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    The Heat Equation on a Metal Ring

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    Let u(x,t) describe the temperature of a thin metal ring with circumference 2pi. For convenience, let's orient the ring so that x spans the interval |-pi, pi|. Suppose that the ring has some internal heating that is angle-dependent, so that u(x, t) satisfies the inhomogeneous heat equation

    u_t = ku_zz + f(x),

    where k is the thermal diffusivity and f(x) describes the internal heating. Furthermore, assume that the temperature of the ring is initially zero.

    a) Because the temperature u(x, t) is parameterized by the angle x, the temperature must be a 2pi-periodic function. This suggests that we should write the temperature as a complex Fourier expansion with time-dependent coefficients,

    u(x, t) = SUM (A_n)e^(inz).

    Substitue this expression into the heat equation, and obtain a differential equation for each (A_n)(t). Using the correct initial conditions, solve each ODE and write down the final solution.

    b) Suppose f(x) = cos^2 (x). Find an explicit solution for u(x, t). (Hint: Expand cos^2(x) into a finite number of complex Fourier modes, and show that all but three of the A_n are always zero.)

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    Solution Summary

    The solution shows how to solve teh Heat equation of a circular ring. The solution is 5 pages long including derivations.