Page 83 - Handbook Of Integral Equations

1.6-2. Kernels Containing Arcsine

x

11. arcsin(λx) – arcsin(λt) y(t) dt = f(x).
a
This is a special case of equation 1.9.2 with g(x) = arcsin(λx).
1 d √
2 2
Solution: y(x)= 1 – λ x f (x) .

x
λ dx
x

12. A arcsin(λx)+ B arcsin(λt) y(t) dt = f(x).
a
For B =–A, see equation 1.6.11. This is a special case of equation 1.9.4 with g(x)=arcsin(λx).
Solution:
sign x d – A x – B

y(x)= arcsin(λx) A+B arcsin(λt) A+B f (t) dt .
t
A + B dx a
x

13. A arcsin(λx)+ B arcsin(µt)+ C y(t) dt = f(x).
a
This is a special case of equation 1.9.6 with g(x)= A arcsin(λx) and h(t)= B arcsin(µt)+ C.
x
n
14. arcsin(λx) – arcsin(λt) y(t) dt = f(x), n =1, 2, ...
a

The right-hand side of the equation is assumed to satisfy the conditions f(a)= f (a)= ··· =
x
f x (n) (a)=0.
Solution:
n+1
1 √ d
y(x)= √ 1 – λ x f(x).
2 2
2 2
n
λ n! 1 – λ x dx
x
15. arcsin(λx) – arcsin(λt) y(t) dt = f(x).
a
Solution:
2 x

2 1 d ϕ(t)f(t) dt 1

y(x)= ϕ(x) √ , ϕ(x)= √ .
π ϕ(x) dx a arcsin(λx) – arcsin(λt) 1 – λ x
2 2
x
y(t) dt
16. √ = f(x).
a arcsin(λx) – arcsin(λt)
Solution:
λ d x ϕ(t)f(t) dt 1
y(x)= √ , ϕ(x)= √ .
π dx a arcsin(λx) – arcsin(λt) 1 – λ x
2 2
x

µ

17. arcsin(λx) – arcsin(λt) y(t) dt = f(x), 0 < µ <1.
a
Solution:
2 x
1 d ϕ(t)f(t) dt

y(x)= kϕ(x) ,
ϕ(x) dx [arcsin(λx) – arcsin(λt)] µ
a
1 sin(πµ)
ϕ(x)= √ , k = .
2 2
1 – λ x πµ
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 61