Cooling Fin project

For the cooling fin project, we combined several physical laws (Newton's and Fourier's) governing the flow of temperature through a rectangular cooling fin and approximated it using the finite difference method. In parts 2, 4, and 5 we experimented with larger fins, and fins made of more conductive materials, and found that these fins could distribute heat more effectively and support higher temperatures. Part 3 showed that positioning the heat source towards the middle of the fin also favored more even heat distribution. Experimenting with different values of M and N in part 2 showed that sparser grids overestimated the fin's temperature but that increasing the point density refined this estimate downwards. In part 6, submerging the copper fin in a denser medium (water) greatly increased the heat loss due to convection, allowing for a fivefold increase in power.

Laplace project

For the Laplace Equation project, we investigaed the special case of PDEs where the Laplacian (the sum of the x and y second derivatives) was zero. First, we recreated Example 8.8 by finding the solution for the Laplace equation with continuous boundary conditions. Next, as a warmup exercise, we looked at the heat distribution across a copper plate with the edge temperatures kept constant. Finally, we moved onto the meat of the problem: finding a solution for the case where the boundary condition on one side was discontinuous with the other three. By comparing our finite difference method approximation with a more precise infinite series derivation, we were able to confirm the accuracy of our solution.

Division of Labor

Daniel primarily worked on the Cooling Fin project and designed the website.
Zak primarily worked on the Laplace Equations project and wrote up the analysis for it.