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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?

How to plot Bloch sphere on maple?...

Ahmed111 130 Maple 18
differential-equations plotting
April 14 2020
0 0

If we have an equation for the generalized Bloch sphere i.e.,

\partial_{t}(u^{2} + v^{2} + w^{2}) = 0,

where u, v and w are functions of x and t and the initial conditions u=v=0, w=-1. Then how to plot this equation on maple?

Is there any way to write the expression...

Ahmed111 130 Maple
simplification
April 09 2020
1 0

Is there any way to write the expression into the ratio of simple determinants? i..e,

 

How to simplify the product of mix combi...

Ahmed111 130 Maple
simplify simplification
March 30 2020
1 2

h1=-i(c1*exp(A)-c2*exp(-A));

h2=c2*exp(A)+c1*exp(-A);

A=(i*a/2)*(x+b*t);

c1=sqrt(a+2*lambda)/a;

c2=sqrt(a-2*lambda)/a;

How to simplify the product of h1 and h2 in terms of trignometric functions?

h1*h2=?

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