I need to solve this Fokker-Planck equation in Mathematica and my attempt to perform this integration is above:

The Fokker-Planck equation that I need to solve.

My first attempt is above:

a = .3;
µ = .1;
c = .2;
σ = 0.1;
b1 = -a s i;
b2 = a s i - µ i + c i - c s i - c i i;
eqns = D[p[s, i, t], t] == 0.5 σ^2 D[p[s, i, t], {i, 2}] - D[b1 p[s, i, t],{s}] - D[(b2 p[s, i, t] p[s, i, t]), {i}];
DSolve[{eqns, p[s, i, 0] == DiracDelta[.3, .7], Limit[p[s, i, t], s -> Infinity] == 0, Limit[p[s, i, t], i -> Infinity] == 0, Limit[D[p[s, i, t],s], s-> Infinity] == 0, Limit[D[p[s, i, t], s], i -> Infinity] == 0}, {p[s,i, t]}, {t, 0, 30}]

But I get nothing from this code. I'm quite sure that's all wrong, How could I solve this equation?

  • $\begingroup$ What is the DiracDelta supposed to do? It does not depend on any of the variables... $\endgroup$ – Marius Ladegård Meyer Jul 24 '18 at 20:07
  • $\begingroup$ It’s a initial condition, the values of $S$ and $I$ in the time $t=t_{0}$ $\endgroup$ – Herr Schrödinger Jul 24 '18 at 20:09
  • $\begingroup$ DiracDelta[x1, x2] is a distribution that, sloppily said, equals infinity when its arguments x1 == x2 == 0. You just have numbers in there. DiracDelta[.3, .7] will forever and ever equal zero. $\endgroup$ – Marius Ladegård Meyer Jul 24 '18 at 20:13
  • $\begingroup$ How can I adjust this initial condition to make it work? Taking away the Dirac Delta? $\endgroup$ – Herr Schrödinger Jul 24 '18 at 20:15
  • $\begingroup$ Maybe you meant DiracDelta[s - 3/10]*DiracDelta[i - 7/10]? We can't help with the implementation of the math if you can't tell us the math :) Maybe add the equations you want to solve in LaTeX form to the question, in addition to the code? $\endgroup$ – Marius Ladegård Meyer Jul 24 '18 at 20:19

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