## 79 Reputation

13 years, 185 days

## Graphical projection question: Projectin...

Can someone help me project ODE solution curves of an ODE onto velocity manifolds (u(x,t), v(x,t))?

Thanks for any help..

Melvin

Help needed to graphically map (x(t), t) trajectories onto velocity manifolds (u, v)

MRB: 16/6/19

The velocity manifolds u(x,t), v(x,t) are depicted in the figure below:

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The above velocity plot above shows classical pairs of negative (red) low and high velocity, and pairs of positive (blue) low and high velocity.  At any given instant there are fast and slow propagating solutions, from which classical trajectories on (x,t) are derived below for each of the red and blue velocity pulse solutions.  The cut away regions of the velocity manifolds represent complex  and

We now solve the velocity ODE to plot a (x,t) trajectory on the red velocity manifold:

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 (1)

If the set of dependent variables is specified as a list, those variables appear in the same order in the output.  Now extract the solution points at given times

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 (2)

Here is a solution at a specified time:

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 (3)

Here is the the solution for the (x,t) trajectories:

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 (4)

We now plot trajectories on (x,t) space:

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We do not generate trajectories in the region  for which regions of the manifolds are no longer real-valued.  The above trajectories are solutions of the above coupled Burger's equations for (u(x,t), v(x,t)). We wish plot these trajectories on their  [t,x(t),D(x(t)] manifolds.

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The following is an example of how to project curves onto a surface, assuming an analytic form for the curves.  However, we wish to project the above two ODE solution trajectory curves onto the surfaces u(x,t) and v(x,t) which are solutions of two PDEs.

Below is an example of projecting a specified curve onto a surface  Our requirement is to project a curve which is an integral solution of the ODE (4) above onto one of the velocity manifolds u(x,y), v(x,y).  Below is an example of an algebraically specified trajectory being projected onto a manifold (surface):

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 (5)
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QUESTION: We wish instead to project the ODE  (rather than algebraic) solution curves onto the surface.   Please someone provide MAPLE code to do that?

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## Can you help solution of the following P...

Maple

I am unable to replicate solution of problem 169 below.  Can you help me debug?

Melvin Brown

Schrdinger PDE on (x,y,t) with initial and boundary conditions. Zero potential: problem number 169

Here is the problem 169 specification and solution from the Nasser list:

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 > x:='x';t:='t';y:='y';hbar:='hbar';
 (1.1)
 > interface(showassumed=0);
 (1.2)
 > pde:=I*diff(f(x,y,t),t)=-hbar^2*(diff(f(x,y,t),x\$2)+diff(f(x,y,t),y\$2))/(2*m);
 (1.3)
 > ic:=f(x,y,0)=sqrt(2)*(sin(2*Pi*x)*sin(Pi*y)+sin(Pi*x)*sin(2*Pi*y));
 (1.4)
 > bc:=f(0,y,t)=0,f(1,y,t)=0,f(x,1,t)=0,f(x,0,t)=0;
 (1.5)
 > cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic,bc],f(x,y,t))),output='realtime'));
 memory used=159.58MiB, alloc change=102.00MiB, cpu time=4.37s, real time=4.23s, gc time=452.40ms
 (1.6)
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 (1.7)
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The problem seems to fail... Why?

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## How do I remove this message? ... Gener...

Maple

I get the following when MAPLE starts...

Warning, .hdb help databases are deprecated, 'C:\Program Files\Maple 2018\lib\OrthogonalExpansions.hdb' will not be used, see ?HelpTools,Migrate help page for more information

I have recently loaded an orthogonal expansions package created for earlier versions of MAPLE.

How can I remove the message?

MRB

## MAPLE and multidimensional Fourier Trans...

Maple

Does MAPLE have capabability to do multidimensional FTs i.e. (x,y)->(u,v)? If not, are there any links to MAPLE packages which meet this requirement that can be recommended?

Melvin Brown

## Can MAPLE 18 solve coupled nonlinear PDE...

Maple 2018

Dear Support

I am attempting to model quantum dynamics, and have defined a coupled set of nonlinear PDEs I would like to solve for coupled solutions u(x,y,t) and v(x,y,t) using MAPLE 18.

I attach an image of part of the worksheet the pair of PDEs...The initial conditions u(x,y,0), v(x,y,0) are a pair of respectively positive and negative 2D gaussians on the x,y, domain.

Before I go any further, please would you check that MAPLE 18 is in principle capable of finding solutions u(x,y,t), v(x,y,t) solutions, and let me know whether it is worth pursuing the solution?  I have had a look at the MAPLE documentation, but am not sure whether MAPLE can solve this system.

As a warm-up, I successfully solved a 1-D system u(x,t), v(x,t) using pdsolve[numeric], but I am not clear whether MAPLE 18 can solve for u(x,y,t), v(x,y,t) either numerically or analytically on the [x,y,t] domain.

I hope you can provide help/guidance. An image the equations in MAPLE is displayed here...

Melvin Brown

UK

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