MATLAB Mathematics - SERC - Index of

Initial Value Problems for ODEs and DAEs
the solution of the partial differential equation is approximated at x k
k = 0, 1, ..., N+1 by
=
kh
for
N
∑
utx (, k ) ≈ c k ()φ t k ( x)
k = 1
Here φ k () x is a piecewise linear function that is 1 at x k and 0 at all the other
. A Galerkin discretization leads to the system of ODEs
x j
Mt ()c′
= Jc where ct () =
c 1 () t
c N () t
and the tridiagonal matrices Mt () and J are given by
and
⎧2h ⁄ 3exp( – t)
if i = j
⎪
M ij = ⎨h ⁄ 6exp( – t)
if i = j ± 1
⎪
⎩ 0 otherwise
⎧ – 2 ⁄ h if i = j
⎪
J ij = ⎨ 1 ⁄ h if i = j ± 1
⎪
⎩ 0 otherwise
The initial values c( 0)
are taken from the initial condition for the partial
differential equation. The problem is solved on the time interval [ 0,
π]
.
In the fem1ode example, the properties
options = odeset('Mass',@mass,'MStateDep','none','Jacobian',J)
indicate that the problem is of the form Mt ()y′ = Jy . The nested function
mass(t) evaluates the time-dependent mass matrix Mt () and J is the constant
Jacobian.
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