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}]
