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Ordinary Differential Equations 189
The exact solution (dashed line in figure 9.2.1) is U(x) := √ 1 + x − 1, x ∈ Ī, corre-
sponding to the data: σ = −1, A ≡ 1
2
and f(x) := 2+x
2 √ 1+x
1 − 2 , x ∈] − 1,1]. We
observe that f ∈ C0 ( Ī) and that U has a singularity in the derivative at x = −1.
Nevertheless, convergence is still achieved. The same experiment, using the collocation
method at the Chebyshev nodes, displays a similar behavior.
The numerical schemes introduced here can be also generalized to other type of
equations. For example, tau and collocation methods applied to problem (9.1.4), give
respectively
(9.2.14)
(9.2.15)
⎧
⎨
⎩
⎧
⎨
−p ′′ n = Πw,n−2f in I,
⎩
pn(−1) = σ1, pn(1) = σ2,
−p ′′ n(η (n)
i ) = f(η (n)
i ) 1 ≤ i ≤ n − 1,
pn(η (n)
0 ) = σ1, pn(η (n)
n ) = σ2,
pn ∈ Pn, n ≥ 2.
The systems corresponding to (9.2.14) an (9.2.15) have been investigated in sections
7.3 and 7.4, respectively. Similar considerations hold for the remaining differential
equations proposed in section 9.1. In all cases, we expect convergence of the sequence of
polynomials pn to the exact solution U, as n tends to infinity. To provide convergence
estimates, it is useful to restate the original problems in an appropriate way. We study
how to do this in the next sections.
9.3 The weak formulation
An effective approach for the study of solutions of differential equations, and how to
approximate them, is the variational method. Here, the problem is conveniently for-
mulated in the so-called weak form. For a given equation, there are several ways to
proceed.