Untitled - Cdm.unimo.it

276 Polynomial Approximation of Differential Equations
(12.3.5)
⎧
⎪⎨
⎪⎩
− q(k) n − q (k−1)
n
h
p (k)
n − p (k−1)
n
h
= − ζ
2 [p(k)
= − ζ
2 [q(k)
n + p (k−1)
n
n + q (k−1)
n
] ′′ − ǫ
4 [Θ(k) n + Θ (k−1)
n
] ′′ − ǫ
4 [Θ(k) n + Θ (k−1)
n
where Θ (k)
n := [p (k)
n ] 2 + [q (k)
n ] 2 , 0 ≤ k ≤ m. The polynomials p (m)
n
][p (k)
][q (k)
n + p (k−1)
n
n + q (k−1)
n
]
and q (m)
n
expected to be the approximations of V (·,T) and W(·,T) respectively. Using the
notations of section 9.9, (12.3.5) is equivalent to
(12.3.6)
⎡
⎣ Mn
−µIn
+ ǫ
⎡
⎣
2ζ
∆(k) n
where ∆ (k)
n := diag (Θ (k)
⎤⎡
µIn
⎦
Mn
0
0 ∆ (k)
n
n +Θ (k−1)
n
⎣ ¯p(k) n
¯q (k)
n
⎤⎡
⎦
⎤ ⎡
⎦ = −⎣
Mn
⎤⎡
−µIn
⎦
⎣ ¯p(k) n + ¯p (k−1)
n
¯q (k)
n + ¯q (k−1)
n
)(η (n)
1 ), · · · ,(Θ (k)
µIn
⎤
Mn
⎣ ¯p(k−1) n
¯q (k−1)
n
⎦ 1 ≤ k ≤ m,
n +Θ (k−1)
n
⎤
⎦
]
are
)(η (n)
n−1 ) and µ := 2/ζh.
The inverse of the matrix on the left-hand side of (12.3.6) can be evaluated once and
for all with the help of (9.9.4).
For any 1 ≤ k ≤ m, we can solve the nonlinear equation (12.3.6) iteratively. One
approach is to construct a sequence of vectors according to the prescription
(12.3.7)
where ∆ (k,j)
n
Θ (k,j)
n
⎡
⎣ ¯p(k,j) n
¯q (k,j)
n
+ ǫ
2ζ
⎡
⎤
⎦ :=
⎡
⎣ ∆(k,j−1) n
⎣ Mn
:= diag (Θ (k,j)
n
−µIn
0
0 ∆ (k,j−1)
n
⎤−1
⎧
µIn ⎨
⎦
⎩
Mn
−
⎡
⎣ Mn
⎤⎡
−µIn
⎦
µIn Mn
⎤⎡
:= [p (k,j)
n ] 2 + [q (k,j)
n ] 2 , j ∈ N, and ¯p (k,0)
n
⎦
⎣ ¯p(k,j−1) n
¯q (k,j−1)
n
+ ¯p (k−1)
n
+ ¯q (k−1)
n
+ Θ (k−1)
n )(η (n)
1 ), · · · ,(Θ (k,j)
n
:= ¯p (k−1)
n , ¯q (k,0)
n
⎤⎫
⎬
⎦
⎭
⎣ ¯p(k−1) n
¯q (k−1)
n
j ≥ 1,
⎤
⎦
+ Θ (k−1)
n )(η (n)
n−1 ) with
:= ¯q (k−1)
n .