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Engineering Mathematica Algebra

How to solve this DAE ?

Jie Li

Posted 10 years ago

I want to use the DAE to find an analytical model. But it always run out of memory (8GB), is there anyway can solve it? DSolve[{a[t] + c[t] - f[t] == 0, d[t] + e[t] + f[t] - i[t] == 0, 2 g[t] + h[t] == 0, a[t] - e[t] == 2 d[t] + e[t] - h[t] == 0, a'[t]/3 + c'[t] == N2 a[t], d'[t]/3 + e'[t]/2 + f'[t] == D(d[t] + e[t])- e[t], g'[t]/3 + h'[t]/2 + i'[t] == M*i[t] -A}, {a[t], d[t], g[t], e[t], h[t], c[t], f[t], i[t]}, t] Where D, M, N, and A is constant. Thanks a lot!

I want to use the DAE to find an analytical model. But it always run out of memory (8GB), is there anyway can solve it?

DSolve[{a[t] + c[t] - f[t] == 0, d[t] + e[t] + f[t] - i[t] == 0, 
  2 g[t] + h[t] == 0, a[t] - e[t] == 2 d[t] + e[t] - h[t] == 0, 
  a'[t]/3 + c'[t] == N*2 a[t], 
  d'[t]/3 + e'[t]/2 + f'[t] == D*(d[t] + e[t])- e[t], 
  g'[t]/3 + h'[t]/2 + i'[t] == M*i[t] -A}, {a[t], d[t], g[t], e[t], 
  h[t], c[t], f[t], i[t]}, t]

Where D, M, N, and A is constant.

Thanks a lot!

POSTED BY: Jie Li

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Bill Simpson

Posted 10 years ago

Since D and N are pre-defined Mathematica function names, do some variable substitutions. Since the resulting system seems to take a long time and since some of your functions are simple linear combinations of others, do some function substitutions. In[1]:= Simplify[{a[t] + c[t] - f[t] == 0, d[t] + e[t] + f[t] - i[t] == 0, 2 g[t] + h[t] == 0, a[t] - e[t] == 0, 2 d[t] + e[t] - h[t] == 0, a'[t]/3 + c'[t] == capN2 a[t], d'[t]/3+e'[t]/2+f'[t] == capD(d[t]+e[t]) - e[t], g'[t]/3+h'[t]/2+i'[t] == capM*i[t]-capA} //. {a[t] -> e[t], a'[t] -> e'[t], h[t] -> -2 g[t], h'[t] -> -2 g'[t], f[t] -> a[t] + c[t], f'[t] -> a'[t] + c'[t], i[t] -> d[t] + 2 a[t] + c[t], i'[t] -> d'[t] + 2 a'[t] + c'[t], g[t] -> - d[t] - e[t]/2, g'[t] -> -d'[t] - e'[t]/2}] Out[1]= {True, True, True, True, True, 6 capN e[t] == 3 c'[t] + e'[t], 6 capD d[t] + 6 (-1 + capD) e[t] == 6 c'[t] + 2 d'[t] + 9 e'[t], capA + c'[t] + (5 d[t])/3 + (7 e'[t])/3 == capM (c[t] + d[t] + 2 e[t])} Then this In[2]:= DSolve[{6 capN e[t] == 3 c'[t]+e'[t], 6 capD d[t]+6 (-1+capD) e[t] == 6 c'[t]+2 d'[t]+9 e'[t], capA + c[t] + (5 d'[t])/3 + (7 e'[t])/3 == capM (c[t] + d[t] + 2 e[t])}, {c[t], d[t], e[t]}, t] solves in a couple of seconds. Unfortunately the result is large and complicated, but with some study you might be able to see how to simplify this greatly. Then you can use that result to determine each of the functions that were eliminated.

Since D and N are pre-defined Mathematica function names, do some variable substitutions. Since the resulting system seems to take a long time and since some of your functions are simple linear combinations of others, do some function substitutions.

In[1]:= Simplify[{a[t] + c[t] - f[t] == 0, d[t] + e[t] + f[t] - i[t] == 0, 2 g[t] + h[t] == 0,
   a[t] - e[t] == 0, 2 d[t] + e[t] - h[t] == 0, a'[t]/3 + c'[t] == capN*2 a[t], 
   d'[t]/3+e'[t]/2+f'[t] == capD*(d[t]+e[t]) - e[t], g'[t]/3+h'[t]/2+i'[t] == capM*i[t]-capA} //.
   {a[t] -> e[t], a'[t] -> e'[t],
    h[t] -> -2 g[t], h'[t] -> -2 g'[t],
    f[t] -> a[t] + c[t], f'[t] -> a'[t] + c'[t], 
    i[t] -> d[t] + 2 a[t] + c[t], i'[t] -> d'[t] + 2 a'[t] + c'[t],
    g[t] -> - d[t] - e[t]/2, g'[t] -> -d'[t] - e'[t]/2}]

Out[1]= {True, True, True, True, True, 6 capN e[t] == 3 c'[t] + e'[t], 
 6 capD d[t] + 6 (-1 + capD) e[t] == 6 c'[t] + 2 d'[t] + 9 e'[t], 
 capA + c'[t] + (5 d[t])/3 + (7 e'[t])/3 == capM (c[t] + d[t] + 2 e[t])}

Then this

In[2]:= DSolve[{6 capN e[t] == 3 c'[t]+e'[t], 6 capD d[t]+6 (-1+capD) e[t] == 6 c'[t]+2 d'[t]+9 e'[t], 
  capA + c[t] + (5 d'[t])/3 + (7 e'[t])/3 == capM (c[t] + d[t] + 2 e[t])}, {c[t], d[t], e[t]}, t]

solves in a couple of seconds. Unfortunately the result is large and complicated, but with some study you might be able to see how to simplify this greatly. Then you can use that result to determine each of the functions that were eliminated.

POSTED BY: Bill Simpson

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