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Question: How find parameter in equation involving exponential ?

Question:How find parameter in equation involving exponential ?

salim-barzani 270 Maple 2024
substitution pde differential-equations duplicate-question
January 20 2025
1

How i can find parameter after substitution in our pde 

> 

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, t))

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

 (2)
> 

declare(f(x, t))

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

 (3)
> 

pde := diff(u(x, t), `$`(x, 3))+6*u(x, t)*(diff(u(x, t), x))+diff(u(x, t), t) = 0

diff(diff(diff(u(x, t), x), x), x)+6*u(x, t)*(diff(u(x, t), x))+diff(u(x, t), t) = 0

 (4)
> 

map(int, diff(diff(diff(u(x, t), x), x), x)+6*u(x, t)*(diff(u(x, t), x))+diff(u(x, t), t) = 0, x)

3*u(x, t)^2+diff(diff(u(x, t), x), x)+int(diff(u(x, t), t), x) = 0

 (5)
> 

pde1 := %

3*u(x, t)^2+diff(diff(u(x, t), x), x)+int(diff(u(x, t), t), x) = 0

 (6)
> 

Y := u(x, t) = 2*(diff(ln(f(x, t)), `$`(x, 2)))

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

 (7)
> 

L := eval(pde1, Y)

3*(2*(diff(diff(f(x, t), x), x))/f(x, t)-2*(diff(f(x, t), x))^2/f(x, t)^2)^2+2*(diff(diff(diff(diff(f(x, t), x), x), x), x))/f(x, t)-8*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))/f(x, t)^2+24*(diff(diff(f(x, t), x), x))*(diff(f(x, t), x))^2/f(x, t)^3-6*(diff(diff(f(x, t), x), x))^2/f(x, t)^2-12*(diff(f(x, t), x))^4/f(x, t)^4-2*(diff(f(x, t), x))*(diff(f(x, t), t))/f(x, t)^2+2*(diff(diff(f(x, t), t), x))/f(x, t) = 0

 (8)
> 

numer(lhs(3*(2*(diff(diff(f(x, t), x), x))/f(x, t)-2*(diff(f(x, t), x))^2/f(x, t)^2)^2+2*(diff(diff(diff(diff(f(x, t), x), x), x), x))/f(x, t)-8*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))/f(x, t)^2+24*(diff(diff(f(x, t), x), x))*(diff(f(x, t), x))^2/f(x, t)^3-6*(diff(diff(f(x, t), x), x))^2/f(x, t)^2-12*(diff(f(x, t), x))^4/f(x, t)^4-2*(diff(f(x, t), x))*(diff(f(x, t), t))/f(x, t)^2+2*(diff(diff(f(x, t), t), x))/f(x, t) = 0))*denom(rhs(3*(2*(diff(diff(f(x, t), x), x))/f(x, t)-2*(diff(f(x, t), x))^2/f(x, t)^2)^2+2*(diff(diff(diff(diff(f(x, t), x), x), x), x))/f(x, t)-8*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))/f(x, t)^2+24*(diff(diff(f(x, t), x), x))*(diff(f(x, t), x))^2/f(x, t)^3-6*(diff(diff(f(x, t), x), x))^2/f(x, t)^2-12*(diff(f(x, t), x))^4/f(x, t)^4-2*(diff(f(x, t), x))*(diff(f(x, t), t))/f(x, t)^2+2*(diff(diff(f(x, t), t), x))/f(x, t) = 0)) = numer(rhs(3*(2*(diff(diff(f(x, t), x), x))/f(x, t)-2*(diff(f(x, t), x))^2/f(x, t)^2)^2+2*(diff(diff(diff(diff(f(x, t), x), x), x), x))/f(x, t)-8*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))/f(x, t)^2+24*(diff(diff(f(x, t), x), x))*(diff(f(x, t), x))^2/f(x, t)^3-6*(diff(diff(f(x, t), x), x))^2/f(x, t)^2-12*(diff(f(x, t), x))^4/f(x, t)^4-2*(diff(f(x, t), x))*(diff(f(x, t), t))/f(x, t)^2+2*(diff(diff(f(x, t), t), x))/f(x, t) = 0))*denom(lhs(3*(2*(diff(diff(f(x, t), x), x))/f(x, t)-2*(diff(f(x, t), x))^2/f(x, t)^2)^2+2*(diff(diff(diff(diff(f(x, t), x), x), x), x))/f(x, t)-8*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))/f(x, t)^2+24*(diff(diff(f(x, t), x), x))*(diff(f(x, t), x))^2/f(x, t)^3-6*(diff(diff(f(x, t), x), x))^2/f(x, t)^2-12*(diff(f(x, t), x))^4/f(x, t)^4-2*(diff(f(x, t), x))*(diff(f(x, t), t))/f(x, t)^2+2*(diff(diff(f(x, t), t), x))/f(x, t) = 0))

