wandering.the.cosmos@gmai science forum beginner
Joined: 09 Jul 2006
Posts: 1

Posted: Sun Jul 09, 2006 4:20 am Post subject:
How to solve this PDE (laplace equation)?



I am looking for suggestions on how to solve this differential equation
in Q dimensions:
nabla^2_a F[a,b,c] = R[a,b]^{2Q} R[a,c]^{2Q}
with boundary condition F > 0 as any of the a or b or c going to
infinity.
Here F[a,b,c] is a function of 3 Qdimensional vectors a,b,c, nabla^2_a
is the laplacian with respect to the vector a, and R[x,y] is the
Euclidean distance between x and y
R[x,y] = (sum_{i=1}^Q (x^i  y^i)^2 )^{1/2}
Up to constant factors I believe the solution can be expressed as an
integral
F[a,b,c] propto int R[x,a]^{2Q} R[x,b]^{2Q} R[x,c]^{2Q} d^{Q}x
we can also rewrite this integral in fourier space
F[a,b,c] propto int int exp[i vec{k}.(ac) + i vec{q}.(ab)] / [
vec{k}.vec{k} vec{q}.vec{q} (vec{q}+vec{k}).(vec{q}+vec{k}) ] d^{Q}k
d^{Q}q
but is it possible to perform these integrals explicitly?
I'm also interested if I could solve a generalized version of this
equation  with arbitrary number of variables
nabla^2_{a_1} F[a_1,a_2,...,a_n] = R[a_1,a_2]^{2Q} R[a_1,a_3]^{2Q}
.... R[a_1,a_n]^{2Q}
Thanks! 
