Solving the 3D Helmholtz Equation Directly

[lth]

Homework Statement

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.

Homework Equations

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

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

 

[mighty2000]

ep

Dear lthep,

Thank you for sharing your attempt at solving the problem. It seems like you are on the right track in transforming the Helmholtz equation into Bessel's equation form. From here, you can use the standard solutions for Bessel's equation, which are Bessel functions of the first and second kind (J and Y functions). The fundamental solution for the Helmholtz equation can then be written as a linear combination of these Bessel functions. You can also use the Wronskian to determine the coefficients in the linear combination.

I hope this helps guide you towards finding the fundamental solution without using the Bessel identity. Good luck with your work!