Untitled - Cdm.unimo.it

272 Polynomial Approximation of Differential Equations
Figure 12.2.1 - The function U(x) = e −x sin 7x
1+x 2 , x ∈ [0,4].
In our first experiment, we discretize the equation in a bounded subset of I, i.e.,
the interval ]0,2[. For any n ≥ 2, we use the collocation method at the nodes ˆη (n)
j :=
η (n)
j +1, 0 ≤ j ≤ n, obtained by shifting the Chebyshev Gauss-Lobatto points in [−1,1]
(see (3.1.11)). Thus, we want to find a polynomial pn ∈ Pn such that
(12.2.2)
⎧
⎨
−p ′′ n(ˆη (n)
i ) − p ′ n(ˆη (n)
i ) = exp(−ˆη (n)
i ) f(ˆη (n)
i ), 1 ≤ i ≤ n − 1,
⎩
pn(0) = 0, pn(2) = 0.
To determine a unique solution of (12.2.2), we force the homogeneous Dirichlet boundary
condition at the point x = 2. This condition is suggested by the behavior prescribed
for U at infinity. We note that U(2) ≈ −.04533, so that the limit limn→+∞ pn does
not coincide with the function U. We examine for instance the approximation of the
quantity U ′ (0) = 7. We give in table 12.2.1 the error En := |U ′ (0) − p ′ n(0)|, n ≥ 2,
for different n. As expected En does not decay to zero, but approaches a value of the
same order of magnitude of the error |U(2) − pn(2)|. Of course, we can improve on
these results by increasing the size of the computational domain.