Untitled - Cdm.unimo.it

224 Polynomial Approximation of Differential Equations
(10.2.7) V (−1,t) = V (1,t) = 0, ∀t ∈ [0,T],
where f(x,t) := − 1
2 (1 − x)σ′ 1(t) − 1
2 (1 + x)σ′ 2(t), x ∈ I, t ∈]0,T]. Here, we require
that the functions σ1 and σ2 are differentiable.
We intend to use spectral methods in order to approximate the unknown V . To
this end, for any t ∈ [0,T], the function V (·,t) : I → R is approximated by an
algebraic polynomial pn(·,t) ∈ P 0 n (see (6.4.11)), where the integer n ≥ 2 has been
fixed in advance. For example, we use the collocation method (see section 9.2). The
approximate problem is
(10.2.8)
∂pn
∂t (η(n) i ,t) = ζ ∂2pn ∂x
(η(n)
2 i ,t) + f(η (n)
i ,t), 1 ≤ i ≤ n − 1, ∀t ∈]0,T].
As usual, the nodes η (n)
d (α,β)
i , 1 ≤ i ≤ n − 1, are the zeroes of the polynomial dxP n ,
α > −1, β > −1. Initial and boundary conditions are respectively given by
(10.2.9) pn(η (n)
i ,0) = U0(η (n)
i ) − 1
2 (1−η(n)
i )U0(−1) − 1
2 (1+η(n)
i )U0(1), 0 ≤ i ≤ n,
(10.2.10) pn(η (n)
0 ,t) = pn(η (n)
n ,t) = 0, ∀t ∈ [0,T].
At this point, our problem is transformed into a (n + 1) × (n + 1) linear system of
ordinary differential equations. Actually, with the notations of sections 7.2 and 7.4, we
can write for any t ∈]0,T] (we choose for instance n = 4):
(10.2.11)
d
dt
⎡
pn(η
⎢
⎣
(n)
1 ,t)
pn(η (n)
⎤ ⎡
⎥ ⎢
⎥ ⎢
2 ,t) ⎥ = ζ ⎢
⎦ ⎣
pn(η (n)
3 ,t)
˜d (2)
11
˜d (2)
21
˜d (2)
31
˜d (2)
12
˜d (2)
22
˜d (2)
32
˜d (2)
13
˜d (2)
23
˜d (2)
33
⎤⎡
pn(η
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
(n)
1 ,t)
pn(η (n)
⎤
⎥
2 ,t) ⎥
⎦ +
⎡
f(η
⎢
⎣
(n)
1 ,t)
f(η (n)
⎤
⎥
2 ,t) ⎥
⎦ .
pn(η (n)
3 ,t)
f(η (n)
3 ,t)
Together with condition (10.2.9), this differential system admits a unique solution.
Moreover, following the remarks of section 8.2, the eigenvalues of the matrix in (10.2.11)
are distinct, real and negative, under suitable assumptions on the parameters α and β.