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Help solving system of 3 differential equations?

Posted 2 years ago

Basically, I want to solve a system of 3 differential equations.

Here I am initializing some variables and defining f0 that is g(v,t=0).

a = 0.5;

b = (3^0.5)/2;

ve = 1;

u = 1;

f0[v_] = (a/(Pi)^0.5)*Exp[v^2/ve^2] + (b/(Pi)^0.5)*Exp[(v - u)^2/ve^2];

u1 = Log[((1 - a^2)/a^2)^0.5]/(2*u) + u/2;

u2 = 3*u/2 - Log[((1 - a^2)/a^2)^0.5]/(2*u);

f0[u1]

f0[u2]

Here I am using a system of 3 equations for 3 functions and equating g(v,0)=f0

sol = NDSolve[{ 

D[go[v, t], t] + 
D[D[w[v, t], t], v]/(v^3) + (-3*v^(-4))*D[w[v, t], t] == 0, 

g[v, t] == Pi/2 *(v^2) *D[go[v, t], v],

go[v, 0] == f0[v],

D[w[v, t], t] == 2*g[v, t]*w[v, t],

w[v, 0] == 0},

{go[v, t], W[v, t]},

{v, u1, u2},

{t, 0, 10000000000}]
POSTED BY: Diogo Carvalho
4 Replies
Posted 2 years ago

Eliminate g and don't capitalize w:

a = 1/2; b = Sqrt[3]/2; ve = 1; u = 1;
f0[v_] = (a/Sqrt[Pi])*Exp[v^2/ve^2] +
   (b/Sqrt[Pi])*Exp[(v - u)^2/ve^2];
u1 = Log[((1 - a^2)/a^2)^(1/2)]/(2*u) + u/2;
u2 = 3*u/2 - Log[((1 - a^2)/a^2)^(1/2)]/(2*u); sol = 
 NDSolve[{D[go[v, t], t] + D[D[w[v, t], t], v]/(v^3) +
     (-3*v^(-4))*D[w[v, t], t] == 0,
   go[v, 0] == f0[v],
   D[w[v, t], t] == 2*Pi/2*(v^2)*D[go[v, t], v]*w[v, t],
   w[v, 0] == 0},
  {go[v, t], w[v, t]},
  {v, u1, u2}, {t, 0, 10000000000}]
POSTED BY: Gianluca Gorni

You should list all three unknown functions that you wish to calculate.

POSTED BY: Gianluca Gorni
Posted 2 years ago
dw(v,t)/dt =2g(v,t) w(t,v)

g(t,v)= Pi/2 *(v^2)*d g0(v,t)/dt

d go(v,t)/dt + d/dv[dw/dt *1/v^3)]=0

initial conditions

f0=go(v,0)

w(v,0)=0
POSTED BY: Diogo Carvalho
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