# Question:PDE problem... can't get solution ... probably an IBC fault ...

## Question:PDE problem... can't get solution ... probably an IBC fault ...

Maple

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

Two 1-D coupled Burgers equations - semiclassical case: remove O( ) terms for u(x,t) but retain O( ) terms 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.

 > #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( ) 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;
 (1)

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

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

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 BC term for .

 > 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
 (3)
 > 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
 (4)
 > IBC := IBCu union IBCv;
 (5)
 > pds:=pdsolve({pdeu,pdev},IBC, time = t, range = 0..0.2,numeric);# 'numeric' solution
 (6)

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;
 (7)

Note that this plot also shows that there are regions in which ,  .  Below, the 3D plot of u(x,t),v(x,t) also illustrates the quantum delocalisation of features:

 > T := 3; p1 := pds:-plot3d({[u, shading = zhue], [v, color = red]}, t = 0 .. T, x = -0.1e-2 .. 2,transparency = 0.0, orientation = [-146, 54, 0], title = print("Coupled quantum solution \n u(x, t) zhue, v(x,t) red", numeric),scaling=unconstrained):p1;
 >

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(
`&hbar;`^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(
`&hbar;`;
) 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

﻿