Page 648 - Handbook Of Integral Equations

By setting θ = θ k (k =1, ... , n) and with regard to the formula

n–1
1 θ k ± θ l
cos mθ l sin mθ k = cot , (11)
2 2
m=1
where the sign “plus” is taken for the case in which |k – l| is even and “minus” if |k – l| odd, we
obtain the following system of linear algebraic equations for the approximate values ψ l of the desired
function ψ(x) at the nodes:
n

a kl ψ l = f k , f k = f(cos θ k ), k =1, ... , n,
l=1 (12)
1 1 θ k ± θ l
a kl = cot + K(cos θ k , cos θ l ) .
n sin θ k 2
After solving the system (12), the corresponding approximate solution to Eq. (1) can be found
by formulas (2) and (4).

12.5-2. A Solution Bounded at One Endpoint of the Interval
In this case we set

1 – x
ϕ(x)= ζ(x), (13)
1+ x
where ζ(x) is a bounded function on [–1, 1].
We take the same interpolation nodes as in Section 12.5-1, replace ζ(x) by the polynomial
n
1 l+1 cos nθ sin θ l
L n (ζ; cos θ)= (–1) ζ(cos θ l ) , (14)
n cos θ – cos θ l
l=1
and substitute the result into the singular integral that enters the expression (1). Just as above, we
obtain the following quadrature formula:

n
n–1
n
1 1 ϕ(t) dt 1 – cos θ 1
=2 ζ(cos θ l ) cos mθ l sin mθ – ζ(cos θ l ). (15)
π –1 t – x n sin θ n
l=1 m=1 l=1
This formula is exact for the case in which ζ(t) is a polynomial of order ≤ n – 1in t.
The formula for the second summand on the left-hand side of the equation becomes
1 n

1 1
K(x, t)ϕ(t) dt = (1 – cos θ l )K(cos θ, cos θ l )ζ(cos θ l ). (16)
π n
–1
l=1
This formula is exact if the integrand is a polynomial in t of degree ≤ 2n – 2.
On substituting relations (15) and (16) into Eq. (1) and on setting θ = θ k (k =1, ... , n), with
regard to formula (11), we obtain a system of linear algebraic equations for the approximate values ζ l
of the desired function ζ(x) at the nodes:
n

b kl ζ l = f k , f k = f(cos θ k ), k =1, ... , n,
l=1 (17)
1 θ k θ k ± θ l 2 θ l
b kl = tan cot – 1+2 sin K(cos θ k , cos θ l ) .
n 2 2 2
After solving the system (17), the corresponding approximate solution to Eq. (1) can be found
by formulas (13) and (14).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 631