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Modifying an Algorithm for Multiple Vari...

salim-barzani 1555 Maple 2024
differentiation duplicate-question
April 27 2025
1 8

How to modify the ND procedure to handle derivatives with respect to more than three independent variables for higher-dimensional PDEs, it is work for [x,t] i want  it work for [x,y,z,t] , 

> 

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)
> 

alias(F=F(x, t), G=G(x, t))

F, G

 (2)
> 

with(PDEtools):
undeclare(prime):

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

 (3)
> 

ND := proc(F, G, U)
  local v, w, f, g, a:
  v := op(F):
  if v[1] in U then w := -v[1] else w := v[1] end if:
  if v[2] in U then w := w, -v[2] else w := w, v[2] end if:
  f := op(0, F):
  g := op(0, G):
  a := diff(f(w)*g(v), U);
  convert(subs([w]=~[v], a), diff)
end proc:
> 

ND(F, G, [x]);
ND(F, G, [t]);

-(diff(F, x))*G+F*(diff(G, x))

  

-(diff(F, t))*G+F*(diff(G, t))

 (4)
> 

ND(F, F, [x]);
ND(F, F, [x, x]);

0

  

2*F*(diff(diff(F, x), x))-2*(diff(F, x))^2

 (5)
> 

ND(F, G, [x$3]);

-(diff(diff(diff(F, x), x), x))*G+3*(diff(diff(F, x), x))*(diff(G, x))-3*(diff(F, x))*(diff(diff(G, x), x))+F*(diff(diff(diff(G, x), x), x))

 (6)
> 

ND(F, F, [x$3, t]);

2*F*(diff(diff(diff(diff(F, t), x), x), x))-2*(diff(diff(diff(F, x), x), x))*(diff(F, t))-6*(diff(diff(diff(F, t), x), x))*(diff(F, x))+6*(diff(diff(F, x), x))*(diff(diff(F, t), x))

 (7)
> 

NULL
> 

NULL
> 

#if i collect P1+P1+...+P7 it must get equation 26 in paper so i want define the up proc to open but is not for (3+1) dimesnion,
> 

P1 := 9*ND(F, F, [x, t])

18*F*(diff(diff(F, t), x))-18*(diff(F, x))*(diff(F, t))

 (8)
> 

NULL
> 

P2 := -5*ND(F, F, [`$`(x, 3), y])

0

 (9)
> 

P3 := ND(F, F, [`$`(x, 6)])

2*F*(diff(diff(diff(diff(diff(diff(F, x), x), x), x), x), x))-12*(diff(diff(diff(diff(diff(F, x), x), x), x), x))*(diff(F, x))+30*(diff(diff(diff(diff(F, x), x), x), x))*(diff(diff(F, x), x))-20*(diff(diff(diff(F, x), x), x))^2

 (10)
> 

P4 := -5*ND(F, F, [`$`(y, 2)])

0

 (11)
> 

P5 := alpha*ND(F, F, [`$`(x, 2)])

alpha*(2*F*(diff(diff(F, x), x))-2*(diff(F, x))^2)

 (12)
> 

P6 := beta*ND(F, F, [x, y])

0

 (13)
> 

P7 := gamma*ND(F, F, [x, z])

0

 (14)

Download define.mw

How build up a system of equations of th...

salim-barzani 1555 Maple 2024
parameters substitution pde
April 26 2025
2 12

I try to construct a system of coefficient but  i don't know why distribute of them is not working, beside this there is any other way for build this kind of systems 

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

_local(gamma)
> 

``

`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 := 9*(diff(u(x, y, z, t), t, x))+diff(u(x, y, z, t), `$`(x, 6))-5*(diff(u(x, y, z, t), `$`(x, 3), y)+diff(u(x, y, z, t), `$`(y, 2)))+15*((diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), `$`(x, 3)))+(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), `$`(x, 4)))-(diff(u(x, y, z, t), x))*(diff(u(x, y, z, t), x, y))-(diff(u(x, y, z, t), `$`(x, 2)))*(diff(u(x, y, z, t), y)))+45*(diff(u(x, y, z, t), x))^2*(diff(u(x, y, z, t), `$`(x, 2)))+alpha*(diff(u(x, y, z, t), `$`(x, 2)))+beta*(diff(u(x, y, z, t), x, y))+delta*(diff(u(x, y, z, t), x, z))

