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salim-barzani

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there is any way to use explicate for re...

salim-barzani 1560 Maple 2024
pde explicit differential-equations
February 28 2025
1 1

I want to remove the Lambert function (LambertW) from my equation, but I don't know how. I tried using the explicit option, but it didn't work. How can I express the equation without LambertW?

> 

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(u(x, y, z, t))

u(x, y, z, t)*`will now be displayed as`*u

 (2)
> 

declare(f(x, y, z, t))

f(x, y, z, t)*`will now be displayed as`*f

 (3)
> 

pde := diff(diff(u(x, y, z, t), t)+6*u(x, y, z, t)*(diff(u(x, y, z, t), x))+diff(u(x, y, z, t), `$`(x, 3)), x)-lambda*(diff(u(x, y, z, t), `$`(y, 2)))+diff(alpha*(diff(u(x, y, z, t), x))+beta*(diff(u(x, y, z, t), y))+gamma*(diff(u(x, y, z, t), z)), x)

diff(diff(u(x, y, z, t), t), x)+6*(diff(u(x, y, z, t), x))^2+6*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x))+diff(diff(diff(diff(u(x, y, z, t), x), x), x), x)-lambda*(diff(diff(u(x, y, z, t), y), y))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+gamma*(diff(diff(u(x, y, z, t), x), z))

 (4)
> 

pde_nonlinear, pde_linear := selectremove(proc (term) options operator, arrow; not has((eval(term, u(x, y, t) = a*u(x, y, t)))/a, a) end proc, expand(pde))

0, diff(diff(u(x, y, z, t), t), x)+6*(diff(u(x, y, z, t), x))^2+6*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x))+diff(diff(diff(diff(u(x, y, z, t), x), x), x), x)-lambda*(diff(diff(u(x, y, z, t), y), y))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+gamma*(diff(diff(u(x, y, z, t), x), z))

 (5)
> 

thetai := t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i]; eval(pde_linear, u(x, y, z, t) = exp(thetai)); eq15 := isolate(%, w[i])

t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i]

  

w[i]*k[i]*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])+12*k[i]^2*(exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i]))^2+k[i]^4*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])-lambda*l[i]^2*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])+alpha*k[i]^2*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])+beta*k[i]*l[i]*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])+gamma*k[i]*r[i]*exp(t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i])

  

w[i] = -(t*k[i]^4+gamma*t*k[i]*r[i]+alpha*t*k[i]^2+beta*t*k[i]*l[i]-lambda*t*l[i]^2+LambertW(12*t*k[i]*exp(-(t*k[i]^4+alpha*t*k[i]^2+beta*t*k[i]*l[i]+gamma*t*k[i]*r[i]-lambda*t*l[i]^2-x*k[i]^2-y*k[i]*l[i]-z*k[i]*r[i]-eta[i]*k[i])/k[i]))*k[i])/(t*k[i])

 (6)
> 

sol := solve(eq15, w[i], explicit)

-(t*k[i]^4+gamma*t*k[i]*r[i]+alpha*t*k[i]^2+beta*t*k[i]*l[i]-lambda*t*l[i]^2+LambertW(12*t*k[i]*exp(-(t*k[i]^4+alpha*t*k[i]^2+beta*t*k[i]*l[i]+gamma*t*k[i]*r[i]-lambda*t*l[i]^2-x*k[i]^2-y*k[i]*l[i]-z*k[i]*r[i]-eta[i]*k[i])/k[i]))*k[i])/(t*k[i])

 (7)
 

NULL

Download remove.mw

How define the PDE which contain integr...

salim-barzani 1560 Maple 2024
pde differential-equations duplicate-question
February 21 2025
0 10

i try find some part of solution of this kind of pde but i can't get results my openion is maybe this pde is wronge when i defined 

pde.mw

Moderator- make different from the ques...

salim-barzani 1560 MaplePrimes
February 21 2025
0 1

 Why you delete my question  most of my equation are same but demand of the questions are different, i can make a 100 account and each time i asked by one of them, and right now most of my question taged like they dublicated but they are don't ، i am not jobles and sick to post a dublicate question, You started a riot.

How check this pde ?...

salim-barzani 1560 Maple 2024
pde differential-equations pdetest
February 20 2025
1 1

my solution is a little bit long which when i click on pdetest command i wait at least more than a hour but still is runing and i don't get any result and not give me error , which i don't know my result is true or not so How i can find that my pde by this solution it will be zero or not ?

