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    Partial Differential Equations - Heat Equation

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    The causal solution f(x t) of the one-dimensional diffusion equation with source term f

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    The causal solution of the one-dimensional diffusion equation with source term f

    (1)
    is given by

    (2)

    where is the causal Green function satisfying

    (3)

    a) Change variables in (3) to , . Letting with the Heaviside step function , show (3) is solved if satisfies

    (4)
    Solve by Fourier transforming with respect to the position variable. Hence obtain an expression for
    b) The temperature of a long uniform insulating metal rod satisfies

    A heating element is used to create an initial temperature distribution Writing obtain an equation for , and combining (1), (2) and the answer to a) identify the temperature variation at the center of the rod, .

    Solution:
    a) After changing the variables, equation (3) becomes:
    (5)
    and putting
    (6)
    one yields:
    (7)
    where the following property of Dirac delta distribution was used:
    (7')
    The other partial derivative will be:
    ...

    Solution Summary

    The inhomogeneous heat equation (partial differential equation) in 1D can be solved by means of Green function

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