Advanced Search

Question: How find Auxiliary solution function by using series?

Question:How find Auxiliary solution function by using series?

salim-barzani 1070 Maple
substitution pde series differential-equations
March 26 2025
4

in a lot of paper i see that they just use the Auxiliary function without mention any detail but now i have to find out how i can reach this function, always i used u=Rdiff(ln(f),x#1,2) or u=Rdiff(ln(f),y,x)  (eq17) in mw. and it is answer for me untill now without knowing finding, but i have to figure out how they reach this in more than 1000 paper i didn't see any explanation about that they just used just in one of the paper mentioned something  like a series which i think they used this series but again is so complicated for undrestanding , i will put some problem picture and now i want to know how find them  eq17 for any equation based on the series in last picture mentioned

 

second example

third example which is so  different from other and i don't know how author reach this point 

i have to find this auxiliary function by using something like series  as mentioned in other question? how i can use this series for finding my auxiliary function u= u_0+R*diff(ln(f),x)  


 

#picture one

NULL

> 

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.

  
> 

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), `$`(x, 3)), x)+diff(alpha*(diff(u(x, y, z, t), x))+beta*(diff(u(x, y, z, t), y))+delta*(diff(u(x, y, z, t), z)), x)+mu*(diff(u(x, y, z, t), `$`(t, 2)))

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)+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))+mu*(diff(diff(u(x, y, z, t), t), t))

 (4)
> 

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

diff(diff(u(x, y, z, t), t), x)+diff(diff(diff(diff(u(x, y, z, t), x), x), 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))+mu*(diff(diff(u(x, y, z, t), t), t)), 6*(diff(u(x, y, z, t), x))^2+6*u(x, y, z, t)*(diff(diff(u(x, y, z, t), x), x))

 (5)
> 

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

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

  

k[i]^2*w[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+k[i]^4*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+alpha*k[i]^2*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+beta*k[i]^2*l[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+delta*k[i]^2*r[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+mu*k[i]^2*w[i]^2*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])

  

w[i] = (1/2)*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu

 (6)
> 

eqf := f(x, y, z, t) = 1+eval(exp(thetai), eq15)

f(x, y, z, t) = 1+exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i])

 (7)
> 

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

u(x, y, z, t) = R*((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)

 (8)
> 

eval(eq17, eqf); simplify(eval(pde, %)); sort([solve(%, R)]); eq17 := eval(eq17, R = simplify(%[2]))

u(x, y, z, t) = R*(k[i]^2*exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i])/(1+exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))-k[i]^2*(exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))^2/(1+exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))^2)

  

12*R*k[i]^6*exp(((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu)*(exp(((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu)-3*exp((1/2)*((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu)+1)*(R-2)/(1+exp((1/2)*((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu))^6

  

[0, 2]

  

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

 (9)
> 

eq19 := eval(eq17, eqf)

u(x, y, z, t) = 2*k[i]^2*exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i])/(1+exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))-2*k[i]^2*(exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))^2/(1+exp(k[i]*((1/2)*t*(-1+(-4*beta*mu*l[i]-4*delta*mu*r[i]-4*mu*k[i]^2-4*alpha*mu+1)^(1/2))/mu+y*l[i]+z*r[i]+x)+eta[i]))^2

 (10)
> 

simplify(eq19)

u(x, y, z, t) = 2*k[i]^2*exp((1/2)*((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu)/(1+exp((1/2)*((1+(-4*beta*l[i]-4*delta*r[i]-4*k[i]^2-4*alpha)*mu)^(1/2)*t*k[i]+((2*y*l[i]+2*z*r[i]+2*x)*mu-t)*k[i]+2*eta[i]*mu)/mu))^2

 (11)
> 

pdetest(eq19, pde)

0

 (12)
> 

#second example
> 

NULL
> 

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`

 (13)
> 

declare(u(x, y, t))

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

 (14)
> 

declare(f(x, y, t))

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

 (15)
> 

pde := diff(u(x, y, t), x, t)+alpha*(diff(u(x, y, t), `$`(x, 4))+6*(diff(u(x, y, t), x))*(diff(u(x, y, t), `$`(x, 2))))+beta*(diff(u(x, y, t), `$`(y, 2)))+a*(diff(u(x, y, t), `$`(x, 2)))+b*(diff(u(x, y, t), x, y))

diff(diff(u(x, y, t), t), x)+alpha*(diff(diff(diff(diff(u(x, y, t), x), x), x), x)+6*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), x)))+beta*(diff(diff(u(x, y, t), y), y))+a*(diff(diff(u(x, y, t), x), x))+b*(diff(diff(u(x, y, t), x), y))