9*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), y))-5*(diff(diff(u(x, y, z, t), y), y))+15*(diff(diff(u(x, y, z, t), x), x))*(diff(diff(diff(u(x, y, z, t), x), x), x))+15*(diff(u(x, y, z, t), x))*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-15*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), y))-15*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), y))+45*(diff(u(x, y, z, t), x))^2*(diff(diff(u(x, y, z, t), x), x))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+delta*(diff(diff(u(x, y, z, t), x), z))

 (4)
> 

``
> 

oppde := [op(expand(pde))]; u_occurrences := map(proc (i) options operator, arrow; numelems(select(has, [op([op(i)])], u)) end proc, oppde); linear_op_indices := ListTools:-SearchAll(1, u_occurrences); pde_linear := add(oppde[[linear_op_indices]]); pde_nonlinear := expand(simplify(expand(pde)-pde_linear))

9*(diff(diff(u(x, y, z, t), t), x))+diff(diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), y))-5*(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))+delta*(diff(diff(u(x, y, z, t), x), z))

  

15*(diff(diff(u(x, y, z, t), x), x))*(diff(diff(diff(u(x, y, z, t), x), x), x))+15*(diff(u(x, y, z, t), x))*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-15*(diff(u(x, y, z, t), x))*(diff(diff(u(x, y, z, t), x), y))-15*(diff(diff(u(x, y, z, t), x), x))*(diff(u(x, y, z, t), y))+45*(diff(u(x, y, z, t), x))^2*(diff(diff(u(x, y, z, t), x), x))

 (5)
> 

H := 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)

 (6)
> 

H1 := int(pde_linear, x)

(diff(u(x, y, z, t), z))*delta+alpha*(diff(u(x, y, z, t), x))+beta*(diff(u(x, y, z, t), y))-5*(int(diff(diff(u(x, y, z, t), y), y), x))+9*(diff(u(x, y, z, t), t))+diff(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x), x)-5*(diff(diff(diff(u(x, y, z, t), x), x), y))

 (7)
> 

L := eval(H1, H) = 0

-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2)+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t) = 0

 (8)
> 

numer(lhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))*denom(rhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0)) = numer(rhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))*denom(lhs(240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^3-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))/f(x, y, z, t)^2-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)^2+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))/f(x, y, z, t)^2+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^2-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)^3+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))/f(x, y, z, t)^4+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^3-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3/f(x, y, z, t)^4+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4/f(x, y, z, t)^5-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^4+beta*(2*(diff(diff(f(x, y, z, t), x), y))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))/f(x, y, z, t)^2)-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2/f(x, y, z, t)^2+60*(diff(diff(f(x, y, z, t), x), x))^3/f(x, y, z, t)^3-240*(diff(f(x, y, z, t), x))^6/f(x, y, z, t)^6-10*(diff(diff(f(x, y, z, t), y), y))/f(x, y, z, t)+10*(diff(f(x, y, z, t), y))^2/f(x, y, z, t)^2+18*(diff(diff(f(x, y, z, t), t), x))/f(x, y, z, t)+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))/f(x, y, z, t)-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))/f(x, y, z, t)+(2*(diff(diff(f(x, y, z, t), x), z))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), z))/f(x, y, z, t)^2)*delta+alpha*(2*(diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-2*(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2) = 0))

2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x))*alpha+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y))*beta+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))*f(x, y, z, t)^4-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^4+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^4+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-240*(diff(f(x, y, z, t), x))^6-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), y))*beta-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), z))*delta+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3+2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (9)
> 

simplify(2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y))*beta+720*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+60*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-60*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2-18*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), t))*f(x, y, z, t)^4-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4-30*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(f(x, y, z, t), x), y))*f(x, y, z, t)^4+10*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^4+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4+2*f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x))*alpha-240*(diff(f(x, y, z, t), x))^6-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), y))*beta-2*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4*(diff(f(x, y, z, t), z))*delta+240*(diff(diff(diff(f(x, y, z, t), x), x), x))*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*f(x, y, z, t)^3-60*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3 = 0)

