(*by Nasser M. Abbasi *) u = Manipulate[ Plot3D[f[x, t, \[Beta], d], {x, minX, maxX}, {t, .01, simTime}, PlotRange -> {Automatic, {0, maxTime}, {0, maxF}}, AxesLabel -> {"x", "t", "f(x,t)"}, PlotLabel -> Style[ "Solution to \!\(\*FractionBox[\(\(\[PartialD]f\)\(\\\ \)\), \(\[PartialD]t\)]\) = \ -\[Beta]\!\(\*FractionBox[\(\(\[PartialD]f\)\(\\\ \)\), \(\[PartialD]x\)]\)+ D\!\(\*FractionBox[\(\*SuperscriptBox[\(\[PartialD]\), \ \(2\)]\\\ f\), \(\[PartialD]\*SuperscriptBox[\(x\), \(2\)]\)]\) \nEinstein-Wiener process", Small], ImagePadding -> 55, ImageSize -> 500, PlotPoints -> ControlActive[{50, Round[simTime/10] + 5}, {50, Round[simTime/10] + 5}], Mesh -> ControlActive[10, 10], MaxRecursion -> 0] , {{\[Beta], .4, "\[Beta] (Drift)="}, -1, 1, .01, AppearanceElements -> All, ContinuousAction -> False}, {{d, .5, "D (Diffusion)="}, .1, 5, .01, AppearanceElements -> All, ContinuousAction -> True}, {{simTime, .5, "simulation time="}, .1, maxTime, .1, AppearanceElements -> All, SynchronousUpdating -> True}, Delimiter, {{maxF, .1, "max F="}, .01, 3, .01, AppearanceElements -> All, ContinuousAction -> True}, AutorunSequencing -> {{3, 80}}, Initialization :> { minX = -300; maxX = 300; maxTime = 500; f[x_, t_, \[Beta]_, Diffusion_] := Module[{\[Mu], \[Sigma]}, \[Mu] = \[Beta] t; \[Sigma] = Sqrt[2 Diffusion t]; 1/(\[Sigma] Sqrt[2 \[Pi]]) Exp[-(1/2) ((x - \[Mu])/\[Sigma])^2] ] } ]