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6.2: Inhomogeneous Heat Equation

  • Page ID
    2157
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    Here we consider the initial value problem for \(u=u(x,t)\), \(u\in C^\infty(\mathbb{R}^n\times R_+)\),

    \begin{eqnarray*}
    u_t-\triangle u&=&f(x,t)\ \ \mbox{in}\ x\in\mathbb{R}^n,\ t\ge0,\\
    u(x,0)&=&\phi(x),
    \end{eqnarray*}

    where \(\phi\) and \(f\) are given. From

    $$ \widehat{u_t-\triangle u}=\widehat{f(x,t)} \]

    we obtain an initial value problem for an ordinary differential equation:

    \begin{eqnarray*}
    \frac{d\widehat{u}}{dt}+|\xi|^2\widehat{u}&=&\widehat{f}(\xi,t)\\
    \widehat{u}(\xi,0)&=&\widehat{\phi}(\xi).
    \end{eqnarray*}

    The solution is given by

    $$\widehat{u}(\xi,t)=e^{-|\xi|^2 t}\widehat{\phi}(\xi)+\int_0^t\ e^{-|\xi|^2(t-\tau)}\widehat{f}(\xi,\tau)\ d\tau.\]

    Applying the inverse Fourier transform and a calculation as in the proof of Theorem 5.1, step (vi), we get}

    \begin{eqnarray*}
    u(x,t)&=&(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}\Big(e^{-|\xi|^2t}\widehat{\phi}(\xi)\\
    &&\ \ +\int_0^t\ e^{-|\xi|^2(t-\tau)}\widehat{f}(\xi,\tau)\ d\tau\Big)\ d\xi.
    \end{eqnarray*}

    From the above calculation for the homogeneous problem and calculation as in the proof of Theorem 5.1, step (vi), we obtain the formula

    \begin{eqnarray*}
    u(x,t)&=&\frac{1}{(2\sqrt{\pi t})^n}\int_{\mathbb{R}^n}\ \phi(y)e^{-|y-x|^2/(4t)}\ dy\\
    & &+\int_0^t \int_{\mathbb{R}^n}\ f(y,\tau)\frac{1}{\left(2\sqrt{\pi(t-\tau)}\right)^n}\ e^{-|y-x|^2/(4(t-\tau))}\ dy\ d\tau.
    \end{eqnarray*}

    This function \(u(x,t)\) is a solution of the above inhomogeneous initial value problem provided

    $$\phi\in C(\mathbb{R}^n),\ \ \sup_{\mathbb{R}^n}|\phi(x)|<\infty\]

    and if

    $$f\in C(\mathbb{R}^n\times[0,\infty)),\ \ M(\tau):=\sup_{\mathbb{R}^n}|f(y,\tau)|<\infty,\ 0\le\tau<\infty.\]

    Contributors and Attributions


    This page titled 6.2: Inhomogeneous Heat Equation is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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