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214 Polynomial Approximation of Differential Equations
(9.7.6)
⎧
⎪⎨
−ǫ p ′′
ǫ,n + pǫ,n =
⎪⎩
pǫ,n(−1) = 0, pǫ,n(1) = 1,
(n + 2ν) cn−1 cn
+
(n + ν)(2n + 2ν − 1) n x
u ′ n
in I,
where ck, 0 ≤ k ≤ n, are the Fourier coefficients of pǫ,n and un := P (ν,ν)
n , ν > −1.
In fact, the polynomial u ′ n vanishes at the points η (n)
i , 1 ≤ i ≤ n − 1, giving the
collocation scheme. The corrective term on the right-hand side of the first equation in
(9.7.6) is obtained by equating the coefficients of the monomials x n and x n−1 to the
respective coefficients of the left-hand side. The Chebyshev case is treated in a similar
way.
We can solve (9.7.6) in the frequency space by recalling (2.3.8) and (7.1.3). Due to
the positivity of the quantities involved in these computations, we still recover ck > 0,
0 ≤ k ≤ n. Therefore, inequalities (9.7.4) and (9.7.5) also hold when pǫ,n is the
approximation associated with the collocation method.
The same analysis can be developed for the problem
⎧
⎨
(9.7.7)
⎩
Uǫ(−1) = 0, Uǫ(1) = 1.
−ǫ U ′′
ǫ + U ′ ǫ = 0 in I, ǫ > 0,
For this example, the results given in canuto(1988) in the Chebyshev case point out a
different behavior of the approximating polynomials, depending on the parity of their
degree.
9.8 Nonlinear equations
In a large number of practical applications, physics events are modelled by the solution
of nonlinear differential equations. Of course, spectral methods can be employed in