2*(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-12*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), x)))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-10*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-20*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+10*(-24*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+24*(diff(f(x, y, z, t), x))*f(x, y, z, t)^3*(diff(diff(f(x, y, z, t), x), x))+(diff(f(x, y, z, t), y))*f(x, y, z, t)^4)*(diff(diff(diff(f(x, y, z, t), x), x), x))+30*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+60*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3-540*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2+2*(alpha*f(x, y, z, t)^5+360*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-30*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3*(diff(f(x, y, z, t), x))+15*f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), y)))*(diff(diff(f(x, y, z, t), x), x))+2*(beta*f(x, y, z, t)^5-30*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3)*(diff(diff(f(x, y, z, t), x), y))+18*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5+2*(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5-10*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5-240*(diff(f(x, y, z, t), x))^6+60*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-2*(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-2*f(x, y, z, t)^4*(beta*(diff(f(x, y, z, t), y))+(diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+10*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (10)
> 

F1 := %*(1/2)

(diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x))*f(x, y, z, t)^5-6*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+15*(2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3-f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), x)))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-5*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))*f(x, y, z, t)^5-10*(diff(diff(diff(f(x, y, z, t), x), x), x))^2*f(x, y, z, t)^4+5*(-24*(diff(f(x, y, z, t), x))^3*f(x, y, z, t)^2+24*(diff(f(x, y, z, t), x))*f(x, y, z, t)^3*(diff(diff(f(x, y, z, t), x), x))+(diff(f(x, y, z, t), y))*f(x, y, z, t)^4)*(diff(diff(diff(f(x, y, z, t), x), x), x))+15*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))*f(x, y, z, t)^4+30*(diff(diff(f(x, y, z, t), x), x))^3*f(x, y, z, t)^3-270*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^2+(alpha*f(x, y, z, t)^5+360*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-30*(diff(f(x, y, z, t), y))*f(x, y, z, t)^3*(diff(f(x, y, z, t), x))+15*f(x, y, z, t)^4*(diff(diff(f(x, y, z, t), x), y)))*(diff(diff(f(x, y, z, t), x), x))+(beta*f(x, y, z, t)^5-30*(diff(f(x, y, z, t), x))^2*f(x, y, z, t)^3)*(diff(diff(f(x, y, z, t), x), y))+9*(diff(diff(f(x, y, z, t), t), x))*f(x, y, z, t)^5+(diff(diff(f(x, y, z, t), x), z))*delta*f(x, y, z, t)^5-5*(diff(diff(f(x, y, z, t), y), y))*f(x, y, z, t)^5-120*(diff(f(x, y, z, t), x))^6+30*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))*f(x, y, z, t)^2-(diff(f(x, y, z, t), x))^2*alpha*f(x, y, z, t)^4-f(x, y, z, t)^4*(beta*(diff(f(x, y, z, t), y))+(diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+5*(diff(f(x, y, z, t), y))^2*f(x, y, z, t)^4 = 0

 (11)
> 

collect(F1, {alpha, beta, f(x, y, z, t)})

