Untitled - Cdm.unimo.it

244 Polynomial Approximation of Differential Equations
gear(1971), lapidus and seinfeld(1971), jain(1984). Here, we only present some
basic algorithms.
Our purpose here is to solve first-order differential systems of the form
(10.6.1)
⎧
⎪⎨
d
dt
⎪⎩
¯p(t) = D¯p(t) + ¯ f(t),
¯p(0) = ¯p0,
t ∈]0,T],
where ¯p(t) ∈ R n , t ∈ [0,T], is the unknown. In (10.6.1), D is a n × n matrix,
¯p0 ∈ R n is an initial guess and ¯ f(t) is a given vector of R n , ∀t ∈]0,T]. Systems like
the one considered above were derived in sections 10.2, 10.3 and 10.5.
The most straightforward time-discretization method is the Euler method. We
subdivide the interval [0,T] in m ≥ 1 equal parts of size h := T/m > 0. Then,
for any m ≥ 1, we construct the vector ¯ph ≡ ¯p (m)
h ∈ Rn , according to the recursion
formula
(10.6.2)
⎧
⎨
⎩
¯p (j)
h := (I + hD)¯p (j−1)
h + h ¯ f(tj−1) 1 ≤ j ≤ m,
¯p (0)
h := ¯p0,
where I denotes the n × n identity matrix and tj := jh, 0 ≤ j ≤ m. We assume that
D admits a diagonal form and we denote by λi, 1 ≤ i ≤ n, its eigenvalues. Then, we
have the following result.
Theorem 10.6.1 - Let the eigenvalues of the matrix D satisfy Reλi < 0, 1 ≤ i ≤ n.
Then, if ¯p(t), t ∈ [0,T], is the solution of problem (10.6.1), and ¯ph is obtained by
(10.6.2), it is possible to find a constant C > 0 such that
(10.6.3) ¯ph − ¯p(T)R n ≤ Ch sup
for any h satisfying
(10.6.4) 0 < h < ˜ h := min
1≤i≤n
t∈]0,T[
d
2
¯p
(t)
dt2
−2Reλi
|λi| 2
.
R n
,