Re: differential equation


Robert Israel a écrit :

In article <1160487017.805735.50270@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Nico <nicograb@xxxxxxxx> wrote:

alainverghote@xxxxxxxx ha scritto:



you may try functions f(x,y,z) = d*exp(a*x+b*y+c*z)
with a^2+b^2+c^2 = A . a,b,c,d constant numbers,
and study functions g such as laplacian g(x,y,z) = 0
(sum of trig.) ,


The question now is:

Is it true that "laplacian f(x,y,z) = A * f(x,y,z) "
implies
" there exists a,b,c,d constant numbers such that f =
d*exp(a*x+b*y+c*z) + g(x,y,z) with a^2+b^2+c^2 = A and laplacian g
(x,y,z) = 0 ?

No, it neither implies nor is implied by that.

You might look up "Helmholtz equation". Solutions include, but are
not limited to, linear combinations of functions exp(a x + b y + c z)
with a^2 + b^2 + c^2 = A (including complex values of a, b and c).
I'm not sure what Alain had in mind about laplacian g(x,y,z) = 0,
except that harmonic functions are the case A = 0 of your equation.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Dear Robert, bonjour,

" laplacian g(x,y,z) = 0,
except that harmonic functions are the case A = 0 of your equation "
It is exactly what I did mean .
Well, we may rely on laplacian properties :
linearity : Lap(f+g)=Lap(f) +Lapg(g) and any
opposition : Lap(g( -x,y,z))=Lap(g(-x,-y,z) ....Lap(g(x,y,z))

Alain

.

Relevant Pages

  • Re: differential equation
    ... and study functions g such as laplacian g= 0 ... it neither implies nor is implied by that. ... You might look up "Helmholtz equation". ... I'm not sure what Alain had in mind about laplacian g= 0, ...
    (sci.math)
  • Re: differential equation
    ... and study functions g such as laplacian g= 0 ... (sum of trig.) ...
    (sci.math)
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