Diffusion Equation by Method of Images

[secret_user]

Homework Statement

I need help in solving a problem I was assigned from Numerical Methods for Physics, 2nd Ed., by Garcia. We are asked to create a solution, by hand, for the diffusion equation, using the method of images. In particular, we have a 1-dimensional bar, centered at x = 0, of length = L. Our initial condition is a Dirac delta heat source at x = L/4. We have Neumann boundary conditions (the spatial derivatives of the temperature at the ends of the bar are zero).

Homework Equations

We know that the Gaussian is a solution, so we use Gaussians in our method of images.

The Attempt at a Solution

I was able to solve the situation for a Dirac delta heat source at x = 0. Basically, I drew a picture of the spatial derivative of the Gaussian, centered at x = 0. For the Neumann boundary conditions, I know that I just add identical images of the original Gaussian at increments of nL (n being an index). However, for the case where the Gaussian is centered at x = L/4, I am having trouble creating a series solution using the method of images. I've tried several variations of pictures (derivatives of Gaussians along an x-axis), but none of them give me the spatial derivative to be zero at the end points (x = +-L/2). I would greatly appreciate any help on this problem.

 

[Jhero]

Dear fellow scientist,

I understand your struggle with solving the diffusion equation using the method of images for a Gaussian centered at x = L/4. This can be a challenging problem, but with the right approach, it can be solved.

First, let's review the method of images. This method involves creating a series of images of the original source, each with a weight that depends on the distance from the original source. For a Dirac delta heat source at x = 0, the images are simply copies of the original source at increments of nL. However, for a Gaussian source centered at x = L/4, the images become more complicated.

To start, let's look at the Gaussian solution you mentioned. The Gaussian can be represented by the following equation:

G(x) = A*exp(-x^2/(2*sigma^2))

Where A is the amplitude and sigma is the standard deviation. Now, let's consider the Gaussian centered at x = L/4. This can be represented by the following equation:

G(x) = A*exp(-(x-L/4)^2/(2*sigma^2))

To satisfy the Neumann boundary conditions, we need the spatial derivative of this Gaussian to be zero at x = +-L/2. This means that the derivative of the Gaussian at these points should be equal to the derivative of the original Gaussian at x = L/4.

To achieve this, we need to add images of the Gaussian at increments of nL, each with a weight that depends on the distance from the original source. However, since the original source is now shifted to x = L/4, the weights for each image will also be shifted. The general formula for the weight of the nth image can be represented as follows:

w_n = (-1)^n * exp(-(nL-L/4)^2/(2*sigma^2))

Using this formula, we can create a series of images that will satisfy the Neumann boundary conditions. However, since the number of images is infinite, we need to truncate the series and only consider a finite number of images. This can be done by choosing an appropriate value for sigma.

I hope this explanation helps you in solving the problem. If you have any further questions, please don't hesitate to ask. Good luck!