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182 Polynomial Approximation of Differential Equations
where σ ∈ R , and U :
explicitly, i.e.
(9.1.2) U(x) = σ +
Ī → R is the unknown. Of course, we know the solution
x
−1
f(s) ds, ∀x ∈ Ī.
We are concerned with the convergence analysis of polynomial approximations of U.
We give an overview of different techniques in the coming sections.
A generalization of (9.1.1) is
(9.1.3)
⎧
⎨U
′ + AU = f in ] − 1,1],
⎩
U(−1) = σ,
where A : Ī → R is a given continuous function.
Other elementary problems can be studied. We are mainly interested in boundary-
value problems. Two typical examples of second-order equations are
(9.1.4)
(9.1.5)
⎧
⎨ −U ′′ = f in I,
⎩
U(−1) = σ1, U(1) = σ2,
⎧
⎨ −U ′′ + µU = f in I,
⎩
U ′ (−1) = σ1, U ′ (1) = σ2,
where σ1, σ2 ∈ R and µ > 0. In particular, (9.1.4) and (9.1.5) are known as Dirichlet
problem and Neumann problem respectively.
Within the framework of fourth-order problems, we consider for instance the equation
(9.1.6)
with σi ∈ R, 1 ≤ i ≤ 4.
⎧
U
⎪⎨
⎪⎩
IV = f in I,
U(−1) = σ1, U(1) = σ2,
U ′ (−1) = σ3, U ′ (1) = σ4,