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Mats Granvik

12:17

Can someone help me, what answers do you get for the following two one liners?

Limit[Sum[Log[2*k], {k, 1, n}] - Sum[Log[2*k - 1], {k, 1, n + 1/2}],
n -> Infinity]

Limit[Sum[Log[2*k], {k, 1, n}] - Sum[Log[2*k - 1], {k, 1, n}],
n -> Infinity]

1 hour later…

Michael E2

13:44

@MatsGranvik Version V13.2.1 Mac OS ARM:

1/2 Log[\[Pi]/2]

\[Infinity]

Mats Granvik

14:06

@MichaelE2 Thanks! I got the same answers in Version 8.0 for Microsoft Windows (64-bit).

1 hour later…

psimeson

15:32

I have set up a PDE as follows:
J[x_, t_] :=
1/2 ((-1 + 2 \[Mu] - 2 x (-1 + \[Mu] + \[Nu])) P[x, t] + (-1 + x) x \!\(\*SuperscriptBox[\(P\), TagBox[RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}],Derivative],MultilineFunction->None]\[x, t]);
fpe = D[P[x, t], t] + D[J[x, t], x] == 0;
ic = P[x, 0] == Exp[-((x - 0.7)^2/0.01)];
bc1 = J[0, t] == 0;
bc2 = J[1, t] == 0;
asol = DSolve[{fpe, ic, bc1, bc2}, P[x, t], {x, t}]
However, mathematica is unable to solve. This paper has the exact solution: https://arxiv.org/pdf/q-bio/0508045.pdf (Page # 10 (iii)). What am I missing?

2 hours later…

Nasser

17:21

@psimeson you have syntax errors, screen shot below. Can you write the pde using normal mathematica 1-D plain text without any fancy tablet 2D entries to make it easier to see/copy?

2 hours later…

psimeson

19:31

@Nasser Sorry about that.
J[x_,t_]:= P[x, t] (\[Mu] (1 - x) - \[Nu] x) - D[(x (1 - x)/2) P[x, t], x]

Nasser

@psimeson your BC were defined wrong btw. But after fixing these., Mathematica V 13.2.1 can not solve it analytically.  Here is the corrected code

J[x_,t_]:=P[x,t] (mu* (1-x)-v*x)-D[(x (1-x)/2) P[x,t],x]
fpe = D[P[x, t], t] + D[J[x, t], x] == 0
ic = P[x, 0] == Exp[-((x - 7/10)^2/(1/100))];
bc1 = (J[x, t]/.x->0) == 0;
bc2 = (J[x, t]/.x->1) == 0;
asol = DSolve[{fpe, ic, bc1, bc2}, P[x, t], {x, t}]

psimeson

I used the corrected code but v13.1 cannot solve it either

Also P[x,t] is supposed to be normalized and using that:
norm = Integrate[P[x, t], {x, 0, 1}] == 1; didn't help either

Nasser

also try to avoid using real numbers with exact solver. i.e. replace 0.7 by 7/10 and so on.

psimeson

19:49

@Nasser I will keep that in mind. Thanks. Do you know why Mathematica can't solve it?

Nasser

@psimeson if this is supposed to have analytical solution, then it means simply that current DSolve does have know how to do it. May be in the next version of Mathematica it will. There are many pde's that can be solved analytically but DSolve do not solve now. But DSolve improves with each version to the next. It is possible for this one that the initial condition function you gave is hard to integrate. I did not look into your PDE more to analyze it.

I changed initial condition to zero, and it still could not solve it. so it is not the IC.

psimeson

@Nasser this equation has a solution as shown in this paper: arxiv.org/pdf/q-bio/0508045.pdf. It's in terms of JacobiP .

Nasser

Only way I found it can solve it is if you make IC=0 and define values for mu and v. Then it gives the trivial solution ofcourse (since IC=0 and BC=0), but other than that, it can not solve it.  I'd say wait for 13.3 and try again.

J[x_,t_]:=P[x,t] (mu* (1-x)-v*x)-D[(x (1-x)/2) P[x,t],x]
fpe = D[P[x, t], t] + D[J[x, t], x] == 0
ic = P[x, 0] ==0;(* Exp[-((x - 7/10)^2/(1/100))];*)
bc1 = (J[x, t]/.x->0) == 0;
bc2 = (J[x, t]/.x->1) == 0;
asol = DSolve[{fpe/.{mu->3,v->4}, ic, bc1, bc2}, P[x, t], {x, t}]

psimeson

The problem is that I can't seem to solve it for the stationary case:
fpe = D[J[x, t], x] == 0
also couldn't solve it using NDSolve. If we set mu = 0.4 and v = 0.4.

It's a Fokker-Planck equation (i.e. a diffusion equation with drift) and I am surprised that mathematica can't solve this one. I think I am just missing something. Anyway, thank you @Nasser

Nasser

20:32

@psimeson if you like others to look at it, you could post this at the main forum. May someone will have an idea or a workaround.

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