2*f(x, t)^2*(3*(diff(diff(f(x, t), x), x))^2+f(x, t)*(diff(diff(diff(diff(f(x, t), x), x), x), x))+f(x, t)*(diff(diff(f(x, t), t), x))-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t))) = 0

 (9)
> 

PP := simplify(2*f(x, t)^2*(3*(diff(diff(f(x, t), x), x))^2+f(x, t)*(diff(diff(diff(diff(f(x, t), x), x), x), x))+f(x, t)*(diff(diff(f(x, t), t), x))-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t))) = 0)

2*f(x, t)^2*(3*(diff(diff(f(x, t), x), x))^2+f(x, t)*(diff(diff(diff(diff(f(x, t), x), x), x), x))+f(x, t)*(diff(diff(f(x, t), t), x))-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t))) = 0

 (10)
> 

%/(2*f(x, t)^2)

3*(diff(diff(f(x, t), x), x))^2+f(x, t)*(diff(diff(diff(diff(f(x, t), x), x), x), x))+f(x, t)*(diff(diff(f(x, t), t), x))-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t)) = 0

 (11)
> 

collect(%, f)

(diff(diff(diff(diff(f(x, t), x), x), x), x)+diff(diff(f(x, t), t), x))*f(x, t)+3*(diff(diff(f(x, t), x), x))^2-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t)) = 0

 (12)
> 

pde2 := %

(diff(diff(diff(diff(f(x, t), x), x), x), x)+diff(diff(f(x, t), t), x))*f(x, t)+3*(diff(diff(f(x, t), x), x))^2-4*(diff(diff(diff(f(x, t), x), x), x))*(diff(f(x, t), x))-(diff(f(x, t), x))*(diff(f(x, t), t)) = 0

 (13)
> 

N = 1

N = 1

 (14)
> 

S := f(x, t) = a[0]+a[1]*exp(t*n[1]+x*k[1])

f(x, t) = a[0]+a[1]*exp(t*n[1]+x*k[1])

 (15)
> 

A := eval(pde2, S)

(a[1]*k[1]^4*exp(t*n[1]+x*k[1])+a[1]*n[1]*k[1]*exp(t*n[1]+x*k[1]))*(a[0]+a[1]*exp(t*n[1]+x*k[1]))-a[1]^2*k[1]^4*(exp(t*n[1]+x*k[1]))^2-a[1]^2*k[1]*(exp(t*n[1]+x*k[1]))^2*n[1] = 0

 (16)
> 

simplify((a[1]*k[1]^4*exp(t*n[1]+x*k[1])+a[1]*n[1]*k[1]*exp(t*n[1]+x*k[1]))*(a[0]+a[1]*exp(t*n[1]+x*k[1]))-a[1]^2*k[1]^4*(exp(t*n[1]+x*k[1]))^2-a[1]^2*k[1]*(exp(t*n[1]+x*k[1]))^2*n[1] = 0)

a[0]*a[1]*exp(t*n[1]+x*k[1])*k[1]*(k[1]^3+n[1]) = 0

 (17)
> 

%/exp(t*n[1]+x*k[1])

(k[1]^3+n[1])*k[1]*a[1]*a[0] = 0

 (18)
> 

PPP := %

(k[1]^3+n[1])*k[1]*a[1]*a[0] = 0

 (19)
> 

Co := solve(PPP, {a[0], a[1], k[1], n[1]})

{a[0] = a[0], a[1] = a[1], k[1] = k[1], n[1] = -k[1]^3}, {a[0] = a[0], a[1] = a[1], k[1] = 0, n[1] = n[1]}, {a[0] = a[0], a[1] = 0, k[1] = k[1], n[1] = n[1]}, {a[0] = 0, a[1] = a[1], k[1] = k[1], n[1] = n[1]}

 (20)
> 

case1 := Co[1]

