Untitled - Cdm.unimo.it

228 Polynomial Approximation of Differential Equations
A special case worth mentioning is I ≡ R in (10.2.1) and (10.2.2). Of course,
we no longer have boundary conditions. These are supplanted by the condition that
U tends to zero at infinity with a specified minimum rate. We derive a collocation
type approximation in this case. According to escobedo and kavian(1987), first set
V (x,t) := U( √ 4ζxe2ζt ,e4ζt − 1), x ∈ R, t ∈]0, ˆ T], where ˆ T := 1
4ζln(T + 1). Then, it
is clear that V satisfies the equation
(10.2.20)
∂V
(x,t) = ζ
∂t
2 ∂ V
(x,t) + 2x∂V
∂x2 ∂x (x,t)
, ∀x ∈ R, ∀t ∈]0, ˆ T],
(10.2.21) V (x,0) = U( 4ζx,0) = U0( 4ζx), ∀x ∈ R.
Assuming that the solution V decays as fast as e −x2
at infinity, we approximate by
Hermite functions (see section 6.7). Let ξ (n)
i , 1 ≤ i ≤ n, be the zeroes of Hn. For any
t ∈]0, ˆ T], we want to find Pn(·,t) ∈ Sn−1 such that
(10.2.22)
∂Pn
∂t (ξ(n)
i ,t) = ζ
2 ∂ Pn
(ξ(n)
∂x2 i ,t) + 2ξ (n)
i
(10.2.23) Pn(ξ (n)
i ,0) = U0( 4ζξ (n)
i ), 1 ≤ i ≤ n.
∂Pn
∂x (ξ(n)
i ,t) , 1 ≤ i ≤ n,
We can formulate the same problem in the space of polynomials by setting pn := Pne x2
(see section 7.4). This is equivalent to a system of ordinary differential equations. The
corresponding matrix has non-positive eigenvalues, as pointed out in section 8.2, which
is very important for proving convergence of Pn to V for n → +∞. Actually,
recalling formula (3.4.1), the proof follows the same arguments as the case when I is
bounded. The theory is presented in funaro and kavian(1988). Finally, we can get
an approximation of the unknown U with the change of variables: x → x[4ζ(t+1)] −1/2 ,
t → 1
4ζln(t + 1).
We remark that the direct approximation of the unknown U by the Hermite-
collocation method applied to equation (10.2.1) with I ≡ R, is unstable. This is due
to the fact that the quantity
R ψ′ (ψv) ′ dx is not in general positive for ψ ∈ H 1 v(R),
where v(x) := e x2
, x ∈ R.