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Mathematics LibreTexts

6.4: Initial-Boundary Value Problems

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    Consider the initial-boundary value problem for \(c=c(x,t)\)

    c_t&=&D\triangle c\ \ \mbox{in}\ \Omega\times (0,\infty)\\
    c(x,0)&=&c_0(x)\ \ x\in\overline{\Omega}\\
    \frac{\partial c}{\partial n}&=& 0\ \ \mbox{on}\ \partial\Omega\times (0,\infty).

    Here is \(\Omega\subset\mathbb{R}^n\), \(n\) the exterior unit normal at the smooth parts of \(\partial\Omega\), \(D\) a positive constant and \(c_0(x)\) a given function.

    Remark. In application to diffusion problems, \(c(x,t)\) is the concentration of a substance in a solution, \(c_0(x)\) its initial concentration and \(D\) the coefficient of diffusion.
    The first Fick's rule says that

    \[w=D\partial c/\partial n,\]

    where \(w\) is the flow of the substance through the boundary \(\partial\Omega\). Thus according to the Neumann boundary condition (\ref{sol3}), we assume that there is no flow through the boundary.