On Spectral Methods for Volterra Type Integral Equations and the ...

830 T. TANG, X. XU AND J. CHENG
Proof. Following the notations of (3.3), we let
N∑
(K(x, s), φ(s)) N,s = K(x, s(x, θ j ))φ(s(x, θ j ))ω j . (4.3)
j=0
Then the numerical scheme (2.7) can be written as
which gives
u i + 1 + x i
(K(x i , s), U N (s)) N,s = g(x i ), (4.4)
2
u i + 1 + x i
2
∫ 1
−1
K(x i , s(x i , θ))U N (s(x i , θ))dθ
= g(x i ) + J 1 (x i ), 1 ≤ i ≤ N, (4.5)
where
J 1 (x) = 1 + x
2
Using Lemma 3.1 gives
∫ 1
−1
It follows from (4.5), (2.2) and (2.4) that
u i +
K(x, s(x, θ))U N (s(x, θ))dθ − 1 + x (K(x, s), U N (s)) N,s . (4.6)
2
|J 1 (x)| ≤ CN −m |K(x, s(x, ·))| ˜Hm,N(I) ‖UN ‖ L 2 (I). (4.7)
∫ xi
−1
K(x i , s)U N (s)ds = g(x i ) + J 1 (x i ), 1 ≤ i ≤ N. (4.8)
Multiplying F j (x) on both sides of (4.8) and summing up from 0 to N yield
(∫ x
) (∫ x
)
U N (x) + I N K(x, s)u(s)ds + I N K(x, s)e(s)ds
−1
−1
= I N (g) + I N (J 1 ), (4.9)
where U N is defined by (4.1), the interpolation operator I N is defined by (3.4), e denotes the
error function, i.e.,
e(x) = U N (x) − u(x), x ∈ [−1, 1]. (4.10)
It follows from (4.9) and (2.1) that
U N (x) + I N (g − u) + I N
(∫ x
= I N (g) + I N (J 1 ),
−1
)
K(x, s)e(s)ds
which gives
Consequently,
e(x) + (u − I N u)(x) + I N
(∫ x
−1
)
K(x, s)e(s)ds = I N (J 1 ). (4.11)
∫ x
e(x) + K(x, s)e(s)ds = I N (J 1 ) + J 2 (x) + J 3 (x), (4.12)
−1