Question: How find parameters of with two independent variable ?

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)

NULL

T := f(x, t) = g(x, t)^2+h(x, t)^2+a[0]

T1 := g(x, t) = t*n[1]+x*k[1]

T2 := h(x, t) = t*n[2]+x*k[2]

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

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

(14)

L3 := eval(pde2, L2)

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

(15)

L4 := collect(L3, [x, t], 'distributed')

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

(16)

eqs := {coeffs(L4, [x, t])}

Error, invalid arguments to coeffs

 

NULL

NULL

ans := solve(eqs, vars)

{a[2] = a[2], a[3] = a[3], a[4] = 0, a[5] = a[5], a[7] = a[7]}

(17)

NULL

eqI := ans

{a[2] = a[2], a[3] = a[3], a[4] = 0, a[5] = a[5], a[7] = a[7]}

(18)

eqpsi := eval(L2, eqI)

f(x, t) = (t*a[2]+a[3])^2+a[5]^2*t^2+a[7]

(19)

eqphi := eval(Y, eqpsi)

w(x, t) = 0

(20)

simplify(eval(pde, eqphi))

 

NULL

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