> 

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(u(x, y, t))

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

 (2)
> 

declare(f(x, y, t))

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

 (3)
> 

pde := diff(u(x, y, t), t)-(diff(diff(u(x, y, t), `$`(x, 4))+5*u(x, y, t)*(diff(u(x, y, t), `$`(x, 2)))+(5/3)*u(x, y, t)^3+5*(diff(u(x, y, t), x, y)), x))-5*u(x, y, t)*(diff(u(x, y, t), y))+5*(int(diff(u(x, y, t), `$`(y, 2)), x))-5*(diff(u(x, y, t), x))*(int(diff(u(x, y, t), y), x))

diff(u(x, y, t), t)-(diff(diff(diff(diff(diff(u(x, y, t), x), x), x), x), x))-5*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), x))-5*u(x, y, t)*(diff(diff(diff(u(x, y, t), x), x), x))-5*u(x, y, t)^2*(diff(u(x, y, t), x))-5*(diff(diff(diff(u(x, y, t), x), x), y))-5*u(x, y, t)*(diff(u(x, y, t), y))+5*(int(diff(diff(u(x, y, t), y), y), x))-5*(diff(u(x, y, t), x))*(int(diff(u(x, y, t), y), x))

 (4)
> 

pde_nonlinear, pde_linear := selectremove(proc (term) options operator, arrow; has((eval(term, u(x, y, t) = a*u(x, y, t)))/a, a) end proc, pde)

-5*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), x))-5*u(x, y, t)*(diff(diff(diff(u(x, y, t), x), x), x))-5*u(x, y, t)^2*(diff(u(x, y, t), x))-5*u(x, y, t)*(diff(u(x, y, t), y))-5*(diff(u(x, y, t), x))*(int(diff(u(x, y, t), y), x)), diff(u(x, y, t), t)-(diff(diff(diff(diff(diff(u(x, y, t), x), x), x), x), x))-5*(diff(diff(diff(u(x, y, t), x), x), y))+5*(int(diff(diff(u(x, y, t), y), y), x))

 (5)
> 

thetai := t*w[i]+y*p[i]+x

t*w[i]+y*p[i]+x

 (6)
> 

eqw := w[i] = -5*p[i]^2

w[i] = -5*p[i]^2

 (7)
> 

Bij := proc (i, j) options operator, arrow; (-6*p[i]-6*p[j])/(p[i]-p[j])^2 end proc

proc (i, j) options operator, arrow; (-6*p[i]-6*p[j])/(p[i]-p[j])^2 end proc

 (8)
> 

NULL
> 

theta1 := normal(eval(eval(thetai, eqw), i = 1)); theta2 := normal(eval(eval(thetai, eqw), i = 2))

-5*t*p[1]^2+y*p[1]+x

  

-5*t*p[2]^2+y*p[2]+x

 (9)
> 

eqf := f(x, y, t) = (-5*t*p[1]^2+y*p[1]+x)*(-5*t*p[2]^2+y*p[2]+x)-(6*(p[1]+p[2]))/(p[1]-p[2])^2

f(x, y, t) = (-5*t*p[1]^2+y*p[1]+x)*(-5*t*p[2]^2+y*p[2]+x)-6*(p[1]+p[2])/(p[1]-p[2])^2

 (10)
> 

eq17 := u(x, y, t) = 6*(diff(diff(f(x, y, t), x), x))/f(x, y, t)-6*(diff(f(x, y, t), x))^2/f(x, y, t)^2

u(x, y, t) = 6*(diff(diff(f(x, y, t), x), x))/f(x, y, t)-6*(diff(f(x, y, t), x))^2/f(x, y, t)^2

 (11)
> 

eqt := eval(eq17, eqf)

u(x, y, t) = 12/((-5*t*p[1]^2+y*p[1]+x)*(-5*t*p[2]^2+y*p[2]+x)-6*(p[1]+p[2])/(p[1]-p[2])^2)-6*(-5*t*p[1]^2-5*t*p[2]^2+y*p[1]+y*p[2]+2*x)^2/((-5*t*p[1]^2+y*p[1]+x)*(-5*t*p[2]^2+y*p[2]+x)-6*(p[1]+p[2])/(p[1]-p[2])^2)^2

 (12)
> 

``
> 

pdetest(eqt, pde)

NULL

Download test.mw

How by apply limit remove the constant...

salim-barzani 1560 Maple 2024
limit duplicate-question
February 16 2025
0 5

How apply long wave limit for removing the constant k in such function , i need a general formula 

Limiting process from eq 12 to Bij

> 

restart
> 

NULL

Eq 12.

> 

eij := ((-3*k[i]*(k[i]-k[j])*l[j]+beta)*l[i]^2-(2*(-3*k[j]*(k[i]-k[j])*l[j]*(1/2)+beta))*l[j]*l[i]+beta*l[j]^2)/((-3*k[i]*(k[i]+k[j])*l[j]+beta)*l[i]^2-(2*(3*k[j]*(k[i]+k[j])*l[j]*(1/2)+beta))*l[j]*l[i]+beta*l[j]^2)

((-3*k[i]*(k[i]-k[j])*l[j]+beta)*l[i]^2-2*(-(3/2)*k[j]*(k[i]-k[j])*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)/((-3*k[i]*(k[i]+k[j])*l[j]+beta)*l[i]^2-2*((3/2)*k[j]*(k[i]+k[j])*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)

 (1)
> 

eval(eij, k[j] = k[i]); series(%, k[i], 3); convert(%, polynom); eval(%, k[j] = k[i]); Bij := %

(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)/((-6*k[i]^2*l[j]+beta)*l[i]^2-2*(3*k[i]^2*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)

  

series(1+((6*l[i]^2*l[j]+6*l[i]*l[j]^2)/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2))*k[i]^2+O(k[i]^4),k[i],4)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

 (2)
> 

NULL

NULL

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