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138 Polynomial Approximation of Differential Equations
Different boundary conditions can be examined. For B > 0, typical conditions are
obtained by imposing p ′ (−1) = σ1, p ′ (1) = σ2. The last two rows in (7.3.6) have to be
changed accordingly.
One can also assume that A and B are non constant functions in I. This implies
that the matrix relative to the system (7.3.5) is full. Moreover, the expression of the
Fourier coefficients of the term Πw,n−2(Ap ′ +Bp), n ≥ 2, is quite involved (see section
2.4). As we see in the next section, it is more natural to treat these kind of problems
in the physical space. Starting from a different approach, further methods to impose
boundary conditions in the frequency space will be analyzed in section 9.4. Suggestions
for the treatment of the Laguerre and Hermite cases are given in section 9.5.
7.4 Boundary conditions in the physical space
It is standard to work with Gauss-Lobatto nodes in the Jacobi case and Gauss-Radau
nodes in the Laguerre case, since these actually include the boundary points of the
domain
Ī. Applications of the Gauss-Radau nodes in (3.1.13) and (3.1.14) have been
considered in canuto and quarteroni(1982b) for boundary conditions on one side of
the interval [−1,1].
We give a survey of the cases that can occur in the discretization of linear boundary-
value problems in one dimension. The study of the generating equations and of the
behavior of the approximated solutions, when the number of nodes increases, are the
subject of chapter nine.
As usual, we first consider the Jacobi case. The problem of inverting the derivative
operator in the space of polynomials can be formulated as follows. Let us assume that
ξ = −1 (similar arguments apply when ξ = 1). Let q be a polynomial in Pn−1, n ≥ 1.
We are concerned with finding p ∈ Pn such that p ′ = q and p(ξ) = σ, where σ ∈ R.
Evaluating these expressions at the nodes, we end up with the following set of equations: