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Separation of Variables I: Dirichlet Boundaries - Partial Differential Equations | Lecture 6
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We are finally in a position to solve the heat equation! In this lecture we solve the heat equation with Dirichlet, or fixed, boundary conditions. We introduce the method of separation of variables and show that it results in an infinite series solutions whose coefficients are determined by the initial condition. Identifying these coefficients leads to a discussion of the orthogonality of sine modes, which in turn presents the Fourier sine series. Different initial conditions are illustrated, as well as a description of the long-time behaviour.
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