(-f(x, y, z, t)^4*(diff(f(x, y, z, t), x))^2+f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), x)))*alpha+(-(diff(f(x, y, z, t), y))*f(x, y, z, t)^4*(diff(f(x, y, z, t), x))+f(x, y, z, t)^5*(diff(diff(f(x, y, z, t), x), y)))*beta+((diff(diff(f(x, y, z, t), x), z))*delta+diff(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x), x)-5*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), y))+9*(diff(diff(f(x, y, z, t), t), x))-5*(diff(diff(f(x, y, z, t), y), y)))*f(x, y, z, t)^5+(-6*(diff(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x), x))*(diff(f(x, y, z, t), x))-15*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-10*(diff(diff(diff(f(x, y, z, t), x), x), x))^2+5*(diff(f(x, y, z, t), y))*(diff(diff(diff(f(x, y, z, t), x), x), x))+15*(diff(diff(diff(f(x, y, z, t), x), x), y))*(diff(f(x, y, z, t), x))+15*(diff(diff(f(x, y, z, t), x), y))*(diff(diff(f(x, y, z, t), x), x))-((diff(f(x, y, z, t), z))*delta+9*(diff(f(x, y, z, t), t)))*(diff(f(x, y, z, t), x))+5*(diff(f(x, y, z, t), y))^2)*f(x, y, z, t)^4+(30*(diff(f(x, y, z, t), x))^2*(diff(diff(diff(diff(f(x, y, z, t), x), x), x), x))-30*(diff(f(x, y, z, t), x))^2*(diff(diff(f(x, y, z, t), x), y))+120*(diff(f(x, y, z, t), x))*(diff(diff(f(x, y, z, t), x), x))*(diff(diff(diff(f(x, y, z, t), x), x), x))-30*(diff(f(x, y, z, t), x))*(diff(f(x, y, z, t), y))*(diff(diff(f(x, y, z, t), x), x))+30*(diff(diff(f(x, y, z, t), x), x))^3)*f(x, y, z, t)^3+(-120*(diff(f(x, y, z, t), x))^3*(diff(diff(diff(f(x, y, z, t), x), x), x))+30*(diff(f(x, y, z, t), x))^3*(diff(f(x, y, z, t), y))-270*(diff(diff(f(x, y, z, t), x), x))^2*(diff(f(x, y, z, t), x))^2)*f(x, y, z, t)^2+360*(diff(diff(f(x, y, z, t), x), x))*(diff(f(x, y, z, t), x))^4*f(x, y, z, t)-120*(diff(f(x, y, z, t), x))^6 = 0

 (12)
> 

NULL
> 

T := f(x, y, z, t) = g(x, y, z, t)^2+h(x, y, z, t)^2+a[11]
> 

T1 := g(x, y, z, t) = t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5]
> 

T2 := h(x, y, z, t) = t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10]
> 

L2 := subs({T1, T2}, T)

f(x, y, z, t) = (t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11]

 (13)
> 

L3 := eval(F1, L2)

30*(2*a[1]^2+2*a[6]^2)^3*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3-270*(2*a[1]^2+2*a[6]^2)^2*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^2+(alpha*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5+360*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^4*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])-30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])+15*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4*(2*a[1]*a[2]+2*a[6]*a[7]))*(2*a[1]^2+2*a[6]^2)+(beta*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^3)*(2*a[1]*a[2]+2*a[6]*a[7])+9*(2*a[1]*a[4]+2*a[6]*a[9])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5+(2*a[1]*a[3]+2*a[6]*a[8])*delta*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-5*(2*a[2]^2+2*a[7]^2)*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^5-120*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^6+30*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^3*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^2-(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])^2*alpha*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4-((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4*(beta*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])+(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[3]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[8])*delta+18*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[4]+18*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[9])*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[1]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[6])+5*(2*(t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])*a[2]+2*(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])*a[7])^2*((t*a[4]+x*a[1]+y*a[2]+z*a[3]+a[5])^2+(t*a[9]+x*a[6]+y*a[7]+z*a[8]+a[10])^2+a[11])^4 = 0

 (14)
> 

L4 := collect(L3, [x, y, z, t], 'distributed')

Warning,  computation interrupted

  

` `

 (15)

Download systems.mw

How i can find the parameter in PDE ?...

salim-barzani 1555 Maple 2024
parameters substitution pde
April 19 2025
0 6

How i can get this special parameter i try to do substitution in another mw file but stilli can't reach this parameter and without this parameter my PDE is not give me zero so i have to find this r[i] parameter, some letter of my mw file are not similar to paper but r[i]=l[i] as mention is paper al clear and i found all structure just this remain, i am looking for equation (14), thanks for any help 

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

_local(gamma)

Warning, A new binding for the name `gamma` has been created. The global instance of this name is still accessible using the :- prefix, :-`gamma`.  See ?protect for details.

