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Question: How to verify the solution of PDEs?

Question:How to verify the solution of PDEs?

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


 

``

> 

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)?

``


 

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