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Question: test solution of PDE

Question:test solution of PDE

salim-barzani 160 Maple 2021
pde differential-equations pdetest duplicate-question
July 21 2024
0
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restart
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with(PDEtools)
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undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

 (1)
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with(DEtools)
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NULL
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with(DifferentialAlgebra)
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"with(Student[ODEs][Solve]): "
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with(IntegrationTools)
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with(inttrans)
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with(PDEtools)
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with(Physics)
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with(PolynomialTools)
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with(RootFinding)
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with(SolveTools)
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with(LinearAlgebra)
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with(sumtools)
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pde := I*(diff(psi(x, t), t))+alpha*(diff(psi(x, t), `$`(x, 2)))+(beta[3]*abs(psi(x, t))+beta[4]*abs(psi(x, t))^2)*psi(x, t)+gamma*(diff(abs(psi(x, t))^2, `$`(x, 2)))*psi(x, t)/abs(psi(x, t)) = 0
> 

case1 := {k = k, lambda = sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3], w = -(9*alpha*k^2*beta[4]+2*beta[3]^2)/(9*beta[4]), A[0] = -beta[3]/(3*beta[4]), A[1] = beta[3]/(3*beta[4]), B[1] = 0}
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" psi(x,t):=U(xi)*exp(I*(-k*x+w*t+theta))"

proc (x, t) options operator, arrow, function_assign; U(xi)*exp(I*(-k*x+w*t+theta)) end proc

 (2)
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" U(xi):=-(beta[3] (cosh(xi)-sinh(xi)))/(3 beta[4] cosh(xi))"

proc (xi) options operator, arrow, function_assign; -(1/3)*beta[3]*(cosh(xi)-sinh(xi))/(beta[4]*cosh(xi)) end proc

 (3)
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convert(U(xi), trig)

-(1/3)*beta[3]*(cosh(xi)-sinh(xi))/(beta[4]*cosh(xi))

 (4)
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xi := sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*kt+x)

(-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x)

 (5)
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S := psi(x, t)

-(1/3)*beta[3]*(cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x))-sinh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x)))*exp(I*(-k*x+t*w+theta))/(beta[4]*cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x)))

 (6)
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solution := subs(case1, S)

-(1/3)*beta[3]*(cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x))-sinh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x)))*exp(I*(-k*x-(1/9)*(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/beta[4]+theta))/(beta[4]*cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*kt+x)))

 (7)
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pdetest(psi(x, t) = -beta[3]*(cosh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x))-sinh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x-(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/(9*beta[4])+theta))/(3*beta[4]*cosh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x-(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/(9*beta[4])+theta)), pde)

Error, (in pdetest) unable to determine the indeterminate function

  
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NULL

 

 

 

 

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