Explore BrainMass

Explore BrainMass

    Partial Differential Equations - Heat Equation

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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

    See attached

    © BrainMass Inc. brainmass.com March 5, 2021, 1:19 am ad1c9bdddf


    Solution Preview

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

    is given by


    where is the causal Green function satisfying


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

    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, .

    a) After changing the variables, equation (3) becomes:
    and putting
    one yields:
    where the following property of Dirac delta distribution was used:
    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