2 Motivation

Consider an example of the type seen in the earlier material concerning separable solutions of the heat conduction equation. Suppose that $u=u\left(x,t\right)$ is the temperature of a metal bar a distance $x$ from one end and at time $t$ . For the sake of argument let us suppose that the metal bar has length equal to $\ell$ and that the ends are held at constant temperatures $u_L$ at the left and $u_R$ at the right.

Figure 2

We also suppose that the temperature distribution at the initial time is known to be $f\left(x\right)$ , with $f\left(0\right)=u_L$ and $f\left(\ell \right)=u_R$ so that the initial and boundary conditions do not give rise to a conflict at the ends of the bar at the initial time.

This physical situation may be modelled by

$\left.\begin{array}{ccccc}\hfill u_t& \hfill =\hfill & \alpha u_{xx}\hfill & \hfill \hfill & \left(0<x<\ell ,t>0\right)\hfill \\ \hfill u\left(0,t\right)& \hfill =\hfill & u_L\hfill & \hfill \hfill & \left(t>0\right)\hfill \\ \hfill u\left(\ell ,t\right)& \hfill =\hfill & u_R\hfill & \hfill \hfill & \left(t>0\right)\hfill \\ \hfill u\left(x,0\right)& \hfill =\hfill & f\left(x\right)\hfill & \hfill \hfill & \left(0<x<\ell \right)\hfill \end{array}\right\}$

in which $\alpha >0$ is a constant called the thermal diffusivity or simply the diffusivity of the metal. If the bar is made of aluminium then $\alpha =0.86{\text{cm}}^2{\text{s}}^{-1}$ , and if made of copper then $\alpha =1.14{\text{cm}}^2{\text{s}}^{-1}$ .

Using separation of variables and Fourier series (neither of which are required for the remainder of this Section) it can be shown that the solution to the above problem (in the case where $u_L=u_R=0$ ) is

$u\left(x,t\right)={\sum }_{m=1}^{\infty }{B}_m{e}^{-m^2\alpha {\pi }^{2}t∕{\ell }^2}sin\left(m\pi x∕\ell \right),\text{where}{B}_m=\frac{2}{\ell }{\int \; <!--nolimits\; -->}_0^{\ell }f\left(s\right)sin\left(m\pi s∕\ell \right)ds.$

Now, let us be realistic. Any evaluation of $u$ for particular choices of $x$ and $t$ must involve approximating the infinite series that defines $u$ (that is, just taking the first few terms - and care is required if we are to be sure that we have taken enough). Also, in each of the terms we retain in the sum, we need to find $B_m$ by integration. It is not surprising that computation of this procedure is a common approach. So if we (eventually) resort to computation in order to find $u$ , why not start with a computational approach?

(This is not to say that there is no value in the analytic solution involving the $B_m$ . The solution above is of great value, but we simply observe here that there are times when a computational approach is all we may end up needing.)

So, the aim of this Section is to derive methods for obtaining numerical solutions to parabolic problems of the type above. In fact, it is sufficient for our present purposes to restrict attention to that particular problem.