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salim-barzani

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These are questions asked by salim-barzani

How find parameter of series by putting ...

salim-barzani 1640 Maple
parameters substitution pde
March 30 2025
1 2

i did this question before but i didn't get any answer before, but the shape of question is different, the function is different this time i try 3 term like they mention in that paper so there  must be a way for finding R[2],R[1], and R[0] 

Download Find-U-in-PDE.mw

How plot complex function including two ...

salim-barzani 1640 Maple 2024
contourplot substitution plotting
March 29 2025
1 4

why i get error in end and how i can fix this error?

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet); declare(u(x, y, t), quiet); declare(f(x, y, t), quiet)
> 

theta := i -> t*w[i]+y*l[i]+x:
> 

eqf := f(x, y, t) = theta(1)*theta(2)+Bij(1, 2):
> 

eqfcomplex := eval(eqf, l[2] = conjugate(l[1])):
> 

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

equ := eval(eq17, eqfcomplex):
> 

sys := map(normal, {diff(rhs(equ), x), diff(rhs(equ), y)}):
> 

nsys  := map(numer, sys):
nroot := solve(nsys, {x, y}, explicit):
> 

dsys  := map(denom, sys):
droot := solve(dsys, {x, y}, explicit):
> 

{nroot} intersect {droot}

{}

 (1)
> 

compact_ans1 := nroot[1]:
> 

__w := seq(w[i] = (-beta*l[i]^2 - b*l[i] - a), i=1..2):
> 

__Bij := (i,j) -> 12*alpha/(beta*(l[i] - l[j])^2):
> 

eval(eval(compact_ans1, {__w, Bij(1, 2) = __Bij(1, 2)}), l[1]=lambda[1]+I*lambda[2])
assuming lambda[1]::real, lambda[2]::real:
 
> 

ans1 := map(simplify, %, size): # it's up to you to use another simplification strategy
> 

eqp1 := eval(eval(ans1, l[2] = conjugate(l[1])), l[1] = lambda[1]+I*lambda[2])
> 

NULL
> 

# Do the same for the other nroot solutions
> 

eqp := {x = xp+((1/2)*beta*lambda[2]^3+I*(-beta*lambda[1]-b)*lambda[2]^2*(1/2)-((1/2)*beta*conjugate(lambda[1]+I*lambda[2])^2-(1/2)*beta*lambda[1]^2+(1/2)*b*conjugate(lambda[1]+I*lambda[2])+a)*lambda[2]+I*lambda[1]*(conjugate(lambda[1]+I*lambda[2])-lambda[1])*(beta*conjugate(lambda[1]+I*lambda[2])+beta*lambda[1]+b)*(1/2))*t/lambda[2], y = yp-(I*beta*lambda[2]^2+(2*beta*lambda[1]+b)*lambda[2]+I*((conjugate(lambda[1]+I*lambda[2])+lambda[1])*beta+b)*(conjugate(lambda[1]+I*lambda[2])-lambda[1]))*t/(2*lambda[2])}
> 

NULL
> 

vx, vy := diff(eval(x, eqp), t), diff(eval(y, eqp), t); dydx := simplify(vy/vx)
> 

eqfp := dchange(eqp, eqfcomplex, [xp, yp], params = [a, b, alpha, beta, `λ__1`, `λ__2`], simplify); eq17p := dchange(eqp, eq17, [xp, yp], params = [a, b, alpha, beta, `λ__1`, `λ__2`], simplify); eqt := simplify(eval(eq17p, eqfp))
> 

