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I want to solve the Fokker-Planck Equation which is used to describe the evolution of orbital dynamics, but I got an error like

NDSolveValue::litarg: To avoid possible ambiguity, the arguments of the dependent variable in p[t,{Subscript[r, 1],Subscript[r, 2],Subscript[r, 3],Subscript[v, 1],Subscript[v, 2],Subscript[v, 3]}] should literally match the independent variables.

Main source code is given as below:

tmin = 0;
tmax = 60;
x = 
  {Subscript[r, 1], Subscript[r, 2], Subscript[r, 3], 
   Subscript[v, 1], Subscript[v, 2], Subscript[v,3]};
pdf0 = 1/(8 π^3 Sqrt[Det[Σ]])Exp[-(1/2) (x - M).Inverse[Σ].(x - M) ];
sol = 
  NDSolveValue[
    {D[p[t, x], {t}] == LFP[x, p[t, x]], p[0, x] == pdf0},
    p[t, x],
    {t, tmin, tmax},
    {x[[1]], -∞, ∞}, {x[[2]], -∞, ∞}, 
    {x[[3]], -∞, ∞}, {x[[4]], -∞, ∞},
    {x[[5]], -∞, ∞}, {x[[6]], -∞, ∞}]

Note that x is a 6x1 vector. Σ is a 6x6 covariance matrix and M is a 6x1 real vector. As you can see, pdf0 is the PDF of MultiGaussian distribution. LFP is a differential operator. I guess this operator is correctly defined since I have tested it with values. There are two questions:

  1. How to express the partial differential equation with vectorised form?
  2. How to express the natural boundary?
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  • $\begingroup$ see here for posing a pde in vector form. mathematica.stackexchange.com/q/78641/2079 (Actually i,m not sure now if that's relevant. is p a vector valued function?) $\endgroup$
    – george2079
    Commented Dec 17, 2016 at 14:19
  • $\begingroup$ The p[t,x] is a scalar function (time variant PDF). I think I need to specify the size of the inputs x(6x1), t (a scalar) and p(t,x) (a scalar function) to Mathematica, but I do not know how. Could you help me? I will go to the page you recommended. Thanks for your reminding and help. $\endgroup$
    – Walter He
    Commented Dec 18, 2016 at 0:40
  • $\begingroup$ ok, its not a vector equation, disregard first comment. I've never seen that error, but try p[t,Sequence@x] all four places. $\endgroup$
    – george2079
    Commented Dec 18, 2016 at 13:42
  • $\begingroup$ you probably should show LFP if that doesnt fix it. $\endgroup$
    – george2079
    Commented Dec 18, 2016 at 13:53
  • 2
    $\begingroup$ Where's the definition of Σ, M, LFP? $\endgroup$
    – xzczd ♦
    Commented Dec 22, 2016 at 10:50

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