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Question: How find parameter in this equation ?

Question:How find parameter in this equation ?

Arya-S-AA 165 Maple 2024
parameters substitution pde differential-equations duplicate-question duplicate-account
March 02 2025
4

i did try but i don't know the result is not come out? also i am not sure to put equation in eq1 in pde or linear part?

> 

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

diff(diff(u(x, y, t), t), y)+diff(diff(diff(diff(u(x, y, t), x), x), x), y)-3*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), y))-3*(diff(diff(u(x, y, t), x), x))*(diff(u(x, y, t), y))+alpha*(diff(diff(u(x, y, t), x), y))+beta*(diff(diff(u(x, y, t), x), 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)

-3*(diff(u(x, y, t), x))*(diff(diff(u(x, y, t), x), y))-3*(diff(diff(u(x, y, t), x), x))*(diff(u(x, y, t), y)), diff(diff(u(x, y, t), t), y)+diff(diff(diff(diff(u(x, y, t), x), x), x), y)+alpha*(diff(diff(u(x, y, t), x), y))+beta*(diff(diff(u(x, y, t), x), x))

 (5)
> 

eq := u(x, y, t) = -2*(diff(ln(f(x, y, t)), x))

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

 (6)
> 

eq1 := -(1/2)*numer(normal(eval(pde_linear, eq)))

f(x, y, t)^4*(diff(diff(diff(f(x, y, t), x), x), x))*beta+f(x, y, t)^4*(diff(diff(diff(f(x, y, t), x), x), y))*alpha-f(x, y, t)^3*(diff(f(x, y, t), y))*(diff(diff(f(x, y, t), x), x))*alpha-2*f(x, y, t)^3*(diff(diff(f(x, y, t), x), y))*(diff(f(x, y, t), x))*alpha-3*f(x, y, t)^3*(diff(f(x, y, t), x))*(diff(diff(f(x, y, t), x), x))*beta+2*f(x, y, t)^2*(diff(f(x, y, t), y))*(diff(f(x, y, t), x))^2*alpha+2*f(x, y, t)^2*(diff(f(x, y, t), x))^3*beta+(diff(diff(diff(f(x, y, t), t), x), y))*f(x, y, t)^4+(diff(diff(diff(diff(diff(f(x, y, t), x), x), x), x), y))*f(x, y, t)^4-(diff(diff(f(x, y, t), t), x))*(diff(f(x, y, t), y))*f(x, y, t)^3-(diff(diff(diff(diff(f(x, y, t), x), x), x), x))*(diff(f(x, y, t), y))*f(x, y, t)^3-(diff(diff(f(x, y, t), x), y))*(diff(f(x, y, t), t))*f(x, y, t)^3-4*(diff(diff(diff(f(x, y, t), x), x), x))*(diff(diff(f(x, y, t), x), y))*f(x, y, t)^3-(diff(f(x, y, t), x))*(diff(diff(f(x, y, t), t), y))*f(x, y, t)^3-4*(diff(diff(diff(diff(f(x, y, t), x), x), x), y))*(diff(f(x, y, t), x))*f(x, y, t)^3-6*(diff(diff(f(x, y, t), x), x))*(diff(diff(diff(f(x, y, t), x), x), y))*f(x, y, t)^3+2*(diff(f(x, y, t), x))*(diff(f(x, y, t), t))*(diff(f(x, y, t), y))*f(x, y, t)^2+8*(diff(diff(diff(f(x, y, t), x), x), x))*(diff(f(x, y, t), x))*(diff(f(x, y, t), y))*f(x, y, t)^2+6*(diff(diff(f(x, y, t), x), x))^2*(diff(f(x, y, t), y))*f(x, y, t)^2+24*(diff(diff(f(x, y, t), x), x))*(diff(f(x, y, t), x))*(diff(diff(f(x, y, t), x), y))*f(x, y, t)^2+12*(diff(diff(diff(f(x, y, t), x), x), y))*(diff(f(x, y, t), x))^2*f(x, y, t)^2-36*(diff(diff(f(x, y, t), x), x))*(diff(f(x, y, t), x))^2*(diff(f(x, y, t), y))*f(x, y, t)-24*(diff(f(x, y, t), x))^3*(diff(diff(f(x, y, t), x), y))*f(x, y, t)+24*(diff(f(x, y, t), x))^4*(diff(f(x, y, t), y))

 (7)
> 

NULL
> 

T := f(x, y, t) = h*a[10]+m^2+n^2+a[9]
> 

T1 := m = t*a[3]+x*a[1]+y*a[2]+a[4]
> 

T2 := n = t*a[7]+x*a[5]+y*a[6]+a[8]
> 

T3 := h = a[10]*exp(t*p[3]+x*p[1]+y*p[2])
> 

L2 := expand(subs({T1, T2, T3}, T))

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

 (8)
> 

eq9a := eval(eq1, L2)
> 

indets(%)

{alpha, beta, t, x, y, a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], p[1], p[2], p[3], exp(p[1]*x), exp(p[2]*y), exp(p[3]*t)}

 (9)
> 

p2b := subs({exp(p[1]*x) = eX, exp(p[2]*y) = eY, exp(p[3]*t) = eT}, eq9a); indets(%)

{alpha, beta, eT, eX, eY, t, x, y, a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], p[1], p[2], p[3]}

 (10)
> 

p2c := numer(normal(p2b))
> 

eqns := {coeffs(collect(p2c, {eT, eX, eY}, distributed), {eT, eX, eY})}; nops(%)

5

 (11)
> 

solve(eqns, {a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], p[1], p[2], p[3]})
 

NULL

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