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How to solve this PDE (laplace equation)?
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wandering.the.cosmos@gmai
science forum beginner


Joined: 09 Jul 2006
Posts: 1

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

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]^{2-Q} R[a,c]^{2-Q}

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 Q-dimensional 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]^{2-Q} R[x,b]^{2-Q} R[x,c]^{2-Q} d^{Q}x

we can also re-write this integral in fourier space

F[a,b,c] propto int int exp[i vec{k}.(a-c) + i vec{q}.(a-b)] / [
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]^{2-Q} R[a_1,a_3]^{2-Q}
.... R[a_1,a_n]^{2-Q}

Thanks!
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