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Question: Simplifying nonlinear terms differential equations

Question: Simplifying nonlinear terms differential equations

salim-barzani 1395 Maple 2024
simplification differential-equations duplicate-question
May 06 2025
3

How can we eliminate nonlinear terms involving two functions in a differential equation?

> 

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(G(xi)); declare(Q(x, t)); declare(Q1(x, t))

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

  

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

  

Q1(x, t)*`will now be displayed as`*Q1

 (2)
> 

NULL
> 

q := (sqrt(P)+Q(x, t))*exp(I*gamma*P*t); B := (sqrt(P)+Q(x, t))*exp(-I*gamma*P*t); B1 := sqrt(P)+Q(x, t); P+sqrt(P)*(Q1(x, t)+Q(x, t))
> 

GeF := I*(diff(q, t))+alpha[1]*(diff(q, x, x))+alpha[2]*(P+sqrt(P)*(Q1(x, t)+Q(x, t)))*q+alpha[3]*(P+sqrt(P)*(Q1(x, t)+Q(x, t)))^2*q+alpha[4]*(P+sqrt(P)*(Q1(x, t)+Q(x, t)))^3*q+alpha[5]*(diff(P+sqrt(P)*(Q1(x, t)+Q(x, t)), x, x))*q = 0
> 

K := simplify(GeF*exp(-I*gamma*P*t))

(Q(x, t)^4*alpha[4]+3*Q(x, t)^3*Q1(x, t)*alpha[4]+3*(Q1(x, t)^2*alpha[4]+alpha[3])*Q(x, t)^2+Q1(x, t)*(Q1(x, t)^2*alpha[4]+4*alpha[3])*Q(x, t)+Q1(x, t)^2*alpha[3]+alpha[2]-gamma)*P^(3/2)+(6*Q(x, t)^2*alpha[4]+9*Q(x, t)*Q1(x, t)*alpha[4]+3*Q1(x, t)^2*alpha[4]+alpha[3])*P^(5/2)+(P^(1/2)*Q(x, t)*alpha[5]+P*alpha[5]+alpha[1])*(diff(diff(Q(x, t), x), x))+alpha[5]*(P^(1/2)*Q(x, t)+P)*(diff(diff(Q1(x, t), x), x))+P^(7/2)*alpha[4]+I*(diff(Q(x, t), t))+alpha[2]*(Q1(x, t)*Q(x, t)+Q(x, t)^2)*P^(1/2)+(4*P^2*alpha[4]+P*alpha[3])*Q(x, t)^3+(9*P^2*alpha[4]+2*P*alpha[3])*Q1(x, t)*Q(x, t)^2+((6*P^2*alpha[4]+P*alpha[3])*Q1(x, t)^2+4*alpha[4]*P^3+3*alpha[3]*P^2+(2*alpha[2]-gamma)*P)*Q(x, t)+P^2*Q1(x, t)^3*alpha[4]+(3*P^3*alpha[4]+2*P^2*alpha[3]+P*alpha[2])*Q1(x, t) = 0

 (3)
> 

remove(has, K, {Q(x, t)^2, Q(x, t)^3, Q(x, t)^4, Q1(x, t)^2, Q1(x, t)^3, Q1(x, t)^4})

() = 0

 (4)
> 

NULL
> 

NULL
> 

AA := (alpha[2]-gamma)*P^(3/2)+alpha[3]*P^(5/2)+(P*alpha[5]+alpha[1])*(diff(Q(x, t), x, x))+alpha[5]*P*(diff(Q1(x, t), x, x))+P^(7/2)*alpha[4]+I*(diff(Q(x, t), t))+(4*alpha[4]*P^3+3*alpha[3]*P^2+(2*alpha[2]-gamma)*P)*Q(x, t)+(3*P^3*alpha[4]+2*P^2*alpha[3]+P*alpha[2])*Q1(x, t) = 0

(alpha[2]-gamma)*P^(3/2)+P^(5/2)*alpha[3]+(P*alpha[5]+alpha[1])*(diff(diff(Q(x, t), x), x))+P*(diff(diff(Q1(x, t), x), x))*alpha[5]+P^(7/2)*alpha[4]+I*(diff(Q(x, t), t))+(4*alpha[4]*P^3+3*alpha[3]*P^2+(2*alpha[2]-gamma)*P)*Q(x, t)+(3*P^3*alpha[4]+2*P^2*alpha[3]+P*alpha[2])*Q1(x, t) = 0

 (5)
 
> 

test := (alpha[2]-gamma)*P^(3/2)+alpha[3]*P^(5/2)+(P*alpha[5]+alpha[1])*0+P*alpha[5]*0+P^(7/2)*alpha[4]+I*0 = 0

(alpha[2]-gamma)*P^(3/2)+P^(5/2)*alpha[3]+P^(7/2)*alpha[4] = 0

 (6)
 

NULL

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