Partial differential equations

This module computes and plots solutions of the wave equation

$$u_{tt} = c^2 u_{xx}$$

and the heat (or diffusion) equation

$$u_t = k u_{xx} .$$

Solutions are found on an interval $0 \leq x \leq L$, for times $0 \leq t \leq T$. The method is separation of variables.

When you select Partial differential equations from the Iode main menu, the Partial differential equations window opens up, showing two graphs. The top graph shows a 3D plot of a solution of either the wave or heat equation. Try rotating this 3D plot, by dragging the mouse across the graph...

The equation, boundary conditions and initial conditions are written above the top graph.

The bottom graph is a cross-section of the 3D solution graph: either a t-snapshot, which shows the solution as a function of $x$ at some time $t$, or else a x-section, which shows the solution as a function of $t$ at some position $x$. Try stepping through these snapshots and sections with the ``arrow'' buttons, or else enter a coordinate value into the box.

Controls

You can rotate the 3D plot (the top graph), by dragging the mouse across it.

The following control buttons relate to the bottom graph only.

• Plot t-snapshots: these are graphs of the solution as a function of $x$, at various computed times $t$. The computed times are determined by the resolution (see Options below).
• Plot x-sections: these are graphs of the solution as a function of $t$, at various computed positions $x$. The computed times are determined by the resolution (see Options below).
• Current coordinate: You can step through the snapshots and sections using the ``arrow'' buttons, or else you can enter a coordinate value inside the box (Iode will round your input to the nearest computed value of the coordinate).

Equation menu

• Enter equation and boundary conditions. First you choose the type of differential equation: Wave, Heat or Other. The Other option lets you call on user-created modules for handling other differential equations. You can create such modules by copying and modifying the files wave.m or heat.m in your Iode directory.

  Next you choose the boundary conditions:

  • Dirichlet: $u(0,t)=0$ and $u(L,t)=0$ for all $t$,
  • Neumann: $u_x(0,t)=0$ and $u_x(L,t)=0$) for all $t$,
  • Periodic: $u(0,t)=u(L,t)$ and $u_x(0,t)=u_x(L,t)$ for all $t$, or
  • Other e.g., you might create a module for mixed boundary conditions. To create such a boundary conditions module, just copy and suitably modify the file dirichlet.m.
• Enter parameters and initial data: this will prompt you for the wavespeed $c$ (if you are working with the wave equation) or the thermal diffusivity $k$ (if you are working with the heat equation). Then it prompts you for the length $L$ of the interval $0 \leq x \leq L$ on which you are solving the equation, and for the duration $T$ of the time interval $0 \leq t \leq T$ on which you want to examine the solution.

  Finally you are asked to enter the intial data, which consists of the initial displacement $u(x,0)=f(x)$ and the initial velocity $u_t(x,0)=g(x)$ (if working with the wave equation) or the initial temperature/concentration function $u(x,0)=f(x)$ (if working with the heat/diffusion equation). As always, functions must be entered using valid Matlab syntax (such as sin(x) for $\sin x$).

  Iode has some built-in functions that make for interesting initial data: hat, triangle and bump. See Appendix C for details.

Remark 5.1   Iode computes its approximate solutions by separation of variables. For example, for the heat equation with Dirichlet boundary conditions Iode will use

$$u(x,t) = \sum_{n=1}^N b_n e^{-(n \pi/L)^2 k t} \sin (\frac{n\pi x}{L})$$

where $N$ is the value of top harmonic (which can be changed using the Options below), and where the $b_n$ are the Fourier sine coefficients of the initial temperature $f(x)$. For the wave equation with Dirichlet boundary conditions the approximate solution is

$$u(x,t) = \sum_{n=1}^N \left[ b_n \cos (\frac{n\pi c t}{L}) + B_n \frac{L}{n\pi c} \sin (\frac{n\pi c t}{L}) \right] \sin (\frac{n\pi x}{L}) ,$$

where the $b_n$ are the Fourier sine coefficients of the initial displacement $f(x)$ and the $B_n$ are the Fourier sine coefficients of the initial velocity $g(x)$.

Options menu

• Change resolution. You can enter the number of points to be plotted in each coordinate direction. This is a merely a question of ``display resolution''. Note that this number is the number of points in the interval $0\leq x < L$ (resp. $0\leq t < T$).

  The remaining option is more important mathematically, for it lets you change the top harmonic (i.e., the number of terms) used in computing the approximate series solution. Increasing top harmonic will yield a more accurate solution, though at some computational cost.