3-D Helmholtz (1 Viewer)

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lth

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1. The problem statement, all variables and given/known data

Given 3D Helmholtz eqn.
u_xx + U-yy + U_zz + Lamda*u = 0 ,Lamda > 0.

We are asked to "Calculate the fundamental solution directly (without using the Bessel identity for J_1/2 given)"
where:
Bessel identity given is w(r)=C_n*r^-(n-2)/2*J-(n-2)/2*(Lamda^1/2*r) ,n=odd ... and in this case n=3.

2. Relevant equations

fundamental soln is u = (1/4*Pi*r_PQ)*Cos(Lamda^1/2*r_PQ)


3. The attempt at a solution

After transforming to r-space and setting delta functions on RHS
u_rr + (2/r) * u_r + lambda * u = delta_3(r)

A second transformation from letting u(r) = r^1/2*v(theta) ;theta =Lamda^1/2*r
yields:

d^2v/d{theta}^2 + (1/theta)*dv/d{theta} +[1-1/4*theta]*v =0
and this is Bessels eqn. form.

This is from Kevorkian PDE text p2.3.4 and I feel like i am running in circles with this material and trying to figure how to solve the fundamental solution without using Bessels identity. Can someone give me some insight into what to do next?

thanks, lth
 

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