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Re: Linear Differential Equations with Matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg59461] Re: [mg59413] Linear Differential Equations with Matrices
• From: Selwyn Hollis <sh2.7183 at earthlink.net>
• Date: Wed, 10 Aug 2005 02:56:53 -0400 (EDT)
• References: <200508090730.DAA19044@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On Aug 9, 2005, at 3:30 AM, David Boily wrote:

> How can I solve this system with mathematica:
>
> <<LinearAlgebra`MatrixManipulation`
>
> A = {{-F1, 0, 0}, {0, -F2, 0}, {1, -1, 0}}
> Q = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}}
> Z = ZeroMatrix[3]
>
> DSolve[{P'[t] + Transpose[A].P[t] + P[t].A + Q == Z, R'[t] +
> Transpose[A].R[t] + P[t] == Z, P[T]==Z, R[T]==Z}, {P, R}, t]
>
> where P and R are 3x3 matrix functions of t. Mathematica tells me:
>
> Solve::eqf: {{-F1, 0, 1}, <<2>>} . R[t] + <<2>> == {{0, <<2>>}, <<2>>}
>      is not a well-formed equation.
>
> shouldn't this be possible without going the long way and defining
>
> P[t_]:={{p11[t], p12[t], p13[t]}, {p21[t], p22[t], p23[t]}, {p31 
> [t],p32[t], p33[t]}}
> R[t_]:={{r11[t], r12[t], r13[t]}, {r21[t], r22[t], r23[t]}, {r31 
> [t],r32[t], r33[t]}}
>
> Thanks,
>
> David Boily
> Centre for Intelligent Machines
> McGill University
> Montreal, Quebec
>


It would be nice, but unless I'm mistaken the only way to do it is to  
provide DSolve with a flat list of scalar equations. Anyhow, the  
following does the trick:

<< "LinearAlgebra`MatrixManipulation`"
A = {{-F1, 0, 0}, {0, -F2, 0}, {1, -1, 0}};
Q = {{0, 0, 0}, {0, 0, 0}, {0, 0, 1}};
Z = ZeroMatrix[3];

P[t_] := {{p11[t], p12[t], p13[t]}, {p21[t], p22[t], p23[t]}, {p31 
[t], p32[t], p33[t]}}
R[t_] := {{r11[t], r12[t], r13[t]}, {r21[t], r22[t], r23[t]}, {r31 
[t], r32[t], r33[t]}}

deqns = Flatten[ Thread /@ Flatten[ Thread /@
        {P'[t] + Transpose[A].P[t] + P[t].A + Q == Z,
         R'[t] + Transpose[A].R[t] + P[t] == Z,
         P[T] == Z, R[T] == Z} ] ];

vars = Flatten[{P[t], R[t]}];

solnrules = DSolve[deqns, vars, t]//First//Simplify

MatrixForm[ Psoln = P[t] /. solnrules ]

MatrixForm[ Rsoln = R[t] /. solnrules ]


----------------------
Selwyn Hollis
http://www.appliedsymbols.com

• Follow-Ups:
  • Re: Re: Linear Differential Equations with Matrices
    • From: Chris Chiasson <chris.chiasson@gmail.com>

• References:
  • Linear Differential Equations with Matrices
    • From: David Boily <dsboily@fastmail.ca>