3D Plotting for the solution of stochastic differential equation

Question

I have the following solution of the SDE: $$U(x,t)=-6+12 * \tanh \left[x+\left(B(t)-\frac{t^{2}}{2}\right)+\int_{0}^{t} e^{s^{2}} d s\right]^{2}$$

Where $B(t)$ is white noise.

In the following code and Figure 1, the solution is drawn without noise

Plot3D[-6 + 12*Tanh[x + (0 - t^2/2) + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(
\*SuperscriptBox[\(E\), 
SuperscriptBox[\(s\), \(2\)]] \[DifferentialD]s\)\)]^2, {t, 0, 
  4}, {x, -4, 4}]

In the following code and Figure 2, the solution is drawn with noise

tmax = 10;
samplesPerSec = 10;
ω = 2;
σ = 1;
noise = Interpolation[
   Normal[RandomFunction[
      WhiteNoiseProcess[σ], {0, tmax*samplesPerSec}]][[1]]];

Plot3D[-6 + 12*Tanh[x + (noise[t*samplesPerSec] - t^2/2) + \!\(
\*SubsuperscriptBox[\(∫\), \(0\), \(t\)]\(
\*SuperscriptBox[\(E\), 
SuperscriptBox[\(s\), \(2\)]] \[DifferentialD]s\)\)]^2, {t, 0, 
  4}, {x, -4, 4}]

The question is how can I get a figure that contains more noise as in Figure 3

Is there a better way to add more Noise to the solution and draw it?

Answer 1

1. You can replace the integral in the plot with 1/2 Sqrt[\[Pi]] Erfi[t] - it will be faster and it will let you increase the sample rate.

2. You can increase the PlotPoints to get a better plot.

3. Correct me if I'm wrong but I don't think you don't need WhiteNoiseProcess here - you can just draw from a NormalDistribution directly.

σ = 1;
Plot3D[-6 + 12*Tanh[x + (RandomVariate[NormalDistribution[0, σ]] - t^2/2) + 1/2 Sqrt[π]Erfi[t]]^2,
  {t, 0, 4}, {x, -4, 4}, PlotPoints -> 50, PlotTheme -> "Classic"]