  
> 

NULL
> 

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

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

 (1)
> 

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

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

 (2)
> 

pde1 := a*(diff(u(x, y, z, t), x, t))-((a^4-6*a^2*b^2+b^4)*(1/16))*(diff(u(x, y, z, t), `$`(x, 4)))-(1/4)*(3*(-a^2+b^2))*(diff(u(x, y, z, t)^2, `$`(x, 2)))+alpha*(diff(u(x, y, z, t), `$`(x, 2)))+beta*(diff(u(x, y, z, t), x, y))+delta*(diff(u(x, y, z, t), x, z))+lambda*(diff(u(x, y, z, t), `$`(z, 2)))+mu*(diff(u(x, y, z, t), y, z))+mu^2*(diff(u(x, y, z, t), `$`(y, 2)))/(4*lambda)

a*(diff(diff(u(x, y, z, t), t), x))-(1/16)*(a^4-6*a^2*b^2+b^4)*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-(3/4)*(-a^2+b^2)*(2*(diff(u(x, y, z, t), x))^2+2*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x)))+alpha*(diff(diff(u(x, y, z, t), x), x))+beta*(diff(diff(u(x, y, z, t), x), y))+delta*(diff(diff(u(x, y, z, t), x), z))+lambda*(diff(diff(u(x, y, z, t), z), z))+mu*(diff(diff(u(x, y, z, t), y), z))+(1/4)*mu^2*(diff(diff(u(x, y, z, t), y), y))/lambda

 (3)
> 

Tr := {beta = alpha, delta = alpha, mu = 2*lambda}

{beta = alpha, delta = alpha, mu = 2*lambda}

 (4)
> 

pde := subs(Tr, pde1)

a*(diff(diff(u(x, y, z, t), t), x))-(1/16)*(a^4-6*a^2*b^2+b^4)*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))-(3/4)*(-a^2+b^2)*(2*(diff(u(x, y, z, t), x))^2+2*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x)))+alpha*(diff(diff(u(x, y, z, t), x), x))+alpha*(diff(diff(u(x, y, z, t), x), y))+alpha*(diff(diff(u(x, y, z, t), x), z))+lambda*(diff(diff(u(x, y, z, t), z), z))+2*lambda*(diff(diff(u(x, y, z, t), y), z))+lambda*(diff(diff(u(x, y, z, t), y), y))

 (5)
> 

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

a*(diff(diff(u(x, y, z, t), t), x))-(1/16)*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))*a^4+(3/8)*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))*a^2*b^2-(1/16)*(diff(diff(diff(diff(u(x, y, z, t), x), x), x), x))*b^4+alpha*(diff(diff(u(x, y, z, t), x), x))+alpha*(diff(diff(u(x, y, z, t), x), y))+alpha*(diff(diff(u(x, y, z, t), x), z))+lambda*(diff(diff(u(x, y, z, t), z), z))+2*lambda*(diff(diff(u(x, y, z, t), y), z))+lambda*(diff(diff(u(x, y, z, t), y), y)), (3/2)*(diff(u(x, y, z, t), x))^2*a^2-(3/2)*(diff(u(x, y, z, t), x))^2*b^2+(3/2)*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x))*a^2-(3/2)*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x))*b^2

 (6)
> 

NULL
> 

eq17 := u(x, y, z, t) = (-a^4+6*a^2*b^2-b^4)*((diff(diff(f(x, y, z, t), x), x))/f(x, y, z, t)-(diff(f(x, y, z, t), x))^2/f(x, y, z, t)^2)/(2*a^2-2*b^2)
> 

``NULL
> 

betai := k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i]

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

 (7)
> 

W := w[i] = ((a^4-6*a^2*b^2+b^4)*k[i]^2-16*lambda*l[i]^2+(-32*lambda*r[i]-16*alpha)*l[i]-16*lambda*r[i]^2-16*alpha*r[i]-16*alpha)/(16*a)
> 

AA := A[12] = (16*(l[1]-l[2]+r[1]-r[2])^2*lambda+3*(k[1]-k[2])^2*(a^2+2*a*b-b^2)*(a^2-2*a*b-b^2))/(16*(l[1]-l[2]+r[1]-r[2])^2*lambda+3*(k[1]+k[2])^2*(a^2+2*a*b-b^2)*(a^2-2*a*b-b^2))
> 

