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    Heat Equation : Moving Source - Dirac Impulse Function

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    Use superposition to solve:

    with boundary conditions:

    and initial condition

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    https://brainmass.com/math/calculus-and-analysis/heat-equation-moving-source-dirac-impulse-function-32916

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    Use superposition to solve:

    with boundary conditions:

    and initial condition

    .

    Solution:

    Let's consider a more general problem of heat equation of form:
    ( 1)
    with initial condition:
    ( 2)
    The solution of this equation can be expressed as a superposition of 2 elemantary solutions:
    ( 3)
    where
    u1 = the general solution of homogenous associated equation of (1)
    u2 = a particular solution of non-homogenous equation (1)
    There are several methods to determine the solution of (1), classical and modern, using the theory of distributions.
    I prefer to use the theory of distributions, so that I will try to sketch the steps to determine the complete solution.
    We will use the following definitions and integral transforms:

    1) The "fundamental solution" of (1), denoted by (E) is the solution of associated equation:
    ...

    Solution Summary

    A heat equation with a moving source is investigated using the Dirac Impulse Function. The solution is detailed and well presented.

    $2.19

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