(adsbygoogle = window.adsbygoogle || []).push({}); 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: 3-D Helmholtz

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**