Page 130 - Handbook Of Integral Equations

2.1-2. Kernels Quadratic in the Arguments x and t

x

2
7. y(x)+ A x y(t) dt = f(x).
a
This is a special case of equation 2.1.50 with λ = 2 and µ =0.
Solution:
x

3
2
3

y(x)= f(x) – A x exp 1 A(t – x ) f(t) dt.
3
a
x
8. y(x)+ A xty(t) dt = f(x).
a
This is a special case of equation 2.1.50 with λ = 1 and µ =1.
Solution:
x
1 3 3
y(x)= f(x) – A xt exp A(t – x ) f(t) dt.
3
a
x

2
9. y(x)+ A t y(t) dt = f(x).
a
This is a special case of equation 2.1.50 with λ = 0 and µ =2.
Solution:
x

3
3
2

y(x)= f(x) – A t exp 1 A(t – x ) f(t) dt.
3
a
x
2
10. y(x)+ λ (x – t) y(t) dt = f(x).
a
This is a special case of equation 2.1.34 with n =2.
Solution:
x

y(x)= f(x) – R(x – t)f(t) dt,
a
√ √ √
1/3
2
2
R(x)= ke –2kx – ke kx cos 3 kx – 3 sin 3 kx , k = 1 λ .
3 3 4
x

2
2
11. y(x)+ A (x – t )y(t) dt = f(x).
a
2
This is a special case of equation 2.9.5 with g(x)= Ax .
Solution: x
1

y(x)= f(x)+ u (x)u (t) – u (x)u (t) f(t) dt,

2
2
1
1
W a
where the primes denote differentiation with respect to the argument specified in the parenthe-
ses; u 1 (x), u 2 (x) is a fundamental system of solutions of the second-order linear homogeneous
ordinary differential equation u +2Axu = 0; and the functions u 1 (x) and u 2 (x) are ex-
xx
pressed in terms of Bessel functions or modified Bessel functions, depending on the sign of
the parameter A:
For A >0,
√
√
W =3/π, u 1 (x)= xJ 1/3 8 Ax 3/2 , u 2 (x)= xY 1/3 8 Ax 3/2 .
9 9
For A <0,
√
8 3/2 √
8 3/2
3
W =– , u 1 (x)= xI 1/3 |A| x , u 2 (x)= xK 1/3 |A| x .
2 9 9
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 109