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MathGroup Archive 2004

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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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