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238 Polynomial Approximation of Differential Equations
(10.5.2) U(±1,t) = 0, ∀t ∈]0,T],
(10.5.3) U(x,0) = U0(x),
∂U
∂t (x,0) = Û0(x), ∀x ∈] − 1,1[,
where ζ > 0 and U0 :] − 1,1[→ R, Û0 :] − 1,1[→ R are given initial data. Equation
(10.5.1) is of hyperbolic type. The solution U represents the displacement of a vibrat-
ing string of length equal 2, fixed at the endpoints. Other oscillatory phenomena are
described by the wave equation. Examples are collected in baldock and bridgeman
(1981).
From the numerical point of view, we can apply the same techniques of section
10.2 to obtain the semi-discrete spectral approximation of problem (10.5.1), (10.5.2),
(10.5.3). This time, after discretization in the variable x, we get an initial-value second-
order differential system in the variable t.
In the Legendre case, stability results and error estimates can be obtained from a
weak formulation of the problem. The reader is addressed to lions and magenes(1972),
chapter 5, theorem 2.1, for results concerning this aspect of the theory. To our knowl-
edge, no analysis is currently available for the other Jacobi cases.
Another approach consists in writing the solution as a suitable superposition of two
waves travelling in opposite directions. Actually, one checks that the general solution of
(10.5.1) is given by U(x,t) = τ(x − ζt) + υ(x + ζt), where τ and υ can be uniquely de-
termined by the initial data (10.5.3) (d’Alembert solution). Thus, U can be decomposed
into the sum of two functions, which are solutions of equations (10.3.1) and (10.3.20)
respectively, with some prescribed initial guess. Namely, we have U = V + W, where
V and W solve the following linear symmetric hyperbolic system:
(10.5.4)
⎛ ⎞
V (x,t)
∂
⎝ ⎠ =
∂t
W(x,t)
⎛ ⎞
−ζ 0
⎝ ⎠
0 ζ
∂
⎛ ⎞
V (x,t)
⎝ ⎠ , x ∈] − 1,1[, t ∈]0,T].
∂x
W(x,t)