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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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    © BrainMass Inc. brainmass.com December 24, 2021, 11:32 pm ad1c9bdddf
    https://brainmass.com/physics/partial-differential-equation/partial-differential-equations-heat-equation-579736

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    SOLUTION This solution is FREE courtesy of BrainMass!

    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:
    (8)
    By replacing (7) and (8) in (5), we will have:
    (9)
    which can be written also as
    (10)
    This expression is equivalent to the following equations simultaneously satisfied:
    (11)
    (proof of statement (4))
    One applies now the Fourier transform on the homogeneous equation (11) with respect to variable "z", using the formula:
    (12)
    and the property of derivative:
    (13)
    It follows that
    (14)
    and (15)
    Applying the Fourier transform to the initial condition, we will get:
    (16)
    where the Fourier transform applied to Dirac delta distribution yields "1".
    The equation (11) becomes a simple ordinary differential equation of 1'st order:
    (17)
    whose general solution is:
    (18)
    By applying the initial condition, it follows that C0 = 1, so that
    (19)
    The inverse of Fourier transform (12) is
    (20)
    and, if applied to (19), yields:

    In the last integral, the variable change:
    (21)
    yields
    (22)
    Substituting in (6), the expression of Green function will be found:
    (23)
    or
    (24)
    b) It is required to solve the p.d. equation:
    (25)
    with the initial condition
    (26)
    One considers the function:
    (27)
    whose partial derivatives are:
    (28)
    (29)
    It follows that

    Since the last parenthesis is null (according to (25)), the equation to be solved is:
    (30)
    which is a inhomogeneous p.d. equation of general form
    (31)
    The solution of this kind of equation is expressed by formula:
    (32)
    where the Green function G is given by (24) and
    (33)
    Replacing with the appropriate expressions, the integral (32) becomes:
    (34)
    Using again the property (7') of Dirac distribution as well as the property
    (35)
    the integral (34) becomes:

    (36)
    Comparing to (27), one deduces that
    (37)
    The temperature in the center of the rod will be given by:
    (38)
    Using a new variable change, the Poisson integral will be again obtained:
    (39)

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

    © BrainMass Inc. brainmass.com December 24, 2021, 11:32 pm ad1c9bdddf>
    https://brainmass.com/physics/partial-differential-equation/partial-differential-equations-heat-equation-579736

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