This is a corrected version of pdeProb2.mw, in which we examine the 1D classical burgers equation, and find an asymptotic steady state in the solution fields u, v which is not reached by a solution via numerical simulation.
NOTE: When generating and displaying PLOTS AT HIGH RESOLUTION, do not use p1 := plot(bla, etc); i.e. do not end with a semicolon. Instead, end with a colon viz p1 := plot(bla, etc): which sends the result to p1 instead of generating an excess memory use message. Then create the plot by executing p1; i.e. end the assigned p1 with a semicolon to display the graphics result.
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We load the MAPLE Physics package from the MapleCloud, in order to support solutions using pdsolve().
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(2) 
Start of definition of problem:
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(3) 
Start of definition of problem:
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with(PDETools); with(CodeTools);with(plots);


(4) 
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Two 1D coupled Burgers equations  semiclassical case O(1), O( ) : retain O(1) only for u(x,t) and O(1), O( ) for v(x,t):
In the quantum case, there are two coupled quantum Burgers equations, which each include the quantum potential terms. As in the classical case above, we apply constant external forces and . Our aim is to display the profiles of and as strings on space.
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#hBar := 'hBar': m := 'm':Fu := 'Fu': Fv := 'Fv': # define constants

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hBar:= 1:m:= 1:Fu:= 0.2:Fv:= 0.1: # set constant values  same as above ...consider reducing

Notice that we set
At O( ) the real quantum potential term is zero, leaving the classical expression:
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pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;


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As in the classical case above, the temporal and spatial derivative are each of order 1; so only one initial condition and one boundary condition are required for this part of the semiclassical equations.
On the otherhand, the imaginary quantum potential equation for v(x,t) has only O( ) terms so together the pair of equations for are semiclassical:
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pdev := diff(v(x,t),t)+u(x,t)/m*(diff(v(x,t),x))hBar*(diff(u(x,t),x$2))/(2*m)+v(x,t)*(diff(u(x,t),x))/m = Fv;


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By inspection of the derivatives in above two equations we now set up the ICs and BCs for and Note that the above second order spatial derivative requires a 1st order derivative boundary condition as defined below.
The quantum initial and boundary conditions are similar to the classical case, but also comprise additional boundary condition terms for and for , notably a 1st derivative reflective BC term for .
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ICu:={u(x,0) = 0.1*sin(2*Pi*x)};# initial conditions for PDE pdeu


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ICv:={v(x,0) = 0.2*sin(Pi*x)};# initial conditions for PDE pdev


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BCu := {u(0,t) = 0.5*(1cos(2*Pi*t)),D[1](u)(1,t) = 0}; # boundary conditions for PDE pdeu: note the reflective derivative term D[1](u)


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BCv := {v(0,t) = 0.5*sin(2*Pi*t), v(1,t)=0.5*sin(2*Pi*t)}; # boundary conditions for PDE pdev


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This set of equations and conditions can now be solved numerically.
The above IC and BC are both at and thus consistent.
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pdu := pdsolve({pdeu,pdev},{IC,BC},numeric, time = t,range = 0..1,spacestep = 1/66,timestep = .1);

Here is the 3D plot of u(x,t):
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T := 3; p1 := pdu:plot3d(u,t=0..T,numpoints = 2000,x=0.0..2, shading = zhue,orientation=[146,54,0],scaling = constrained, title = print("Figure 1",u(x, t), numeric));

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