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How to calculate this?...

Ahmed111 130 Maple 18
norm simplification
August 11 2020
1 2

u=a-(4*i*b)*(a*sinh(c)-2*i*b)/(a+2*i*b*sinh(c)), where b is a complex parameter, a is a real constant and, c=\sqrt(a^2-4*b^2)*(x+t/(2*b^2)), then how to calculate |u|^{2} on maple?

Why Maple is not plotting the solution?...

Ahmed111 130 Maple 18
substitution plot3d rhs
May 29 2020
2 0


 

> 

restart;
> 

with(PDEtools):
> 

with(plot):

Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received plot

  
> 

A1:=Matrix([[phi,(chi),conjugate(phi),conjugate(chi)],
          [chi,(phi),conjugate(chi),conjugate(phi)],
          [lambda*phi,-(lambda)*(chi),
           conjugate(lambda)*conjugate(phi),-conjugate(lambda)*conjugate(chi)],
          [lambda*chi,-(lambda)*(phi),
           conjugate(lambda)*conjugate(chi),-conjugate(lambda)*conjugate(phi)]]);

A1 := Matrix(4, 4, {(1, 1) = phi, (1, 2) = chi, (1, 3) = conjugate(phi), (1, 4) = conjugate(chi), (2, 1) = chi, (2, 2) = phi, (2, 3) = conjugate(chi), (2, 4) = conjugate(phi), (3, 1) = lambda*phi, (3, 2) = -lambda*chi, (3, 3) = conjugate(lambda)*conjugate(phi), (3, 4) = -conjugate(lambda)*conjugate(chi), (4, 1) = lambda*chi, (4, 2) = -lambda*phi, (4, 3) = conjugate(lambda)*conjugate(chi), (4, 4) = -conjugate(lambda)*conjugate(phi)})

 (1)
> 

d1 := LinearAlgebra:-Determinant(A1):
> 

d1; length(%);

conjugate(lambda)^2*conjugate(phi)^2*chi^2-conjugate(lambda)^2*conjugate(phi)^2*phi^2-conjugate(lambda)^2*conjugate(chi)^2*chi^2+conjugate(lambda)^2*conjugate(chi)^2*phi^2+2*conjugate(lambda)*conjugate(phi)^2*chi^2*lambda+2*conjugate(lambda)*conjugate(phi)^2*lambda*phi^2-8*conjugate(lambda)*conjugate(phi)*conjugate(chi)*chi*lambda*phi+2*conjugate(lambda)*conjugate(chi)^2*chi^2*lambda+2*conjugate(lambda)*conjugate(chi)^2*lambda*phi^2+conjugate(phi)^2*chi^2*lambda^2-conjugate(phi)^2*lambda^2*phi^2-conjugate(chi)^2*chi^2*lambda^2+conjugate(chi)^2*lambda^2*phi^2

  

705

 (2)
> 

den:=simplify(d1,size); length(%);

-(-(conjugate(chi)-conjugate(phi))*(chi+phi)*conjugate(lambda)+lambda*(conjugate(chi)+conjugate(phi))*(chi-phi))*(-(conjugate(chi)+conjugate(phi))*(chi-phi)*conjugate(lambda)+lambda*(conjugate(chi)-conjugate(phi))*(chi+phi))

  

333

 (3)
> 

 

> 

con1:=phi=exp(I*lambda*(x-t/(4*lambda^2)-w^2)):con2:=chi=exp(-I*lambda*(x-t/(4*lambda^2)-w^2)):
> 

 

> 

den1:=simplify(dsubs({con1,con2},den));

4*conjugate(lambda)^2*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2-4*conjugate(lambda)^2*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2+8*conjugate(lambda)*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2*lambda+8*conjugate(lambda)*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2*lambda+4*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2*lambda^2-4*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2*lambda^2-16*conjugate(lambda)*lambda

 (4)
> 

plot3d(subs(Re(lambda)=1, Im(lambda)=.2, w=1, rhs(den1)),x=-6..6, t=-6..6)

Warning, inserted missing semicolon at end of statement

  

Error, invalid input: rhs received 4*conjugate(lambda)^2*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2-4*conjugate(lambda)^2*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2+8*conjugate(lambda)*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2*lambda+8*conjugate(lambda)*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2*lambda+4*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*...

