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Mathematics Equation Solving Wolfram Language

[?] Solve a system of non linear equations?

marouane bouadi

Posted 6 years ago

I want to Solve the following set of non linear differential equations. In this context, I want to have the expression of p1, p2,p3,p4 in the stationnary state (t-> Infinity) and I want to add the normalization p1[t]+ p2[t]+p3[t]+p4[t]=1. DSolve[{D[p1[t], t] == a Cos[t] p2[t] - p1[t], D[p2[t], t] == p1[t] + a Cos[t] p3[t] - p2[t], D[p3[t], t] == a Cos[t]p4[t] - p3[t], D[p4[t], t] == (1 - a *Cos[t]) (p2[t] + p3[t] + p4[t]) - p4[t], p1[t] + p2[t] + p3[t] + p4[t] == 1}, {p1, p2, p3, p4}, t] When I used the above instruction , the system does not provide me a solution. Please help me. Thank you in advance.

POSTED BY: marouane bouadi

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marouane bouadi

Posted 6 years ago

Thank you very much for your valuable help. I really appreciate your fruitful discussion

POSTED BY: marouane bouadi

Neil Singer

Neil Singer, AC Kinetics, Inc.

Posted 6 years ago

Somehow I got attributed for Mariusz s post. Mine was far less complete and interesting than His post so Ill take the credit anytime! Sorry Mariusz! I posted that if you only care about steady state, you can set the derivatives all to zero and solve the resulting algebraic equations. Regards Neil

POSTED BY: Neil Singer

Neil Singer

Neil Singer, AC Kinetics, Inc.

Posted 6 years ago

DSolve function can't find symbolic solution that means Mathematica doesn't know a closed-form solution, if it exists. Using AsymptoticDSolveValue we can find a series solution in point $t=0$ of order 3: {P1, P2, P3, P4} = AsymptoticDSolveValue[{D[p1[t], t] == a Cos[t] p2[t] - p1[t], D[p2[t], t] == p1[t] + a Cos[t] p3[t] - p2[t], D[p3[t], t] == a Cos[t] p4[t] - p3[t], D[p4[t], t] == (1 - aCos[t])(p2[t] + p3[t] + p4[t]) - p4[t]}, {p1[t], p2[t], p3[t], p4[t]}, {t, 0, 3}] Plot[Evaluate[{P1, P2, P3, P4} /. C[1] -> 1 /. C[2] -> 1 /. C[3] -> 1 /. C[4] -> 1 /. a -> 1], {t, 0, 1}, PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}]

DSolve function can't find symbolic solution that means Mathematica doesn't know a closed-form solution, if it exists.

Using AsymptoticDSolveValue we can find a series solution in point $t=0$ of order 3:

  {P1, P2, P3, P4} = AsymptoticDSolveValue[{D[p1[t], t] == a Cos[t] p2[t] - p1[t], 
  D[p2[t], t] == p1[t] + a Cos[t] p3[t] - p2[t], D[p3[t], t] == a Cos[t] p4[t] - p3[t], 
  D[p4[t], t] == (1 - a*Cos[t])*(p2[t] + p3[t] + p4[t]) - p4[t]}, {p1[t], p2[t], p3[t], p4[t]}, {t, 0, 3}]

  Plot[Evaluate[{P1, P2, P3, P4} /. C[1] -> 1 /. C[2] -> 1 /. C[3] -> 1 /. C[4] -> 1 /. a -> 1], {t, 0, 1}, 
  PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}]

enter image description here

POSTED BY: Neil Singer

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 6 years ago

DSolve function can't find symbolic solution that means Mathematica doesn't know a closed-form solution, if it exists. Numeric solution: a = 1; {P1, P2, P3, P4} = NDSolveValue[{D[p1[t], t] == a Cos[t] p2[t] - p1[t], D[p2[t], t] == p1[t] + a Cos[t] p3[t] - p2[t], D[p3[t], t] == a Cos[t] p4[t] - p3[t], D[p4[t], t] == (1 - aCos[t])(p2[t] + p3[t] + p4[t]) - p4[t], p1[0] == 1, p2[0] == 1, p3[0] == 1, p4[0] == 1}, {p1, p2, p3, p4}, {t, 0, 100}] Plot[Evaluate[{P1[t], P2[t], P3[t], P4[t]}], {t, 0, 100}, PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}] EDITED 2.4.2018: By suggestion Mr Neil's: sol = Solve[{0 == a Cos[t] p2[t] - p1[t], 0 == p1[t] + a Cos[t] p3[t] - p2[t], 0 == a Cos[t] p4[t] - p3[t], 0 == (1 - a*Cos[t]) (p2[t] + p3[t] + p4[t]) - p4[t], p1[t] + p2[t] + p3[t] + p4[t] == 1}, {p1[t], p2[t], p3[t], p4[t]}] Plot[Evaluate[{p1[t], p2[t], p3[t], p4[t]} /. sol[[1]] /. a -> 1], {t, 0, 100}, PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}] If the system is overdetermined then it had no solutions.

DSolve function can't find symbolic solution that means Mathematica doesn't know a closed-form solution, if it exists.

Numeric solution:

         a = 1; {P1, P2, P3, P4} = 
         NDSolveValue[{D[p1[t], t] == a Cos[t] p2[t] - p1[t], 
         D[p2[t], t] == p1[t] + a Cos[t] p3[t] - p2[t], 
         D[p3[t], t] == a Cos[t] p4[t] - p3[t], 
         D[p4[t], t] == (1 - a*Cos[t])*(p2[t] + p3[t] + p4[t]) - p4[t], 
         p1[0] == 1, p2[0] == 1, p3[0] == 1, p4[0] == 1}, {p1, p2, p3, 
         p4}, {t, 0, 100}]


         Plot[Evaluate[{P1[t], P2[t], P3[t], P4[t]}], {t, 0, 100}, 
         PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}]

enter image description here

EDITED 2.4.2018:

By suggestion Mr Neil's:

      sol = Solve[{0 == a Cos[t] p2[t] - p1[t], 
      0 == p1[t] + a Cos[t] p3[t] - p2[t], 0 == a Cos[t] p4[t] - p3[t], 
      0 == (1 - a*Cos[t]) (p2[t] + p3[t] + p4[t]) - p4[t], 
      p1[t] + p2[t] + p3[t] + p4[t] == 1}, {p1[t], p2[t], p3[t], p4[t]}]

     Plot[Evaluate[{p1[t], p2[t], p3[t], p4[t]} /. sol[[1]] /. a -> 1], {t,
        0, 100}, PlotLegends -> {"p1[t]", "p2[t]", "p3[t]", "p4[t]"}]

enter image description here

If the system is overdetermined then it had no solutions.

POSTED BY: Mariusz Iwaniuk

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