Re: NDSolve w/ vectors requires homogenous system in 5.0
- To: mathgroup at smc.vnet.net
- Subject: [mg42803] Re: [mg42789] NDSolve w/ vectors requires homogenous system in 5.0
- From: Selwyn Hollis <selwynh at earthlink.net>
- Date: Wed, 30 Jul 2003 04:07:29 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
I've just noticed the same problem with nonhomogeneous systems.
For instance, this works:
NDSolve[
{x'[t] == {{-1, 1}, {1, -1}}.x[t], x[0] == {1, 1}},
x, {t, 0, 1} ]
but this doesn't:
NDSolve[
{x'[t] == {{-1, 1}, {1, -1}}.x[t] + {1, 0}, x[0] == {1, 1}},
x, {t, 0, 1} ]
Pretty lame.
-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis
On Saturday, July 26, 2003, at 04:32 AM, Brian Dale wrote:
> It seems that the NDSolve portion works fine, but that the functions
> such as Cross and Plus have trouble working correctly with the
> required mix of numeric and symbolic vectors. The effect of this is
> to restrict vector diff eqts to homogenous systems. Here are my
> results:
>
> The Bloch equation (fundamental equation for magnetic resonance
> imaging) describes the motion of an isochromat in a magnetic field.
> For all parameters = 1 (for simplicity) I can define the following:
>
> b = {1,1,0} represents the magnetic field
> d = DiagonalMatrix[{1,1,1}] represents the signal decay
> r = {0,0,1} represents the signal regrowth
> bd = {{-1,0,-1},{0,-1,1},{1,-1,-1}} represents field and decay
> bdr = {{-1,0,-1,0},{0,-1,1,0},{1,-1,-1,1},{0,0,0,0}} represents
> everything for the augmented coordinate system {x,y,z,Mo}
>
> m0 = {0,0,1} initial magnetization
> ma0 = {0,0,1,1} augmented initial magnetization
>
> The Bloch equation can be written in 3 ways:
>
> eq1:= {m'[t] == Cross[m[t],b] - d.m[t] + r , m[0] == m0}
> eq2:= {m'[t] == bd.m[t] + r , m[0] == m0}
> eq3:= {ma'[t] == bdr.ma[t] , ma[0] == ma0}
>
> NDSolve[eq1, m, {t,0,10}] generates the message Cross::nonn1
> (arguments should be equal-length vectors) and the message
> NDSolve::ndfdmc (inconsistent dimensionality) and fails.
>
> NDSolve[eq2, m, {t,0,10}] generates the message NDSolve::ndfdmc
> (inconsistent dimensionality) and fails.
>
> NDSolve[eq3, ma, {t,0,10}] succeeds.
>
> If I remove the r term from eq1 and eq2 then:
>
> NDSolve[eq1, m, {t,0,10}] still generates the message Cross::nonn1 but
> succeeds (with no regrowth).
>
> NDSolve[eq2, m, {t,0,10}] succeeds (with no regrowth).
>
> The success with eq3 seems to indicate that NDSolve itself is capable
> of solving the equation, but the failure of the others seems to show
> that the equations must be entered such that each term contains an
> unknown vector.
>
> Does anyone have any other experience with this? Perhaps there is an
> easy work-around that doesn't require the augmented coordinate system
> with its wasteful computation of the 4 the coordinate. Any input
> would be appreciated.
>
> -Thanks
> Brian Dale
>
>
>
>
>