# Diffusion Equation by Method of Images

#### secret_user

1. 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).

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

3. 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.

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"Diffusion Equation by Method of Images"

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