F2 := 1+exp(beta[1])+A[1, 2]*exp(beta[1]+beta[2])+exp(beta[2])

1+exp(beta[1])+A[1, 2]*exp(beta[1]+beta[2])+exp(beta[2])

 (8)
> 

NULL
> 

F22 := f(x, y, z, t) = 1+exp((a^4*t*k[1]^3-6*a^2*b^2*t*k[1]^3+b^4*t*k[1]^3-16*lambda*t*k[1]*l[1]^2-32*lambda*t*k[1]*l[1]*r[1]-16*lambda*t*k[1]*r[1]^2+16*a*y*k[1]*l[1]+16*a*z*k[1]*r[1]-16*alpha*t*k[1]*l[1]-16*alpha*t*k[1]*r[1]+16*a*x*k[1]-16*alpha*t*k[1]+16*a*eta[1])/(16*a))+exp((a^4*t*k[2]^3-6*a^2*b^2*t*k[2]^3+b^4*t*k[2]^3-16*lambda*t*k[2]*l[2]^2-32*lambda*t*k[2]*l[2]*r[2]-16*lambda*t*k[2]*r[2]^2+16*a*y*k[2]*l[2]+16*a*z*k[2]*r[2]-16*alpha*t*k[2]*l[2]-16*alpha*t*k[2]*r[2]+16*a*x*k[2]-16*alpha*t*k[2]+16*a*eta[2])/(16*a))
> 

eq := eval(eq17, F22)

u(x, y, z, t) = (-a^4+6*a^2*b^2-b^4)*((k[1]^2*exp((1/16)*(a^4*t*k[1]^3-6*a^2*b^2*t*k[1]^3+b^4*t*k[1]^3-16*lambda*t*k[1]*l[1]^2-32*lambda*t*k[1]*l[1]*r[1]-16*lambda*t*k[1]*r[1]^2+16*a*y*k[1]*l[1]+16*a*z*k[1]*r[1]-16*alpha*t*k[1]*l[1]-16*alpha*t*k[1]*r[1]+16*a*x*k[1]-16*alpha*t*k[1]+16*a*eta[1])/a)+k[2]^2*exp((1/16)*(a^4*t*k[2]^3-6*a^2*b^2*t*k[2]^3+b^4*t*k[2]^3-16*lambda*t*k[2]*l[2]^2-32*lambda*t*k[2]*l[2]*r[2]-16*lambda*t*k[2]*r[2]^2+16*a*y*k[2]*l[2]+16*a*z*k[2]*r[2]-16*alpha*t*k[2]*l[2]-16*alpha*t*k[2]*r[2]+16*a*x*k[2]-16*alpha*t*k[2]+16*a*eta[2])/a))/(1+exp((1/16)*(a^4*t*k[1]^3-6*a^2*b^2*t*k[1]^3+b^4*t*k[1]^3-16*lambda*t*k[1]*l[1]^2-32*lambda*t*k[1]*l[1]*r[1]-16*lambda*t*k[1]*r[1]^2+16*a*y*k[1]*l[1]+16*a*z*k[1]*r[1]-16*alpha*t*k[1]*l[1]-16*alpha*t*k[1]*r[1]+16*a*x*k[1]-16*alpha*t*k[1]+16*a*eta[1])/a)+exp((1/16)*(a^4*t*k[2]^3-6*a^2*b^2*t*k[2]^3+b^4*t*k[2]^3-16*lambda*t*k[2]*l[2]^2-32*lambda*t*k[2]*l[2]*r[2]-16*lambda*t*k[2]*r[2]^2+16*a*y*k[2]*l[2]+16*a*z*k[2]*r[2]-16*alpha*t*k[2]*l[2]-16*alpha*t*k[2]*r[2]+16*a*x*k[2]-16*alpha*t*k[2]+16*a*eta[2])/a))-(k[1]*exp((1/16)*(a^4*t*k[1]^3-6*a^2*b^2*t*k[1]^3+b^4*t*k[1]^3-16*lambda*t*k[1]*l[1]^2-32*lambda*t*k[1]*l[1]*r[1]-16*lambda*t*k[1]*r[1]^2+16*a*y*k[1]*l[1]+16*a*z*k[1]*r[1]-16*alpha*t*k[1]*l[1]-16*alpha*t*k[1]*r[1]+16*a*x*k[1]-16*alpha*t*k[1]+16*a*eta[1])/a)+k[2]*exp((1/16)*(a^4*t*k[2]^3-6*a^2*b^2*t*k[2]^3+b^4*t*k[2]^3-16*lambda*t*k[2]*l[2]^2-32*lambda*t*k[2]*l[2]*r[2]-16*lambda*t*k[2]*r[2]^2+16*a*y*k[2]*l[2]+16*a*z*k[2]*r[2]-16*alpha*t*k[2]*l[2]-16*alpha*t*k[2]*r[2]+16*a*x*k[2]-16*alpha*t*k[2]+16*a*eta[2])/a))^2/(1+exp((1/16)*(a^4*t*k[1]^3-6*a^2*b^2*t*k[1]^3+b^4*t*k[1]^3-16*lambda*t*k[1]*l[1]^2-32*lambda*t*k[1]*l[1]*r[1]-16*lambda*t*k[1]*r[1]^2+16*a*y*k[1]*l[1]+16*a*z*k[1]*r[1]-16*alpha*t*k[1]*l[1]-16*alpha*t*k[1]*r[1]+16*a*x*k[1]-16*alpha*t*k[1]+16*a*eta[1])/a)+exp((1/16)*(a^4*t*k[2]^3-6*a^2*b^2*t*k[2]^3+b^4*t*k[2]^3-16*lambda*t*k[2]*l[2]^2-32*lambda*t*k[2]*l[2]*r[2]-16*lambda*t*k[2]*r[2]^2+16*a*y*k[2]*l[2]+16*a*z*k[2]*r[2]-16*alpha*t*k[2]*l[2]-16*alpha*t*k[2]*r[2]+16*a*x*k[2]-16*alpha*t*k[2]+16*a*eta[2])/a))^2)/(2*a^2-2*b^2)

