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    Partial Differential Equation : Diffusion Equation and Explicit Series Solution

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    Consider the diffusion equation
    ut = ku.xx for 0 < < pi and t > 0 with the boundary conditions
    ux(0, t) = 0 and u(pi, t) = 0
    and the initial condition
    u(x,0) = 1.
    (a) Find the separated solutions satisfying the differential equation and boundary conditions.
    (b) Use these solutions to write an explicit series solution to the differential equation satisfying the boundary conditions and the initial condition.

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    (a) Since this is a diffusion equation, we have k>0. We use separated variable method. Assume that its solution . Then


    By , we have

    Letting ,where then we have

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