Title:

Order of the Convergence of the Partial Differential Equation for Heat Diffusion

ABSTRACT: In this project, the partial differential equation for the Heat Equation was explored using analytical methods to approximate solutions. The results shown on this poster were found using an implicit method known as the Crank Nicholson to obtain graphs on MatLab that illustrate the decay of heat on a bar over time. These results were compared to the theoretical solution found by solving this PDE by making an assumption that it is a product of two functions to use separation of variables. Further on, we are able to solve the left hand side of the equation using the ordinary differential equation method of separation of variables. The theoretical results showed that there should be exponential decay as time increases with faster decay for higher frequency modes. The numerical results illustrate this, given an initial condition of U(x,0) = 10sin(4x) + 4sin(2x) with boundary conditions U(0,t)=U(1,t)=0.

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