 (9)
> 

pdetest(eq, pde)

Download fusion-undon.mw

How make number shorter in equation or f...

salim-barzani 1555 Maple
float floating-point
April 19 2025
1 0

i want to find critical point but becuase the equation is ong the result not shown up i try to put variable and then result show up but the number are to ugly and so long i want make them be 1 number without any decimal can we do that?

short-Dc.mw

How apply substitution and apply limit ...

salim-barzani 1555 Maple 2024
solve parameters substitution
April 16 2025
1 18

As shown in the paper, and in many similar ones, the authors use a particular method that I believe is related to the long wave limit. I’m familiar with other approaches, but the traditional methods haven’t been successful in this case. This author, along with a few others, has tried applying this long wave limit approach, though many papers don’t explicitly mention the substitutions they use to arrive at the lump solution.

I’ve been able to separately find the lump series, but for some of the other solutions, we first need to figure out how to derive this key result. Once that part is clear, the rest should be easier to handle. I've been working through everything step by step and have managed to reproduce many of the solutions from the paper.

Also i don't know how finding (eq17) in paper, which they found by apply long wave limit to (eq7) in paper

additionaly How finding line which i think they found by finding velocity?

Please, if you have any information or insight into how we can obtain this more difficult result, I would really appreciate your help.

> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

_local(gamma)
> 

NULL
> 

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

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

 (1)
> 

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

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

 (2)
> 

alpha := 1; beta := 1; delta := 1; lambda := 1

1

  

1

  

1

  

1

 (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))+delta*(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)-(diff(diff(u(x, y, z, t), y), y))+diff(diff(u(x, y, z, t), x), x)+diff(diff(u(x, y, z, t), x), y)+diff(diff(u(x, y, z, t), x), z)

