Undetermined PDE Error

Question

I'm trying to solve the Fokker-Planck equation

$$\frac{\partial p}{\partial L}(L, \eta)= \frac{\partial}{\partial \eta}\left[\left(\eta^{2}-1\right) \frac{\partial p}{\partial \eta}(L, \eta)\right] = (\eta^2-1)\frac{\partial^2p}{\partial\eta^2}(L,\eta)+2\eta\frac{\partial p}{\partial\eta}(L,\eta), \quad \eta>1$$

with initial condition

$$p(L=0, \eta)=\delta(\eta-1).$$

Ignoring the initial condition my code is

FPE = D[P[L, eta], 
   L] == (1/Lloc)*((eta^2 - 1)*D[P[L, eta], {eta, 2}] + 
     2*eta*D[P[L, eta], eta])

s = NDSolve[{FPE, P[0, eta] == DiracDelta[eta-1]}, P, {L, 0, 30}, {eta, 1, 30}]

to which I get the error "Encountered non-numerical value for a derivative at L == 0." Is there something I have done wrong? I know this FPE has an analytical solution in terms of Legendre polynomials. Any help would be much appreciated. Thanks.

Answer 1

Try

n = 20;
FPE = D[P[L, eta], L] - D[(eta^2 - 1) D[P[L, eta], eta], eta]
s = Quiet @ NDSolve[{FPE == 0, P[0, eta] == D[Tanh[n (eta - 1)], eta]}, P, {L, 0, 30}, {eta, 1, 30}][[1]]
Plot3D[Evaluate[P[L, eta] /. s], {L, 0, 30}, {eta, 1, 30}]

Here the problem is with DiracDelta[]. Use instead an approximation as

DiracDelta[eta] ~ D[Tanh[n eta], eta]

for $n \gg 1$ and $\eta \ge 0$