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Question: Techniques for Simplifying Differential Equations

Question:Techniques for Simplifying Differential Equations

salim-barzani 185 Maple 2024
pde differential-equations
December 11 2024
0

I'm trying to transform a partial differential equation (PDE) into an ordinary differential equation (ODE) as demonstrated in the paper. However, I find some steps confusing and difficult to follow. The process often feels chaotic, and managing the complexity of the equations is overwhelming. Could you suggest an effective and systematic method to handle such transformations more easily?

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

 (1)
> 

declare(Omega(x, t)); declare(U(xi))

Omega(x, t)*`will now be displayed as`*Omega

  

U(xi)*`will now be displayed as`*U

 (2)
> 

tr := {t = tau, x = tau*c[0]+xi, Omega(x, t) = U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))}

{t = tau, x = tau*c[0]+xi, Omega(x, t) = U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))}

 (3)
> 

P1 := diff(Omega(x, t)^m, t)

Omega(x, t)^m*m*(diff(Omega(x, t), t))/Omega(x, t)

 (4)
> 

L1 := PDEtools:-dchange(tr, P1, [xi, tau, U])

(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*(-((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*c[0]+I*U(xi)*(-k*c[0]+w+delta*(diff(W(tau), tau))-delta^2)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))

 (5)
 
> 

pde1 := I*(diff(Omega(x, t)^m, t))+alpha*(diff(Omega(x, t)^m, `$`(x, 2)))+I*beta*(diff(abs(Omega(x, t))^(2*n)*Omega(x, t)^m, x))+m*sigma*Omega(x, t)^m*(diff(W(t), t)) = I*gamma*abs(Omega(x, t))^(2*n)*(diff(Omega(x, t)^m, x))+delta*abs(Omega(x, t))^(4*n)*Omega(x, t)^m

I*Omega(x, t)^m*m*(diff(Omega(x, t), t))/Omega(x, t)+alpha*(Omega(x, t)^m*m^2*(diff(Omega(x, t), x))^2/Omega(x, t)^2+Omega(x, t)^m*m*(diff(diff(Omega(x, t), x), x))/Omega(x, t)-Omega(x, t)^m*m*(diff(Omega(x, t), x))^2/Omega(x, t)^2)+I*beta*(2*abs(Omega(x, t))^(2*n)*n*(diff(Omega(x, t), x))*abs(1, Omega(x, t))*Omega(x, t)^m/abs(Omega(x, t))+abs(Omega(x, t))^(2*n)*Omega(x, t)^m*m*(diff(Omega(x, t), x))/Omega(x, t))+m*sigma*Omega(x, t)^m*(diff(W(t), t)) = I*gamma*abs(Omega(x, t))^(2*n)*Omega(x, t)^m*m*(diff(Omega(x, t), x))/Omega(x, t)+delta*abs(Omega(x, t))^(4*n)*Omega(x, t)^m

 (6)
> 

NULL
> 

L1 := PDEtools:-dchange(tr, pde1, [xi, tau, U])

I*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*(-((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*c[0]+I*U(xi)*(-k*c[0]+w+delta*(diff(W(tau), tau))-delta^2)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+alpha*((U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m^2*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2/(U(xi)^2*(exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2)+(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(diff(U(xi), xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-(2*I)*(diff(U(xi), xi))*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-U(xi)*k^2*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))-(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2/(U(xi)^2*(exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^2))+I*beta*(2*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*n*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*abs(1, U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m/(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))))+m*sigma*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*(diff(W(tau), tau)) = I*gamma*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(2*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m*m*((diff(U(xi), xi))*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau))-I*U(xi)*k*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))/(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))+delta*(abs(U(xi))*exp(-Im(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^(4*n)*(U(xi)*exp(I*(-k*(tau*c[0]+xi)+w*tau+delta*W(tau)-delta^2*tau)))^m

 (7)
> 

``

``

 (8)

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