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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.