I can't seem to get a solution to the following problem. Can anyone see where I am going wrong I thought I had correct IBC s but they may be wrong/ill-posed

Melvin

Download BurgersEqns.mw

Here it is:

#hBar := 'hBar': m := 'm':Fu := 'Fu': Fv := 'Fv': # define constants

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

At O(

`ℏ`^2;

) the real quantum potential term is zero, leaving the classical expression:

pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;

/ d \ / d \

pdeu := |--- u(x, t)| + u(x, t) |--- u(x, t)| = 0.2

\ dt / \ dx /

On the otherhand, the imaginary quantum potential equation for v(x,t) has only O(

`ℏ`;

) terms and so is retained as semiclassical

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;

2

/ d \ / d \ 1 d

pdev := |--- v(x, t)| + u(x, t) |--- v(x, t)| - - ---- u(x, t)

\ dt / \ dx / 2 2

dx

/ d \

+ v(x, t) |--- u(x, t)| = 0.1

\ dx /

By inspection of the derivatives in above equations we now set up the ICs and BCs for

u(x, t);

and

v(x,t).;

The quantum initial and boundary conditions are similar to the classical case, but also comprise additional boundary condition terms for

v;

and for

u;

, notably a 1st derivative BC term for

u;

.

IBCu := {u(x,0) = 0.1*sin(2*Pi*x),u(0,t) = 0.5-0.5*cos(2*Pi*t),D[1](u)(0,t) = 2*Pi*0.1*cos(2*Pi*t)};# IBC for u

IBCu := {u(0, t) = 0.5 - 0.5 cos(2 Pi t),

u(x, 0) = 0.1 sin(2 Pi x),

D[1](u)(0, t) = 0.6283185308 cos(2 Pi t)}

IBCv := {v(x,0) = 0.2*sin((1/2)*Pi*x),v(0,t)=0.2-0.2*cos(2*Pi*t)};# IBC for v

/

IBCv := { v(0, t) = 0.2 - 0.2 cos(2 Pi t),

\

/1 \\

v(x, 0) = 0.2 sin|- Pi x| }

\2 //

IBC := IBCu union IBCv;

/

IBC := { u(0, t) = 0.5 - 0.5 cos(2 Pi t),

\

u(x, 0) = 0.1 sin(2 Pi x), v(0, t) = 0.2 - 0.2 cos(2 Pi t),

/1 \

v(x, 0) = 0.2 sin|- Pi x|,

\2 /

\

D[1](u)(0, t) = 0.6283185308 cos(2 Pi t) }

/

pds:=pdsolve({pdeu,pdev},IBC, time = t, range = 0..0.2,numeric);# 'numeric' solution

pds := _m2606922675232

The following quantum animation is in contrast with the classical case, and illustrates the delocalisation of the wave form caused by the quantum diffusion and advection terms:

T:=2; p1:=pds:-animate({[u, color = green, linestyle = dash], [v, color = red, linestyle = dash]},t = 0..T, gridlines = true, numpoints = 2000,x = 0..0.2):p1;

T := 2

Error, (in pdsolve/numeric/animate) unable to compute solution for t>HFloat(0.0):

matrix is singular

p1