I am trying to solve the linear Schrödinger equation with Fourier transform as follows. Although analytically I was able to solve it and create the graphs I wanted. I have difficulty making the corresponding graph when I solve the problem numerically. Can you explain to me what is my mistake in the following code?

Analytical solution

f[x_] = Exp[-x^2]
u[x_, t_, v_] = 
 Integrate[(Exp[-(I*Pi)/(4) + I*(x - y)^2/(4*v*t)])*
    f[y], {y, -Infinity, Infinity}, 
   Assumptions -> {v > 0, t > 0}]/(Sqrt[4 Pi*v*t])
eik5 = Plot3D[Abs[u[x, t, 0.25]]^2, {x, -25, 25}, {t, 0.05, 30}, 
  PlotRange -> All, PlotPoints -> 250]

enter image description here

Numerical solution

(*  Fourier tranform for the equation  Subscript[u, \
t](x,t)=i*v*Subscript[u, xx](x,t) *)
FourierSinTransform[ D[u[x, t], {t, 1}], x, \[Omega]] == 
 FourierSinTransform[ I*v* D[u[x, t], {x, 2}], x, \[Omega], 
  FourierParameters -> {1, -1}]
ode = D[
\!\(\*OverscriptBox[\(\(\ \ \)\(u\)\), \(^\)]\)[\[Omega], t], {t, 
     1}] == I*v*\[Omega]*(-\[Omega]*
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t] - 2*T0);
sol = DSolve[ode, 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], t]
FourierSinTransform[u[x, 0], x, \[Omega]] == 
 FourierSinTransform[f[x], x, \[Omega], FourierParameters -> {1, -1}]
ic = 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], 0] == 
  FourierSinTransform[f[x], x, \[Omega], FourierParameters -> {1, -1}]
sol = DSolve[{ode, ic}, 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], t]

\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega]_, t_] = -((
  2 E^(-I t v \[Omega]^2) (-T0 + 
     E^(I t v \[Omega]^2)
       T0 + \[Omega] DawsonF[\[Omega]/2]))/\[Omega])
u[x_, t_] = InverseFourierSinTransform[
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], \[Omega], x, 
  FourierParameters -> {1, -1}, Assumptions -> {x > 0}]
T0 = 1;
Plot[Abs[u[x, t]]^2, {x, 0, 20}, PlotRange -> All]

You can download the notebook of my project from Wolfram Community enter image description here


1 Answer 1


Too large in comment:

It is very simple actually. The InverseFourierSinTransform in your notebook did not evaluate. Then you are trying to plot it. Here is screen shot from your notebook near the end

enter image description here

You should really always make sure each command completes OK before using its output in the next command.

V 13.2.1 on windows 10

  • $\begingroup$ Why the code did not evaluate? $\endgroup$ Commented Jun 9, 2023 at 18:41
  • 1
    $\begingroup$ @AthanasiosParaskevopoulos I do not know. Mathematica just could not do it. May be there is a workaround or may be it is too hard. But this is separate issue. You could post separate question on this part only. I am just answering the question on why you are getting these errors from the plot command. $\endgroup$
    – Nasser
    Commented Jun 9, 2023 at 18:43
  • $\begingroup$ Οh ok!!! Thank you. I will do it $\endgroup$ Commented Jun 9, 2023 at 18:44

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