Unformatted text preview: ondorder partial derivative:
∂2u
.
(xi , t) =
∂x2
∂u
.
=n
∂x ∂u
∂x xi +xi+1
,t
2 − ∂u
∂x xi−1 +xi
,t
2 1/n
xi + xi+1
∂u
,t −
2
∂x xi−1 + xi
,t
2 .
= n2 [ui−1 (t) − 2ui (t) + ui+1 (t)].
Thus the partial diﬀerential equation
∂u
∂2u
=
∂t
∂x2
can be approximated by a system of ordinary diﬀerential equations
dui
= n2 (ui−1 − 2ui + ui+1 ).
dt
This is a ﬁrst order linear system which can be presented in vector form as −2 1
0 ···
0 1 −2 1 · · ·
0 du
2
0
1 −2 · · ·
· ,
= n P u, where P = dt
·
·
· ···
·
0
0
· · · · −2
93 the last matrix having n − 1 rows and n − 1 columns. Finally, we can rewrite
this as
du
= Au,
dt where A = n2 P, (4.16) a system exactly like the ones studied in Section 2.5. In the limit as n → ∞
one can use the Mathematica program of §2.6 to check that the eigenvalues of A
approach the eigenvalues of d2 /dx2 as determined in the preceding section, and
the eigenvectors approximate more and more closely the standard orthonormal
basis of sine functions.
One cou...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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