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Question: How collect a fraction which contain derivative? And finding parameters

Question:How collect a fraction which contain derivative? And finding parameters

salim-barzani 1555 Maple 2024
parameters pde differential-equations coefficients duplicate-question
February 15 2025
2

i want construct a series trail function for all pdf not just this one but this is a easy one, also after replacing the function How i can collect variable and make algebraic system for finding the constant of series function like a[20],a[10],a[00]. where i is number of derivative by x and n is number of derivative by t also n=0 and m=2 

> 

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(w(x, t))

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

 (3)
> 

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

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

 (4)
> 

NULL
> 

K := u(x, t) = a[20]*(diff(ln(w(x, t)), `$`(x, 2)))+a[10]*(diff(ln(w(x, t)), x))+a[0]

u(x, t) = a[20]*((diff(diff(w(x, t), x), x))/w(x, t)-(diff(w(x, t), x))^2/w(x, t)^2)+a[10]*(diff(w(x, t), x))/w(x, t)+a[0]

 (5)
> 

K1 := normal(eval(pde, K))

(w(x, t)^4*(diff(diff(w(x, t), x), x))*a[0]*a[10]+w(x, t)^4*(diff(diff(diff(w(x, t), x), x), x))*a[0]*a[20]+w(x, t)^4*(diff(diff(diff(diff(diff(w(x, t), x), x), x), x), x))*delta*a[20]+w(x, t)^4*(diff(diff(diff(diff(w(x, t), x), x), x), x))*delta*a[10]-3*w(x, t)^3*(diff(diff(w(x, t), x), x))^2*delta*a[10]+w(x, t)^3*(diff(diff(w(x, t), x), x))^2*a[10]*a[20]+w(x, t)^3*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))*a[10]^2+w(x, t)^3*(diff(diff(w(x, t), x), x))*(diff(diff(diff(w(x, t), x), x), x))*a[20]^2-w(x, t)^3*(diff(w(x, t), x))^2*a[0]*a[10]-3*w(x, t)^2*(diff(diff(w(x, t), x), x))^2*(diff(w(x, t), x))*a[20]^2+2*w(x, t)^2*(diff(w(x, t), x))^3*a[0]*a[20]-w(x, t)^2*(diff(w(x, t), x))^2*(diff(diff(diff(w(x, t), x), x), x))*a[20]^2+5*w(x, t)*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^3*a[20]^2-6*w(x, t)*(diff(w(x, t), x))^4*delta*a[10]+3*w(x, t)*(diff(w(x, t), x))^4*a[10]*a[20]-w(x, t)^3*(diff(diff(w(x, t), x), x))*(diff(w(x, t), t))*a[20]-a[10]*(diff(w(x, t), x))*(diff(w(x, t), t))*w(x, t)^3-2*w(x, t)^3*(diff(w(x, t), x))*(diff(diff(w(x, t), t), x))*a[20]+2*w(x, t)^2*(diff(w(x, t), t))*(diff(w(x, t), x))^2*a[20]-3*w(x, t)^3*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))*a[0]*a[20]-10*w(x, t)^3*(diff(diff(w(x, t), x), x))*(diff(diff(diff(w(x, t), x), x), x))*delta*a[20]-4*w(x, t)^3*(diff(w(x, t), x))*(diff(diff(diff(w(x, t), x), x), x))*delta*a[10]+w(x, t)^3*(diff(w(x, t), x))*(diff(diff(diff(w(x, t), x), x), x))*a[10]*a[20]-5*w(x, t)^3*(diff(w(x, t), x))*(diff(diff(diff(diff(w(x, t), x), x), x), x))*delta*a[20]+30*w(x, t)^2*(diff(diff(w(x, t), x), x))^2*(diff(w(x, t), x))*delta*a[20]+12*w(x, t)^2*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^2*delta*a[10]-5*w(x, t)^2*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^2*a[10]*a[20]+20*w(x, t)^2*(diff(w(x, t), x))^2*(diff(diff(diff(w(x, t), x), x), x))*delta*a[20]-60*w(x, t)*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^3*delta*a[20]-w(x, t)^2*(diff(w(x, t), x))^3*a[10]^2+24*(diff(w(x, t), x))^5*delta*a[20]+(diff(diff(diff(w(x, t), t), x), x))*w(x, t)^4*a[20]+a[10]*(diff(diff(w(x, t), t), x))*w(x, t)^4-2*(diff(w(x, t), x))^5*a[20]^2)/w(x, t)^5 = 0

