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Math Help - Laplace transform of Diffusion Equation

  1. #1
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    Laplace transform of Diffusion Equation

    D(t)\frac{\partial^{2}C}{\partial x^{2}}=\frac{\partial C}{\partial t},
    I've been using the Laplace transform to solve the above equation with certain boundary conditions. Ive done this successfully when D is constant. However, I have now figured out that D is infact not constant in the system im looking at, it depends on t. The exact form of D(t) is not known, but I know which value it starts for and that it decreases towards another value. My thinking was to choose a D(t) that made the problem solvable. My initial guess was:
    D(t)=D_{1}+D_{2}(1-e^{\frac{-k}{t}})
    which leads me to figure out
    \mathcal{L}\left\{ D(t)\frac{\partial^{2}C}{\partial x^{2}}\right\}
    or
    <br />
\mathcal{L}\left\{ \frac{1}{D(t)}\frac{\partial C}{\partial t}\right\}?
    Is there any easy way to do this, for example some rules that let me isolate \mathcal{L}\left\{D(t)\right\}
    Last edited by TboneWalker; March 1st 2011 at 07:00 AM.
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  2. #2
    A Plied Mathematician
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    I can't say I know exactly how to solve your problem, but I think Chapter 7 of Crank's The Mathematics of Diffusion might be useful.
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