eqt1 := eval(subs({xp = x, yp = y}, eqt), l[1] = lambda[1]+I*lambda[2])
> 

with(plots); lambda[1] := .14; lambda[2] := .68; alpha := -.46; beta := 1.83; a := 1.56; b := -.19; eq := y = (-beta*conjugate(lambda[1]+I*lambda[2])^2-b*conjugate(lambda[1]+I*lambda[2])-beta*lambda[2]^2+I*(2*beta*lambda[1]+b)*lambda[2]+lambda[1]*(beta*lambda[1]+b))*(x+(2*I)*sqrt(3)*lambda[1]*sqrt(alpha/(beta*(lambda[1]+I*lambda[2]-conjugate(lambda[1]+I*lambda[2]))^2))/lambda[2])/((lambda[1]+I*lambda[2])*beta*conjugate(lambda[1]+I*lambda[2])^2+(lambda[1]+I*lambda[2])*b*conjugate(lambda[1]+I*lambda[2])-I*beta*lambda[2]^3+(-beta*lambda[1]-b)*lambda[2]^2+I*(-beta*lambda[1]^2+2*a)*lambda[2]-beta*lambda[1]^3-b*lambda[1]^2); U := proc (x, y, a, b, alpha, beta, `λ__1`, `λ__2`) options operator, arrow; rhs(eqt1) end proc; contour1 := contourplot(eval(U(x, y, a, b, alpha, beta, `λ__1`, `λ__2`), t = -50), x = -200 .. 200, y = -100 .. 100, contours = 30, color = red, grid = [100, 100], transparency = .1); contour2 := contourplot(eval(U(x, y, a, b, alpha, beta, `λ__1`, `λ__2`), t = 0), x = -200 .. 200, y = -100 .. 100, contours = 30, color = green, grid = [100, 100], transparency = .1); contour3 := contourplot(eval(U(x, y, a, b, alpha, beta, `λ__1`, `λ__2`), t = 50), x = -200 .. 200, y = -100 .. 100, contours = 30, color = blue, grid = [100, 100], transparency = .1); trajectory_plot := implicitplot(eq, x = -200 .. 200, y = -200 .. 200, color = black, thickness = 1); T := textplot([[100, 45, "t=50", color = blue], [45, -10, "t=0", color = green], [-100, -45, "t=-50", color = red]], font = [Times, Roman, 16]); display(contour1, contour2, contour3, trajectory_plot, T, labels = ["x", "y"], scaling = constrained, size = [1200, 800])

.14

  

.68

  

-.46

  

1.83

  

1.56

  

-.19

  

y = (.4755583090+0.*I)*(x+(-0.+.1517971372*I)*3^(1/2))

  

proc (x, y, a, b, alpha, beta, lambda__1, lambda__2) options operator, arrow; rhs(eqt1) end proc

  

Error, (in plot/iplot2d) invalid input: Plot:-ColorBar expects its 2nd argument, ymin, to be of type numeric, but received infinity

  

Error, (in plot/iplot2d) invalid input: Plot:-ColorBar expects its 2nd argument, ymin, to be of type numeric, but received infinity

  

Error, (in plot/iplot2d) invalid input: Plot:-ColorBar expects its 2nd argument, ymin, to be of type numeric, but received infinity

  

Error, (in plots:-display) expecting plot structure but received: contour1

  
8

Download line-plot.mw

How determine interval of singularity in...

salim-barzani 1640 Maple 2024
explore singularity plotting
March 27 2025
1 4

in a lot of my function i have a interval which is make my function singular and i don't know how remove this singularity even when i am change a lot of parameter with explore which explore option for plot is a little bit heavy for more than  7 or 8 parameter for running , and i know the shape of the graph is 2-soliton and 1-breather(zig-zag) but i have to see the shape and make my plot have a best shape  there is any idea for fixing this issue?

singular-interval.mw

Why i can't find x&y C-point ?...

salim-barzani 1640 Maple 2024
substitution duplicate-question
March 27 2025
1 1

in some equation i don't have problem but in a lot of them this problem is come up for me and i don't know how fix this issue?

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet); declare(u(x, y, t), quiet); declare(f(x, y, t), quiet)

``

 (1)
> 

thetai := t*w[i]+y*l[i]+x
> 

eqw := w[i] = (-1+sqrt(-4*b*beta*l[i]-4*a*beta+1))/(2*beta)
> 

Bij := proc (i, j) options operator, arrow; -24*alpha*beta/(sqrt(1+(-4*b*l[j]-4*a)*beta)*sqrt(1+(-4*b*l[i]-4*a)*beta)-1+((2*l[i]+2*l[j])*b+4*a)*beta) end proc
> 

