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      <text>25.2.2.1 Heat conduction equation</text>
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<h3>Heat conduction equation</h3>
<p>Consider a long thin bar, or wire, of constant cross-section and of homogeneous material oriented along the \(x\)-axis (see figure below).</p>
<p><br></p>
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    <img src="@@PLUGINFILE@@/workbook25fig3-1%20%282%29.svg" alt="Image of a thin bar of length L" role="presentation">
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<p>Imagine that the bar is thermally insulated laterally and is sufficiently thin that heat flows (by conduction) only in the \(x\)-direction. Then the temperature \(u\) at any point in the bar depends only on the \(x\)-coordinate of the point and the time
    \(t\). By applying the principle of conservation of energy it can be shown that \(u(x,t)\) satisfies the PDE

</p>
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            <td>\(\qquad \dfrac{\partial u}{\partial t}=k\dfrac{\partial^2u}{\partial x^2}\qquad \begin{array}{c}0\le x\le L\\ t\ge 0\end{array}\)</td>
            <td>\((3)\)</td>
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where \( k\) is a positive constant. In fact \(k\), sometimes called the <strong>thermal diffusivity</strong> of the bar, is given by
<p>\(\qquad k=\dfrac{\kappa}{s\rho}\)</p> where
<p>\(\qquad \kappa=\) thermal conductivity of the material of the bar</p>
<p>\(\qquad s=\) specific heat capacity of the material of the bar</p>
<p>\(\qquad \rho=\) density of the material of the bar. </p>The PDE \((3)\) is called the one-dimensional heat conduction equation (or, in other contexts where it arises, the <strong>diffusion equation</strong>).
<p></p>
<hr>
<h4 class="HELM_example">Example</h4>
<p>What is the obvious difference between the wave equation \((1)\) and the heat conduction equation \((3)\)?</p>
<h4 class="HELM_solution">Solution</h4>
<p>Both equations involve second derivatives in the space variable \(x\) but whereas the wave equation has a <i>second</i> derivative in the time variable \(t\) the heat conduction equation has only a <i>first</i> derivative in \(t\). This means that the
    solutions of \((3)\) are quite different in form from those of \((1)\) and we shall study them separately later.
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