0 ≤ a < b < ∞.
2
a
The substitution w(t)= ty(t) leads to an equation of the form 1.3.2:
b
|x – t|w(t) dt = f(x).
a
b

x – t y(t) dt = f(x).
8. 2 2
a
2
This is a special case of equation 3.8.3 with g(x)= x .

d f (x)
x
Solution: y(x)= . The right-hand side f(x) of the equation must satisfy
dx 4x
certain constraints, given in 3.8.3.
a
9. 2 2 β >0.
x – βt y(t) dt = f(x),
0
2
2
This is a special case of equation 3.8.4 with g(x)= x and β = λ .
a

2
10. Ax + Bx – Aλt – Bλ t y(t) dt = f(x), λ >0.
2 2
0
2
This is a special case of equation 3.8.4 with g(x)= Ax + Bx .
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 199