NULL
> 

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

eqf := f(x, y, t) = theta1*theta2+Bij(1, 2)
> 

eqfcomplex := eval(eval(eval(eqf, l[2] = conjugate(l[1])), l[1] = lambda[1]+I*lambda[2]))
> 

eq17 := u(x, y, t) = 2*(diff(f(x, y, t), x))/f(x, y, t); equ := simplify(eval(eq17, eqfcomplex))

u(x, y, t) = 8*(-(1/2)*(-4*b*beta*conjugate(lambda[1]+I*lambda[2])-4*a*beta+1)^(1/2)*(1+((-(4*I)*lambda[2]-4*lambda[1])*b-4*a)*beta)^(1/2)-b*beta*conjugate(lambda[1]+I*lambda[2])+1/2-(b*(lambda[1]+I*lambda[2])+2*a)*beta)*((1/2)*t*(1+((-(4*I)*lambda[2]-4*lambda[1])*b-4*a)*beta)^(1/2)+(1/2)*t*(-4*b*beta*conjugate(lambda[1]+I*lambda[2])-4*a*beta+1)^(1/2)+conjugate(lambda[1]+I*lambda[2])*y*beta+((lambda[1]+I*lambda[2])*y+2*x)*beta-t)/((1+((-(4*I)*lambda[2]-4*lambda[1])*b-4*a)*beta)^(1/2)*(-(-4*b*beta*conjugate(lambda[1]+I*lambda[2])-4*a*beta+1)^(1/2)*((2*y*((lambda[1]+I*lambda[2])*y+x)*beta+t*(b*t-y))*conjugate(lambda[1]+I*lambda[2])+2*x*((lambda[1]+I*lambda[2])*y+x)*beta+((b*(lambda[1]+I*lambda[2])+2*a)*t-(lambda[1]+I*lambda[2])*y-2*x)*t)+4*(I*lambda[2]-conjugate(lambda[1]+I*lambda[2])+lambda[1])*((1/2)*conjugate(lambda[1]+I*lambda[2])*b*y*beta+(a*y-(1/2)*b*x)*beta+(1/4)*b*t-(1/4)*y)*t)-4*t*(-4*b*beta*conjugate(lambda[1]+I*lambda[2])-4*a*beta+1)^(1/2)*(conjugate(lambda[1]+I*lambda[2])*(beta*(-y*((1/2)*b*(lambda[1]+I*lambda[2])+a)+(1/2)*b*x)-(1/4)*b*t+(1/4)*y)+(((I*lambda[1]*lambda[2]+(1/2)*lambda[1]^2-(1/2)*lambda[2]^2)*b+a*(lambda[1]+I*lambda[2]))*y-(1/2)*(lambda[1]+I*lambda[2])*b*x)*beta+(1/4)*(b*t-y)*(lambda[1]+I*lambda[2]))+4*y*beta*b*conjugate(lambda[1]+I*lambda[2])^2*(-((lambda[1]+I*lambda[2])*y+x)*beta+(1/2)*t)+conjugate(lambda[1]+I*lambda[2])*(-4*beta^2*(y^2*(b*(lambda[1]^2-lambda[2]^2+(2*I)*lambda[1]*lambda[2])+2*a*(lambda[1]+I*lambda[2]))+2*x*(b*(lambda[1]+I*lambda[2])+a)*y+b*x^2)+2*beta*(-4*b*(b*(lambda[1]+I*lambda[2])+a)*t^2+2*t*(y*(b*(lambda[1]+I*lambda[2])+a)+b*x)+y*((lambda[1]+I*lambda[2])*y+x))+b*t^2-t*y)+4*(-2*((I*lambda[1]*lambda[2]+(1/2)*lambda[1]^2-(1/2)*lambda[2]^2)*b+a*(lambda[1]+I*lambda[2]))*x*y-(lambda[1]+I*lambda[2])*x^2*b-2*a*x^2+12*alpha)*beta^2+2*beta*(-4*a*(b*(lambda[1]+I*lambda[2])+a)*t^2+t*(y*(b*(lambda[1]^2-lambda[2]^2+(2*I)*lambda[1]*lambda[2])+2*a*(lambda[1]+I*lambda[2]))+2*(b*(lambda[1]+I*lambda[2])+2*a)*x)+x*((lambda[1]+I*lambda[2])*y+x))+((b*(lambda[1]+I*lambda[2])+2*a)*t-(lambda[1]+I*lambda[2])*y-2*x)*t)

 (2)
> 

ans := solve({diff(rhs(equ), x), diff(rhs(equ), y)}, {x, y}, explicit)
 

``

Download critical-point.mw

How find Auxiliary solution function by ...

salim-barzani 1640 Maple
substitution pde series
March 26 2025
0 0

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]))


 

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