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Re: Vector of functions in ordinary differential equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg57140] Re: Vector of functions in ordinary differential equations
*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>
*Date*: Thu, 19 May 2005 03:08:07 -0400 (EDT)
*References*: <d69bi8$5tn$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Thanks for your suggestions.
I noticed that there is appreciable dependence on the Mathematica version (Windows)
Version 4.0: Examples 1 and 2 are ok, example 3 is not
Version 5.0: Examples 1 and 3 (using Thread) are ok, but example 2 is not
Regards,
Wolfgang
Dr. Wolfgang Hintze wrote:
> Considering ordinary differential equations for more than one function
> I'd like to have one differential equation for n functions in terms a
> vector rnv[t] = {x[1,t], x[2,t], ... , x[n,t]} but I did not succeed in
> expressing it in Mathematica.
>
> Any hints are greatly appreciated.
>
> Regards,
> Wolfgang
>
> Here are the details:
>
> 1) There are no problems if I define the independent functions
> explicitly like x[t], y[t], ... and write down a differential eqaution
> for each component.
>
> 2) There are also no problems if the explicit components are collected
> into a vector, e.g. rv[t] := {x[t], y[t]}. Then one differential
> equation for the vector works fine. This is a example for a linear diffeq:
>
> In[103]:=
> rv[t_] := {x[t], y[t]}
>
> In[104]:= rv[t]
>
> Out[104]= {x[t], y[t]}
>
> In[105]:= M = {{a, b}, {c, d}}
>
> Out[105]= {{a, b}, {c, d}}
>
> In[108]:= eq = D[rv[t], t] == M . rv[t]
>
> Out[108]= {Derivative[1][x][t], Derivative[1][y][t]} == {a*x[t] +
> b*y[t], c*x[t] + d*y[t]}
>
> In[110]:= sm = DSolve[eq, rv[t], t];
>
> In[74]:= x[t] /. sm
>
> Out[74]= {1/(2*Sqrt[a^2 + 4*b*c ... [snip]
>
> 3) Now, in the general case, I'd like to have n functions x[1,t],
> x[2,t], ... , x[n,t]: there is the problem
>
> In[91]:= Clear[rv, M, n, x, k, eqm]
>
> In[92]:= n = 2;
>
> In[93]:= rv[t_, n_] := Table[x[k, t], {k, 1, n}]
>
> In[94]:= rv[t, n]
>
> Out[94]= {x[1, t], x[2, t]}
>
> In[95]:= M[n_] := Table[Table[Random[], {n}], {n}]
>
> In[96]:= MatrixForm[M[n]]
>
> Out[96]//MatrixForm=
> MatrixForm[{{0.4129438091150023, 0.19074665264305043},
> {0.31296972444474486, 0.31543944991928186}}]
>
> In[99]:= eqm = D[rv[t, n], t] == M[n] . rv[t, n]
>
> Out[99]=
> {Derivative[0, 1][x][1, t], Derivative[0, 1][x][2, t]} ==
> {0.8654946712834201*x[1, t] + 0.6944612611502227*x[2, t],
> 0.7720429142604556*x[1, t] + 0.016353185110213362*x[2, t]}
>
> In[100]:= DSolve[eqm, rv[t,n], t]
>
> From In[100]:= DSolve::"nvld" : "The description of the equations
> appears to be ambiguous or \
> invalid."
>
> From In[100]:= DSolve::"deqx" : "Supplied equations are not
> differential equations of the \
> given functions."
>
> From In[100]:= DSolve::"deqx" : "Supplied equations are not
> differential equations of the \
> given functions."
>
> Out[100]= DSolve[{Derivative[0, 1][x][1, t], Derivative[0, 1][x][2, t]}
> == {0.8654946712834201*x[1, t] + 0.6944612611502227*x[2, t],
> 0.7720429142604556*x[1, t] + 0.016353185110213362*x[2, t]}, {x[1,
> t], x[2, t]}, t]
>
> From In[111]:= DSolve::"nvld" : "The description of the equations
> appears to be ambiguous or \
> invalid."
>
> From In[111]:= DSolve::"deqx" : "Supplied equations are not
> differential equations of the \
> given functions."
>
> From In[111]:= DSolve::"deqx" : "Supplied equations are not
> differential equations of the \
> given functions."
>
>
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