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Ordinary Differential Equations 217
9.9 Systems of differential equations
Approximation by spectral methods of systems of differential equations in two or more
unknowns can be also taken into consideration. We illustrate a simple example which
will be also useful in section 12.3. Other systems are examined in section 10.5.
We are concerned with finding the solutions V : [−1,1] → R and W : [−1,1] → R of
the differential problem
(9.9.1)
⎧
−V
⎪⎨
⎪⎩
′′ + µW = f in ] − 1,1[,
−W ′′ − µV = g in ] − 1,1[,
V (±1) = W(±1) = 0,
where f and g are given continuous functions and µ ∈ R.
We approximate (9.9.1) by the collocation method. Thus, for n ≥ 1, we have to find
two polynomials pn,qn ∈ Pn such that
(9.9.2)
⎧
⎪⎨
⎪⎩
−p ′′ n(η (n)
i ) + µqn(η (n)
i ) = f(η (n)
i ) 1 ≤ i ≤ n − 1,
−q ′′
n(η (n)
i ) − µpn(η (n)
i ) = g(η (n)
i ) 1 ≤ i ≤ n − 1,
pn(η (n)
0 ) = qn(η (n)
0 ) = 0,
pn(η (n)
n ) = qn(η (n)
n ) = 0.
When n tends to infinity, pn converges to V and qn converges to W. For the proof we
can use the techniques developed in section 9.4. The convergence is uniform in [−1,1].
As usual, the determination of the approximating polynomials is equivalent to
solving a linear system which can be written by defining the vectors
¯pn ≡ pn(η (n)
1 ), · · · ,pn(η (n)
n−1 ) , ¯qn ≡ qn(η (n)
1 ), · · · ,qn(η (n)
n−1 ) ,
¯fn ≡ fn(η (n)
1 ), · · · ,fn(η (n)
n−1 ) , ¯gn ≡ gn(η (n)
1 ), · · · ,gn(η (n)
n−1 ) .