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Question: Maybe not the right place to tell about that?

Question:Maybe not the right place to tell about that?

mmcdara 6149 Maple
pde pdsolve differential-equations
February 02 2019
3

I'm presently interested in PDE and I have just discovered the impressive work Nasser Abbasi,has done, and keeps doing, concerning the solution of PDE benchmarks with Maple and Mathematica.
https://www.12000.org/my_notes/pde_in_CAS/pdse3.htm

It seems that the result Maple returns for the test case below is not the general solution.

  • 4.19 first order PDE of three unknowns
    problem number 19
    (from example 3.5.4, p 212, nonlinear ode’s by Lokenath Debnath, 3rd edition)


It is rather simple to see that any spherical function u(x, y, z) =f(x^2+y^2+z^2) is a solution of the PDE.
Then any function of the form u(x, y, z) = f(x^2+y^2+z^2) * exp(x+y+z) *C1 (C1 being any constant) is also a solution.
Maple returns only the solution u(x, y, z) = exp(C2*(x^2+y^2+z^2)) * exp(x+y+z) * C1


 

> 

restart:
> 

u := (x, y, z) -> f(x^2+y^2+z^2)

proc (x, y, z) options operator, arrow; f(x^2+y^2+z^2) end proc

 (1)
> 

expr := (y-z)*diff(u(x, y, z), x)+(z-x)*diff(u(x, y, z), y)+(x-y)*diff(u(x, y, z), z)

2*(y-z)*(D(f))(x^2+y^2+z^2)*x+2*(z-x)*(D(f))(x^2+y^2+z^2)*y+2*(x-y)*(D(f))(x^2+y^2+z^2)*z

 (2)
> 

simplify(%);

0

 (3)
> 

u := (x, y, z) -> C*f(x^2+y^2+z^2)*exp(x+y+z)

proc (x, y, z) options operator, arrow; C*f(x^2+y^2+z^2)*exp(x+y+z) end proc

 (4)
> 

expr := (y-z)*diff(u(x, y, z), x)+(z-x)*diff(u(x, y, z), y)+(x-y)*diff(u(x, y, z), z)

(y-z)*(2*C*(D(f))(x^2+y^2+z^2)*x*exp(x+y+z)+C*f(x^2+y^2+z^2)*exp(x+y+z))+(z-x)*(2*C*(D(f))(x^2+y^2+z^2)*y*exp(x+y+z)+C*f(x^2+y^2+z^2)*exp(x+y+z))+(x-y)*(2*C*(D(f))(x^2+y^2+z^2)*z*exp(x+y+z)+C*f(x^2+y^2+z^2)*exp(x+y+z))

 (5)
> 

simplify(%);

0

 (6)
> 

pdsolve((y-z)*diff(U(x, y, z), x)+(z-x)*diff(U(x, y, z), y)+(x-y)*diff(U(x, y, z), z)=0, U(x,y,z),'build')

U(x, y, z) = exp((1/2)*x^2*_C2)*exp(_C1*x)*exp((1/2)*y^2*_C2)*exp(_C1*y)*_C3*_C5*_C4*exp((1/2)*z^2*_C2)*exp(_C1*z)

 (7)
> 

combine(rhs(%), exp)

exp((1/2)*x^2*_C2+_C1*x+(1/2)*y^2*_C2+_C1*y+(1/2)*z^2*_C2+_C1*z)*_C3*_C5*_C4

 (8)
> 

 


 

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