It works fine. I just cannot understand why did you use the "op" for initial boundary conditions? what does this "op" do in Maple?

I am grateful of you, Jakubi.

Saeid

Thank you very much Jakubi. Your remarks are so helpful.

There is also another question: Would you please let me know how to plot the probability function p=\psi^*\psi= (\psi1)^2-(\psi2)^2 ? because it is required to compare the numeric "p" with the WKB approximation.

regards,

Saeid

Thanks, let's do it.

Would you please let me know how to apply the initial and boundary conditions to a system of pdes? I think I have to impose the real part of the initial conditions for the real pde and imaginary part of initial conditions for the imaginary pde. I mean that I have to apply 4 initial/boundary conditions for each of the pdes in the system. Is it correct?

Thank you Jakubi, if one put t=.3361035810 in Omega[1] and Omega[2] and then execute the program, the first Error still exists.

Thank you for the information, but it seems that the "pdsolve" command is also missed in the previous post, therefore, the full version of the input code is as what follows:

b := 0.1e-2;
lambda := 100;
beta := (1/2)*sqrt(lambda^2-exp(4*alpha))+(1/4)*Pi;
Omega[1] := -t+(1/2)*arccosh(lambda*exp(-2*alpha));
Omega[2] := t+(1/2)*arccosh(lambda*exp(-2*alpha));

PDE := diff(psi(alpha, phi), [`$`(alpha, 2)])-(diff(psi(alpha, phi), 
[`$`(phi, 2)]))-exp(4*alpha)*psi(alpha, phi) = 0;

IBC := {psi(-100, phi) = 0, psi(100, phi) = 0, 
psi(alpha, .3361035810) = sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)
*(exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))
+exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)
*(lambda^2-exp(4*alpha))^(1/4)), (D[2](psi))(alpha, .3361035810) 
= sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)*((b^2*Omega[1]-I*lambda)
*exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))+(-b^2*Omega[2]-I*lambda)
*exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)
*(lambda^2-exp(4*alpha))^(1/4))};

pds := pdsolve(PDE, IBC, numeric;
module () local INFO; export plot, plot3d, animate, value, settings; 
option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; 
end module;

pds:-plot3d([LinearAlgebra:-HermitianTranspose(psi)*psi], phi = 5);
%;
Error, (in pdsolve/numeric/plot3d) unable to compute solution for phi>.336103580999999984:
solution becomes undefined, problem may be ill posed or method may be ill suited to solution

I would be pleased, if any one can help.

best

 > b := 1;
> print(`output redirected...`); # input placeholder
1
> lambda := 1000;
> print(`output redirected...`); # input placeholder
1000
> beta := (1/2)*sqrt(lambda^2-exp(4*alpha))+(1/4)*Pi;
> print(`output redirected...`); # input placeholder
1 (1/2) 1
- (1000000 - exp(4 alpha)) + - Pi
2 4
> Omega[1] := -.3361035810+(1/2)*arccosh(lambda*exp(-2*alpha));
> print(`output redirected...`); # input placeholder
1
-0.3361035810 + - arccosh(1000 exp(-2 alpha))
2
> Omega[2] := .3361035810+(1/2)*arccosh(lambda*exp(-2*alpha));
> print(`output redirected...`); # input placeholder
1
0.3361035810 + - arccosh(1000 exp(-2 alpha))
2
> PDE := diff(psi(alpha, phi), [`$`(alpha, 2)])-(diff(psi(alpha, phi), [`$`(phi, 2)]))-exp(4*alpha)*psi(alpha, phi) = 0;
> print(`output redirected...`); # input placeholder
/ d / d \\ / d / d \\
|------- |------- psi(alpha, phi)|| - |----- |----- psi(alpha, phi)||
\ dalpha \ dalpha // \ dphi \ dphi //

- exp(4 alpha) psi(alpha, phi) = 0
> IBC := {psi(-100, phi) = 0, psi(100, phi) = 0, psi(alpha, .3361035810) = sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)*(exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))+exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)*(lambda^2-exp(4*alpha))^(1/4)), (D[2](psi))(alpha, .3361035810) = sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)*((b^2*Omega[1]-I*lambda)*exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))+(-b^2*Omega[2]-I*lambda)*exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)*(lambda^2-exp(4*alpha))^(1/4))};
> print(`output redirected...`); # input placeholder
/
|
< psi(-100, phi) = 0, psi(100, phi) = 0, psi(alpha, 0.3361035810) =
|
\

/ / /
1 | (1/2) (3/4) | |
------------------------------- |2 Pi exp(-250 Pi) |exp|
(1/4) \ \ \
4 (1000000 - exp(4 alpha))
2
1 / 1 \ /
- - |-0.3361035810 + - arccosh(1000 exp(-2 alpha))| + I |-336.1035810
2 \ 2 / \

