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