Helmholtz Equation with non-homogeneous b.c.

[HappyEuler2]

Homework Statement

In 2D: (del² + k²) u(r,theta) = 0 with b.c u(R,theta) = f(theta)

Starting from the general solution (by separation of variables) show that the solution can be rewritten as:

$$\int$$ K(r,theta,theta')*f(theta') dtheta' from 0 to 2*Pi

Homework Equations

The general solution is J_m(k*r)*P_m(cos(theta)) where P_m are the Legendre polynomials and J_m are the Bessel functions

The Attempt at a Solution

Ok so I've attempted this on three different occasions, following the professor's instructions to the problem I end up with a relationship J_m(kR) P_m(cos(theta)) = f(theta) where I need to find the values for k, but this would require some type of inversion for the Bessel functions. So I am thinking this isn't the case.

Alternatively, if I try to define the solution u(r,theta) = v(theta) + w(r,theta) I end up getting confused on how to properly deal with the v(theta) term. I know it needs to obey the boundary conditions so I can rewrite the equation as an inhomogeneous helmholtz equation in terms of w(r,theta) and then find the solution as an integral involving the appropriate Green's function.

Any help?

 

[mighty2000]

Hello!

I would approach this problem by first understanding the equations and variables involved. The equation given is a 2D Laplace's equation with a constant k, which can represent the wave number or a diffusion coefficient. The boundary condition given is a function of theta, which suggests that the solution is dependent on the angular coordinate.

To solve this, we can use the method of separation of variables, where we assume the solution can be written as a product of two functions, one dependent on r and the other on theta.

u(r,theta) = R(r)*Θ(theta)

Substituting this into the Laplace's equation, we get:

R''(r)*Θ(theta) + (1/r)*R'(r)*Θ(theta) + k^2*R(r)*Θ(theta) + R(r)*Θ''(theta) = 0

Dividing by R(r)*Θ(theta), we get:

(R''(r) + (1/r)*R'(r) + k^2*R(r)) + (Θ''(theta)/Θ(theta)) = 0

Since the two terms on the left-hand side are independent of each other, they must be equal to a constant, which we can denote as -λ^2.

R''(r) + (1/r)*R'(r) + (k^2 - λ^2)*R(r) = 0

Θ''(theta) + λ^2*Θ(theta) = 0

Now, we can solve these two ordinary differential equations separately. The solution for R(r) is given by the Bessel functions, as mentioned in the problem statement. The solution for Θ(theta) is given by the Legendre polynomials.

We can write the general solution as:

u(r,theta) = Σ[Am*Jm(k*r)*Pm(cos(theta))]

Now, to satisfy the boundary condition u(R,theta) = f(theta), we can substitute r = R in the above equation and equate it to f(theta). This will give us the coefficients Am, which will depend on the function f(theta).

Therefore, the solution can be rewritten as:

u(r,theta) = Σ[Am*Jm(k*r)*Pm(cos(theta))]

= Σ[Am*Jm(k*r)]*