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Unformatted text preview: t = 0 t = . 04 t = . 08 t = . 12 t = . 16 t = . 2 Figure 12.6. Heat Diffusion in a Disk. 12.5. The Fundamental Solution of the Heat Equation. As we learned in Section 4.1, the fundamental solution to the heat equation measures the temperature distribution resulting from a concentrated initial heat source, e.g., a hot soldering iron applied instantaneously at a single point on a metal plate. The physical problem is modeled mathematically by taking a delta function as the initial data along with the relevant homogeneous boundary conditions. Once the fundamental solution is known, one is able to use linear superposition to recover the solution generated by any other initial data. As in our one-dimensional analysis, we shall concentrate on the most tractable case, when the domain is the entire plane: = R 2 . Thus, our first goal is to solve the initial value problem u t = u, u (0 , x, y ) = ( x ) ( y ) , (12 . 121) for t > 0 and ( x, y ) R 2 . The solution u = F ( t, x ; ) = F ( t, x, y ; , ) to this initial value problem is known as the fundamental solution for the heat equation on R 2 . The quickest route to the desired solution relies on the following means of combining solutions of the one-dimensional heat equation to produce solutions of the two-dimensional version....
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