 (4)
> 

thetai := t*w[i]+x*k[i]+y*l[i]+z*r[i]+eta[i]
> 

eq15 := w[i] = -(k[i]^4+k[i]^2+k[i]*l[i]+k[i]*r[i]-l[i]^2)/k[i]
> 

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

A[sj] := (3*k[i]^4*k[j]^2-6*k[i]^3*k[j]^3+(3*k[j]^4+l[j]^2)*k[i]^2-2*k[i]*k[j]*l[i]*l[j]+k[j]^2*l[i]^2)/(3*k[i]^4*k[j]^2+6*k[i]^3*k[j]^3+(3*k[j]^4+l[j]^2)*k[i]^2-2*k[i]*k[j]*l[i]*l[j]+k[j]^2*l[i]^2)
> 

F2 := 1+exp(eta[1])+b[1, 2]*exp(eta[1]+eta[2])+exp(eta[2])

1+exp(eta[1])+b[1, 2]*exp(eta[1]+eta[2])+exp(eta[2])

 (5)
> 

F22 := 1+exp(eta[1])+(3*k[1]^4*k[2]^2-6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)*exp(eta[1]+eta[2])/(3*k[1]^4*k[2]^2+6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)+exp(eta[2])

1+exp(eta[1])+(3*k[1]^4*k[2]^2-6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)*exp(eta[1]+eta[2])/(3*k[1]^4*k[2]^2+6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)+exp(eta[2])

 (6)
> 

NULL
> 

NULL
> 

F222 := exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1])+(3*k[1]^4*k[2]^2-6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)*exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1]-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2])/(3*k[1]^4*k[2]^2+6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)+exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2])

exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1])+(3*k[1]^4*k[2]^2-6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)*exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1]-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2])/(3*k[1]^4*k[2]^2+6*k[1]^3*k[2]^3+(3*k[2]^4+l[2]^2)*k[1]^2-2*k[1]*k[2]*l[1]*l[2]+k[2]^2*l[1]^2)+exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2])

 (7)
> 

indets(F222)

{t, x, y, eta[1], eta[2], k[1], k[2], l[1], l[2], r[1], r[2], exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1]), exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2]), exp(-(t*k[1]^4+t*k[1]^2+t*k[1]*l[1]+t*k[1]*r[1]-t*l[1]^2-x*k[1]^2-y*k[1]*l[1]-eta[1]*k[1])/k[1]-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]-eta[2]*k[2])/k[2])}

 (8)
> 

eq1 := eval(F222, {eta[1] = -1, eta[2] = -1, k[1] = K[1]*epsilon, l[1] = L[1]*epsilon, r[1] = R[1]*epsilon})

exp(-(epsilon^4*t*K[1]^4+epsilon^2*t*K[1]^2+epsilon^2*t*K[1]*L[1]+epsilon^2*t*K[1]*R[1]-epsilon^2*t*L[1]^2-epsilon^2*x*K[1]^2-epsilon^2*y*K[1]*L[1]+epsilon*K[1])/(K[1]*epsilon))+(3*K[1]^4*epsilon^4*k[2]^2-6*K[1]^3*epsilon^3*k[2]^3+(3*k[2]^4+l[2]^2)*K[1]^2*epsilon^2-2*K[1]*epsilon^2*k[2]*L[1]*l[2]+k[2]^2*L[1]^2*epsilon^2)*exp(-(epsilon^4*t*K[1]^4+epsilon^2*t*K[1]^2+epsilon^2*t*K[1]*L[1]+epsilon^2*t*K[1]*R[1]-epsilon^2*t*L[1]^2-epsilon^2*x*K[1]^2-epsilon^2*y*K[1]*L[1]+epsilon*K[1])/(K[1]*epsilon)-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]+k[2])/k[2])/(3*K[1]^4*epsilon^4*k[2]^2+6*K[1]^3*epsilon^3*k[2]^3+(3*k[2]^4+l[2]^2)*K[1]^2*epsilon^2-2*K[1]*epsilon^2*k[2]*L[1]*l[2]+k[2]^2*L[1]^2*epsilon^2)+exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]+k[2])/k[2])

 (9)
> 

G := limit(eq1, epsilon = 0)

exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]+2*k[2])/k[2])+exp(-1)+exp(-(t*k[2]^4+t*k[2]^2+t*k[2]*l[2]+t*k[2]*r[2]-t*l[2]^2-x*k[2]^2-y*k[2]*l[2]+k[2])/k[2])

 (10)

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