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evolution matrix and ODEs, i think....
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48229] evolution matrix and ODEs, i think....
*From*: sean_incali at yahoo.com (sean kim)
*Date*: Wed, 19 May 2004 02:42:04 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
hello group.
I think i'm tryign to use eovlution matrix and matrix exponential to
solve a linear ODE system. ("I think" being the operative phrase.
that's kinda funny)
let's say you have a linear ODE system which can be represented by the
matrix as follows.
c' = A c + b c(0) = c0
where b is a vector [k0 c0 , 0, 0, 0, 0, 0, 0] and
c = [c11, c12, c13, c14, c15, c2, c3]
and
A = {
{-K, 0, 0, 0, 0, 0, 0},
{2 K1, -K, 0, 0, 0, 0, 0},
{0, 2 K1, -K, 0, 0, 0, 0},
{0, 0, 2 K1, -K, 0, 0, 0},
{0, 0, 0, 2 K1, -k2, 0, 0},
{2 K2, 2 K2, 2 K2, 2 K2, 2 k2, -k3, 0},
{0, 0, 0, 0, 0, k3, -kd}
}
How do I implement the particular solution to above system using Mathematica?
i want to get a solution to above using the matrix exponential.
/t
c = e^(A t) + | e^(A t) b0 dt
/0
or in Mathematica notation,
\!\(c[t]\ \[Equal] \ \[ExponentialE]\^\(\(A\)\(\ \)\(t\)\(\ \)\)\ \(\
\[Integral]\_0\%t\ \[ExponentialE]\^\(\(A\)\(\ \)\(t\)\(\ \)\)\ b0\ \
\[DifferentialD]t\)\)
is supposed to be equal to
\!\(\(\[ExponentialE]\^\(\(S\)\(\ \)\(t\)\(\ \)\)\) \((\
I\ + \ N\ t\ + \(\(\ \)\(\(N\^2\) t\)\)\/\(2!\) + \(\(\
\)\(\(N\^3\) \
t\)\)\/\(3!\) + \(\(\ \)\(\(N\^4\) t\)\)\/\(4!\) + \(\(\ \)\(\(N\^5\)
t\)\)\/\
\(5!\) + \(\(\ \)\(\(N\^6\) t\)\)\/\(6!\))\)
A^\(-1\) \((\(\[ExponentialE]\^\(\(-S\)\(\ \)\(t\)\(\ \)\)\) \((\
I\ - \ N\ t\ + \(\(\ \)\(\(N\^2\) t\)\)\/\(2!\) - \(\(\
\)\(\(N\
\^3\) t\)\)\/\(3!\) + \(\(\ \)\(\(N\^4\) t\)\)\/\(4!\) - \(\(\
\)\(\(N\^5\) t\
\)\)\/\(5!\) + \(\(\ \)\(\(N\^6\) t\)\)\/\(6!\))\) + \ I)\) b0\)
Where I is the identity matrix, and N is the nilpotent matrix.( i
think)
above is from using the evolution matrix generated by A. ( its missing
a term i think because c0 = constant?
Does this make sense?
I guess this is more of math/ Mathematica question. I woudl like to start with
the system I have and generated that answer given using Mathematica. ( short
of typing it up)
and plot the solution in Mathematica
Any and all thoughts are so appreciated from all you smart people out
there.
thanks a lot again in advance
sean
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