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278 Polynomial Approximation of Differential Equations
We give results from a numerical experiment corresponding to the following data:
ζ = 1, ǫ = 200, T = .32, n = 40, m = 64. The initial guess is U0(x) :=
ǫ
exp (iπx) cosh 2ζx
, x ∈ I, U0(±1) = 0. The function Θ (k)
n , plotted in figure
12.3.1 for k = 0,8,16,24,32,40,48,56,64, approximates a travelling soliton.
Approximations of the nonlinear Schrödinger equation in R, by a non-overlapping
multidomain method using Laguerre functions as in the previous section, are presented
in de veronico(1991), and de veronico, funaro and reali(1991).
12.4 Zeroes of Bessel functions
Bessel functions frequently arise when solving boundary-value partial differential equa-
tions by the method of separation of variables. For the sake of simplicity, we only
consider Bessel functions Jk with a positive integer index k ≥ 1. These are deter-
mined, up to a multiplicative constant, by the Sturm-Liouville problem (see section
1.1):
(12.4.1)
⎧
⎪⎨
−(x J ′ k (x))′ −
⎪⎩
Jk(0) = 0.
x − k2
Jk(x) = 0, x > 0,
x
Many properties are known for this family of functions (they can be found, for example,
in watson(1966)). In particular, we have |Jk(x)| ≤ 1/ √ 2, k ≥ 1, x > 0.
An interesting issue is to evaluate the zeroes of Bessel functions. It turns out
that Jk, k ≥ 1, has an infinite sequence of positive real zeroes z (k)
j , j ≥ 1, that we
assume to be in increasing order. We describe here one of the techniques available for
the computation of the z (k)
j ’s. We take for instance k = 1. For λ > 0, we define
U(x) := J1( √ λx), x > 0. After substitution in (12.4.1), we find the differential equation
(12.4.2)
⎧
⎨ − U
⎩
′′ (x) − 1
x U ′ (x) + 1
U(x) = λ U(x),
x2 U(0) = 0.
x > 0,