BurgersAndContinuityEqns = {r*ne[r, t]* D[(u[r, t]),t] + ne[r, t]
* u[r, t] * D[(u[r, t]*r),r] == -B D[(r*ne[r, t]),r], r D[(ne[r,
t],r] + D[(r u[r, t] ne[r, t]),r] == 0,
u[r, [Epsilon]] == 0, ne[r, [Epsilon]] == Exp[-(r^2/2)]};

SolnBa =
esol = Block[{[Epsilon] = $MachineEpsilon},
NDSolve[BurgersAndContinuityEqns /. B -> 157.59, {u[r, t],
ne[r, t]}, {r, [Epsilon], 5}, {t, [Epsilon], 0.18},
Method -> {"StiffnessSwitching"}]]