{a[0] = a[0], a[1] = a[1], k[1] = k[1], n[1] = -k[1]^3}

 (21)
> 

F := subs(case1, S)

f(x, t) = a[0]+a[1]*exp(-t*k[1]^3+x*k[1])

 (22)
> 

F1 := eval(Y, F)

u(x, t) = 2*a[1]*k[1]^2*exp(-t*k[1]^3+x*k[1])/(a[0]+a[1]*exp(-t*k[1]^3+x*k[1]))-2*a[1]^2*k[1]^2*(exp(-t*k[1]^3+x*k[1]))^2/(a[0]+a[1]*exp(-t*k[1]^3+x*k[1]))^2

 (23)
> 

pdetest(F1, pde)

0

 (24)
> 

N = 2

N = 2

 (25)
> 

S2 := f(x, t) = a[0]+a[1]*exp(t*n[1]+x*k[1])+a[2]*exp(t*n[2]+x*k[2])+a[3]*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])

f(x, t) = a[0]+a[1]*exp(t*n[1]+x*k[1])+a[2]*exp(t*n[2]+x*k[2])+a[3]*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])

 (26)
> 

eq5 := normal(eval(pde2, S2))

exp(t*n[1]+x*k[1])*a[0]*a[1]*k[1]^4-4*exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]^3*k[2]+6*exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]^2*k[2]^2-4*exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]*k[2]^3+exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]*n[1]-exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]*n[2]-exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[2]*n[1]+exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[2]*n[2]+exp(t*n[1]+x*k[1])*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[1]*a[3]*k[2]*n[2]+exp(t*n[2]+x*k[2])*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[2]*a[3]*k[1]*n[1]+exp(t*n[1]+x*k[1])*a[0]*a[1]*k[1]*n[1]+exp(t*n[2]+x*k[2])*a[0]*a[2]*k[2]^4+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]^4+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[2]^4+exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[1]^4+exp(t*n[1]+x*k[1])*exp(t*n[2]+x*k[2])*a[1]*a[2]*k[2]^4+exp(t*n[1]+x*k[1])*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[1]*a[3]*k[2]^4+exp(t*n[2]+x*k[2])*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[2]*a[3]*k[1]^4+4*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]^3*k[2]+6*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]^2*k[2]^2+4*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]*k[2]^3+exp(t*n[2]+x*k[2])*a[0]*a[2]*k[2]*n[2]+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]*n[1]+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[1]*n[2]+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[2]*n[1]+exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])*a[0]*a[3]*k[2]*n[2] = 0

 (27)
> 

indets(eq5)

{t, x, a[0], a[1], a[2], a[3], k[1], k[2], n[1], n[2], exp(t*n[1]+x*k[1]), exp(t*n[2]+x*k[2]), exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])}

 (28)
> 

eq6 := eval(eq5, {t*n[1]+x*k[1] = X, t*n[2]+x*k[2] = Y}); indets(eq6)

Error, invalid input: exp expects its 1st argument, x, to be of type algebraic, but received u(x,t) = 2*diff(diff(f(x,t),x),x)/f(x,t)-2*diff(f(x,t),x)^2/f(x,t)^2

  

{eq6}

 (29)
> 

``
> 

NULL
> 

NULL
> 

NULL
> 

NULL
> 

S3 := f(x, t) = a[0]+sum(exp(t*n[i]+x*k[i]), i = 1 .. 3)+a[1]*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])+a[2]*exp(t*n[1]+t*n[3]+x*k[1]+x*k[3])+a[3]*exp(t*n[2]+t*n[3]+x*k[2]+x*k[3])+a[4]*exp(t*n[1]+t*n[2]+t*n[3]+x*k[1]+x*k[2]+x*k[3])

f(x, t) = a[0]+exp(t*n[1]+x*k[1])+exp(t*n[2]+x*k[2])+exp(t*n[3]+x*k[3])+a[1]*exp(t*n[1]+t*n[2]+x*k[1]+x*k[2])+a[2]*exp(t*n[1]+t*n[3]+x*k[1]+x*k[3])+a[3]*exp(t*n[2]+t*n[3]+x*k[2]+x*k[3])+a[4]*exp(t*n[1]+t*n[2]+t*n[3]+x*k[1]+x*k[2]+x*k[3])

 (30)
> 

NULL

NULL

> 

eq5 := normal(eval(pde2, S3))
 

``

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