 (16)
> 

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))

diff(diff(u(x, y, t), t), x)+alpha*(diff(diff(diff(diff(u(x, y, t), x), x), x), x))+beta*(diff(diff(u(x, y, t), y), y))+a*(diff(diff(u(x, y, t), x), x))+b*(diff(diff(u(x, y, t), x), y))

  

6*alpha*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), x))

 (17)
> 

thetai := k[i]*(t*w[i]+y*l[i]+x)+eta[i]; eval(pde_linear, u(x, y, t) = 1+exp(thetai)); eq15 := isolate(%, w[i])

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

  

k[i]^2*w[i]*exp(k[i]*(t*w[i]+y*l[i]+x)+eta[i])+alpha*k[i]^4*exp(k[i]*(t*w[i]+y*l[i]+x)+eta[i])+beta*k[i]^2*l[i]^2*exp(k[i]*(t*w[i]+y*l[i]+x)+eta[i])+a*k[i]^2*exp(k[i]*(t*w[i]+y*l[i]+x)+eta[i])+b*k[i]^2*l[i]*exp(k[i]*(t*w[i]+y*l[i]+x)+eta[i])

  

w[i] = -alpha*k[i]^2-beta*l[i]^2-b*l[i]-a

 (18)
> 

eqf := f(x, y, t) = 1+eval(exp(thetai), eq15)

f(x, y, t) = 1+exp(k[i]*((-alpha*k[i]^2-beta*l[i]^2-b*l[i]-a)*t+l[i]*y+x)+eta[i])

 (19)
> 

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

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

 (20)
> 

eval(eq17, eqf); simplify(eval(pde, %)); sort([solve(%, R)]); eq17 := eval(eq17, R = simplify(%[2]))

[0, 2]

  

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

 (21)
> 

eq19 := eval(eq17, eqf)

u(x, y, t) = 2*k[i]*exp(k[i]*((-alpha*k[i]^2-beta*l[i]^2-b*l[i]-a)*t+l[i]*y+x)+eta[i])/(1+exp(k[i]*((-alpha*k[i]^2-beta*l[i]^2-b*l[i]-a)*t+l[i]*y+x)+eta[i]))

 (22)
> 

M := eval(rhs(eq19), i = 1)

2*k[1]*exp(k[1]*(t*(-alpha*k[1]^2-beta*l[1]^2-b*l[1]-a)+y*l[1]+x)+eta[1])/(1+exp(k[1]*(t*(-alpha*k[1]^2-beta*l[1]^2-b*l[1]-a)+y*l[1]+x)+eta[1]))

 (23)
> 

simplify(eq19)

u(x, y, t) = 2*k[i]*exp(-alpha*t*k[i]^3+((-beta*l[i]^2-b*l[i]-a)*t+y*l[i]+x)*k[i]+eta[i])/(1+exp(-alpha*t*k[i]^3+((-beta*l[i]^2-b*l[i]-a)*t+y*l[i]+x)*k[i]+eta[i]))

 (24)
> 

pdetest(eq19, pde)

0

 (25)
> 

#third example which is so different and really i don't know how the author reach this point? which is diff(arctan(f),x)?
> 

NULL
> 

restart
> 

with(PDEtools)
> 

with(LinearAlgebra)
> 

NULL
> 

with(SolveTools)
> 

_local(gamma)
> 

undeclare(prime)

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

 (26)
> 

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

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

 (27)
> 

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

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

 (28)
> 

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

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

 (29)
> 

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

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

 (30)
> 

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

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

  

k[i]*w[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+k[i]^3*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+alpha*k[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+beta*k[i]*l[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+delta*k[i]*r[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+lambda*k[i]^2*w[i]*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])+mu*k[i]^2*w[i]^2*exp(k[i]*(t*w[i]+y*l[i]+z*r[i]+x)+eta[i])

  

w[i] = (1/2)*(-lambda*k[i]-1+(-4*beta*mu*k[i]*l[i]-4*delta*mu*k[i]*r[i]+lambda^2*k[i]^2-4*mu*k[i]^3-4*alpha*mu*k[i]+2*lambda*k[i]+1)^(1/2))/(mu*k[i])

 (31)
> 

eqf := f(x, y, z, t) = 1+eval(exp(thetai), eq15)

f(x, y, z, t) = 1+exp(k[i]*((1/2)*(-lambda*k[i]-1+(-4*beta*mu*k[i]*l[i]-4*delta*mu*k[i]*r[i]+lambda^2*k[i]^2-4*mu*k[i]^3-4*alpha*mu*k[i]+2*lambda*k[i]+1)^(1/2))*t/(mu*k[i])+l[i]*y+r[i]*z+x)+eta[i])

 (32)
> 

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

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

 (33)
> 

eval(eq17, eqf); simplify(eval(pde, %)); sort([solve(%, R)]); eq17 := eval(eq17, R = simplify(%[2]))


 

Download F-series.mw

Thanks for any help!
This has been branched into the following page(s):
Please Wait...