  
> 

NULL
> 

``

 

``


 

Download 23May(1).mw

How to common a factor appears in many t...

Ahmed111 130 Maple 18
collect simplification
May 21 2020
1 0

> den := -(-(conjugate(chi)-conjugate(phi))*(chi+phi)*conjugate(lambda)+lambda*(conjugate(chi)+conjugate(phi))*(chi-phi))*(-(conjugate(chi)+conjugate(phi))*(chi-phi)*conjugate(lambda)+lambda*(conjugate(chi)-conjugate(phi))*(chi+phi));

> phi:=exp(I*lambda*(x-t/(4*lambda^2)-w^2)):chi:=exp(-I*lambda*(x-t/(4*lambda^2)-w^2)):

> den1:=simplify(dsubs({phi,chi},den));

> dsubs({exp((1/4*I)*(4*lambda^2*w^2-4*lambda^2*x+t)/lambda), exp(-(1/4*I)*(4*lambda^2*w^2-4*lambda^2*x+t)/lambda)}, 4*conjugate(lambda)^2*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2-4*conjugate(lambda)^2*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2+8*abs(lambda)^2*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2+8*abs(lambda)^2*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2+4*cos((1/4)*(4*w^2*lambda^2-4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda+t)/lambda)^2*lambda^2-4*cos((1/4)*(-4*w^2*lambda^2+4*x*lambda^2+conjugate((4*lambda^2*w^2-4*lambda^2*x+t)/lambda)*lambda-t)/lambda)^2*lambda^2-16*abs(lambda)^2)

 

Since "cos(...)" appears in every term in last equation (except a last one), how to common it? 

How to verify the solution of PDEs?...

Ahmed111 130 Maple 18
pde pdsolve differential-equations
May 21 2020
1 0


 

``

> 

restart:
> 

with(PDEtools):
> 

with(LinearAlgebra):
> 

 

> 

alias(f=f(x,t),g=g(x,t));

f, g

 (1)
> 

 

> 

 

> 

eq1:=diff(f,x)=-I*eta*f +I*exp(-I*t)*g;

diff(f, x) = -I*eta*f+I*exp(-I*t)*g

 (2)
> 

eq2:=diff(g,x)=-I*eta*g +I*exp(I*t)*f;

diff(g, x) = -I*eta*g+I*exp(I*t)*f

 (3)
> 

eq3:=diff(f,t)=(I*eta^2-I/2)*f +I*eta*exp(-I*t)*g;

diff(f, t) = (I*eta^2-(1/2)*I)*f+I*eta*exp(-I*t)*g

 (4)
> 

eq4:=diff(g,t)=(-I*eta^2+I/2)*g +I*eta*exp(I*t)*f;

diff(g, t) = (-I*eta^2+(1/2)*I)*g+I*eta*exp(I*t)*f

 (5)
> 

#### The solution of (2)-(5) is
> 

eq5:=f=I*(c1*exp(A)-c2*exp(-A))*exp(-i*t/2);

f = I*(exp(A)*c1-c2*exp(-A))*exp(-(1/2)*i*t)

 (6)
> 

eq6:=g=(c2*exp(A)-c1*exp(-A))*exp(i*t/2);

g = (c2*exp(A)-c1*exp(-A))*exp((1/2)*i*t)

 (7)
> 

#### where
> 

c1=sqrt(h-sqrt(h^2-1))/sqrt(h^2-1);c2=sqrt(h+sqrt(h^2-1))/sqrt(h^2-1);A=sqrt(h^2-1)*(x+I*h*t);

c1 = (h-(h^2-1)^(1/2))^(1/2)/(h^2-1)^(1/2)

  

c2 = (h+(h^2-1)^(1/2))^(1/2)/(h^2-1)^(1/2)

  

A = (h^2-1)^(1/2)*(x+I*h*t)

 (8)
> 

#### How to verify (6) and (7) is the solution of (2)-(5)?

``


 

Download verification.mw

How can we remove c1 from the expression...

Ahmed111 130 Maple
solve
April 27 2020
1 0

(-(2*I)*c1*eta+c1*sqrt(-e0^2-4*eta^2))/e0 = c2, ((2*I)*c2*eta+c2*sqrt(-e0^2-4*eta^2))/e0 = -c1

How can we remove c1 from the expression of c2 and similarly c2 from c1?

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