[Date Index]
[Thread Index]
[Author Index]
Vector of functions in ordinary differential equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg57109] Vector of functions in ordinary differential equations
*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>
*Date*: Mon, 16 May 2005 01:29:50 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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."
Prev by Date:
**Re: Re: Re: Mathematica Notebook Organiztion**
Next by Date:
**Re: Re: NIntegrate-FindRoot acting up in version 5.1**
Previous by thread:
**Perron Number Tiling Systems -- from Mathematica Information Center**
Next by thread:
**Re: Vector of functions in ordinary differential equations**
| |