\
1 (1/2) 1 \|
+ 500 arccosh(1000 exp(-2 alpha)) - - (1000000 - exp(4 alpha)) - - Pi|| +
2 4 //

/ 2
| 1 / 1 \ /
exp|- - |0.3361035810 + - arccosh(1000 exp(-2 alpha))| - I |336.1035810
\ 2 \ 2 / \

\
1 (1/2) 1 \|
+ 500 arccosh(1000 exp(-2 alpha)) - - (1000000 - exp(4 alpha)) - - Pi||
2 4 //

\\ /
|| 1 | (1/2)
||, D[2](psi)(alpha, 0.3361035810) = ------------------------------- |2
// (1/4) \
4 (1000000 - exp(4 alpha))

/
(3/4) |/
Pi exp(-250 Pi) ||(-0.3361035810 - 1000. I)
\\

/
1 \ |
+ - arccosh(1000 exp(-2 alpha))| exp|
2 / \
2
1 / 1 \ /
- - |-0.3361035810 + - arccosh(1000 exp(-2 alpha))| + I |-336.1035810
2 \ 2 / \

\
1 (1/2) 1 \|
+ 500 arccosh(1000 exp(-2 alpha)) - - (1000000 - exp(4 alpha)) - - Pi|| +
2 4 //

/
/ 1 \ |
|(-0.3361035810 - 1000. I) - - arccosh(1000 exp(-2 alpha))| exp|
\ 2 / \
2
1 / 1 \ /
- - |0.3361035810 + - arccosh(1000 exp(-2 alpha))| - I |336.1035810
2 \ 2 / \

\
1 (1/2) 1 \|
+ 500 arccosh(1000 exp(-2 alpha)) - - (1000000 - exp(4 alpha)) - - Pi||
2 4 //

\\\
|||
|| >
//|
/
> pds := pdsolve(PDE, IBC, numeric);
> print(`output redirected...`); # input placeholder
module () local INFO; export plot, plot3d, animate, value, settings; option

`Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; end module
> pds:-plot3d([LinearAlgebra:-HermitianTranspose(psi)*psi], phi = 5);
> %;
Error, (in pdsolve/numeric/plot3d) unable to compute solution for phi>.336103580999999984:
solution becomes undefined, problem may be ill posed or method may be ill suited to solution

In this code I used a Dirichlet boundary conditions for the spatial independent varialbe and complex Cauchy boundary conditions for the time independent variable. I myself guess that the problem is with the comlex Cauchy BC, but it is inevitable, because the time (\phi)  BC used here, is the WKB approximation of the solution of the same PDE and my goal is to compare the exact numerical solution with the WKB approximation and see how good is the WKB solution.

So, this error means that the initial/boundary conditions are incomplete. am I right?

thanks

Saeid

Ok, thanks jakubi, please find more details below: (The Maple worksheet pasted as a plain text format)

> b := 0.1e-2;
> lambda := 100;
> beta := (1/2)*sqrt(lambda^2-exp(4*alpha))+(1/4)*Pi;
> Omega[1] := -t+(1/2)*arccosh(lambda*exp(-2*alpha));
> Omega[2] := t+(1/2)*arccosh(lambda*exp(-2*alpha));
 

> PDE := diff(psi(alpha, phi), [`$`(alpha, 2)])-(diff(psi(alpha, phi), [`$`(phi, 2)]))-exp(4*alpha)*psi(alpha, phi) = 0;

> IBC := {psi(-100, phi) = 0, psi(100, phi) = 0, psi(alpha, .3361035810) = sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)*(exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))+exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)*(lambda^2-exp(4*alpha))^(1/4)), (D[2](psi))(alpha, .3361035810) = sqrt((1/2)*Pi)*b*Pi^(1/4)*exp(-(1/4)*lambda*Pi)*((b^2*Omega[1]-I*lambda)*exp(-(1/2)*b^2*Omega[1]^2+I*(lambda*Omega[1]-beta))+(-b^2*Omega[2]-I*lambda)*exp(-(1/2)*b^2*Omega[2]^2-I*(lambda*Omega[2]-beta)))/(2*sqrt(b)*(lambda^2-exp(4*alpha))^(1/4))};
 

module () local INFO; export plot, plot3d, animate, value, settings; option

`Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; end module
 

> pds:-plot([LinearAlgebra:-HermitianTranspose(psi)*psi], phi = 5);
%;
Error, (in pdsolve/numeric/plot) unable to compute solution for phi>.336103580999999984:
solution becomes undefined, problem may be ill posed or method may be ill suited to solution

Thank you all in advance,

Saeid