 (6)
> 

K2 := expand(%)

(diff(diff(w(x, t), x), x))*a[0]*a[10]/w(x, t)+(diff(diff(diff(w(x, t), x), x), x))*a[0]*a[20]/w(x, t)+(diff(diff(diff(diff(diff(w(x, t), x), x), x), x), x))*delta*a[20]/w(x, t)+(diff(diff(diff(diff(w(x, t), x), x), x), x))*delta*a[10]/w(x, t)-3*(diff(diff(w(x, t), x), x))^2*delta*a[10]/w(x, t)^2+(diff(diff(w(x, t), x), x))^2*a[10]*a[20]/w(x, t)^2+(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))*a[10]^2/w(x, t)^2+(diff(diff(w(x, t), x), x))*(diff(diff(diff(w(x, t), x), x), x))*a[20]^2/w(x, t)^2-(diff(w(x, t), x))^2*a[0]*a[10]/w(x, t)^2-3*(diff(diff(w(x, t), x), x))^2*(diff(w(x, t), x))*a[20]^2/w(x, t)^3+2*(diff(w(x, t), x))^3*a[0]*a[20]/w(x, t)^3-(diff(w(x, t), x))^2*(diff(diff(diff(w(x, t), x), x), x))*a[20]^2/w(x, t)^3+5*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^3*a[20]^2/w(x, t)^4-6*(diff(w(x, t), x))^4*delta*a[10]/w(x, t)^4+3*(diff(w(x, t), x))^4*a[10]*a[20]/w(x, t)^4-(diff(diff(w(x, t), x), x))*(diff(w(x, t), t))*a[20]/w(x, t)^2-a[10]*(diff(w(x, t), x))*(diff(w(x, t), t))/w(x, t)^2-2*(diff(w(x, t), x))*(diff(diff(w(x, t), t), x))*a[20]/w(x, t)^2+2*(diff(w(x, t), t))*(diff(w(x, t), x))^2*a[20]/w(x, t)^3+24*(diff(w(x, t), x))^5*delta*a[20]/w(x, t)^5+20*(diff(w(x, t), x))^2*(diff(diff(diff(w(x, t), x), x), x))*delta*a[20]/w(x, t)^3-60*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^3*delta*a[20]/w(x, t)^4-3*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))*a[0]*a[20]/w(x, t)^2-10*(diff(diff(w(x, t), x), x))*(diff(diff(diff(w(x, t), x), x), x))*delta*a[20]/w(x, t)^2-4*(diff(w(x, t), x))*(diff(diff(diff(w(x, t), x), x), x))*delta*a[10]/w(x, t)^2+(diff(w(x, t), x))*(diff(diff(diff(w(x, t), x), x), x))*a[10]*a[20]/w(x, t)^2-5*(diff(w(x, t), x))*(diff(diff(diff(diff(w(x, t), x), x), x), x))*delta*a[20]/w(x, t)^2+30*(diff(diff(w(x, t), x), x))^2*(diff(w(x, t), x))*delta*a[20]/w(x, t)^3+12*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^2*delta*a[10]/w(x, t)^3-5*(diff(diff(w(x, t), x), x))*(diff(w(x, t), x))^2*a[10]*a[20]/w(x, t)^3-(diff(w(x, t), x))^3*a[10]^2/w(x, t)^3+(diff(diff(diff(w(x, t), t), x), x))*a[20]/w(x, t)+a[10]*(diff(diff(w(x, t), t), x))/w(x, t)-2*(diff(w(x, t), x))^5*a[20]^2/w(x, t)^5 = 0

 (7)

NULL

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