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Question: How apply conjugate for getting result of pdetest ?

Question:How apply conjugate for getting result of pdetest ?

salim-barzani 600 Maple 2024
pde differential-equations conjugate pdetest
February 13 2025
2

i did pdetest without conjugate like the paper did i get zero but when i did pde test  with conjugate i didn't where is my problem 
i will do without conjugate but how change p[2]=conjugate(p[1])

> 

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 := -4*(diff(u(x, y, z, t), x, t))+diff(u(x, y, z, t), `$`(x, 3), z)+3*alpha*(diff(u(x, y, z, t), `$`(y, 2)))+4*(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), x, z))+2*(diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), z))

-4*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(u(x, y, z, t), x), x), x), z)+3*alpha*(diff(diff(u(x, y, z, t), y), y))+4*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), z))+2*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), z))

 (4)
> 

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

4*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), z))+2*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), z)), -4*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(u(x, y, z, t), x), x), x), z)+3*alpha*(diff(diff(u(x, y, z, t), y), y))

 (5)
> 

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

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

 (6)
> 

eqw := w[i] = 3*alpha*p[i]^2*(1/4)

w[i] = (3/4)*alpha*p[i]^2

 (7)
> 

Bij := proc (i, j) options operator, arrow; 4/((p[i]-p[j])^2*alpha) end proc

proc (i, j) options operator, arrow; 4/((p[i]-p[j])^2*alpha) end proc

 (8)
> 

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

(3/4)*alpha*t*p[1]^2+y*p[1]+x+z

  

(3/4)*alpha*t*p[2]^2+y*p[2]+x+z

 (9)
> 

eqf := f(x, y, z, t) = theta1*theta2+4/((p[1]-p[2])^2*alpha)

f(x, y, z, t) = ((3/4)*alpha*t*p[1]^2+y*p[1]+x+z)*((3/4)*alpha*t*p[2]^2+y*p[2]+x+z)+4/((p[1]-p[2])^2*alpha)

 (10)
> 

eq17 := u(x, y, z, t) = 2*(diff(ln(f(x, y, z, t)), x))

u(x, y, z, t) = 2*(diff(f(x, y, z, t), x))/f(x, y, z, t)

 (11)
> 

eqt := eval(eq17, eqf)

u(x, y, z, t) = 2*((3/4)*alpha*t*p[2]^2+y*p[2]+2*x+2*z+(3/4)*alpha*t*p[1]^2+y*p[1])/(((3/4)*alpha*t*p[1]^2+y*p[1]+x+z)*((3/4)*alpha*t*p[2]^2+y*p[2]+x+z)+4/((p[1]-p[2])^2*alpha))

 (12)
> 

NULL
> 

pdetest(eqt, pde)

0

 (13)
> 

NULL

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