# Notes on Diffy Qs - Sage demos for section 1.9

Press the Evaluate button below to launch the Sage demonstration. You may have to wait a little before the graph appears. Be patient. To change the function, you should edit the code and press the Evaluate button again.

## Plotting Example 1.9.1: Transport equation

Let us work through the solution to $$u_x + u_t = 0,$$ the transport equation, for $$-\infty < x < \infty ,$$ $$t \geq 0,$$ with the initial condition $$u(x,0) = e^{-x^2} .$$ That is, Example 1.9.2 in the book with $$\alpha=1$$ and $$f(x) = e^{-x^2} .$$

The characteristic curves are the sets where $$\xi = \text{constant},$$ so we need to write $$\xi$$ as a function of $$x$$ and $$t$$ and plot the level sets of this function. We found the characteristic coordinates $$(\xi,s)$$ to be given by $$\xi=x-t,$$ $$s=t .$$ So we find the level sets of $$x-t .$$ The $$x$$ axis is the horizontal one and $$t$$ the vertical one.

In the book, we solved the equation to find

$$\displaystyle u(x,t) = e^{-(x-t)^2} .$$

Before graphing $$u$$ in three dimensions, we draw the values of $$u$$ as colors and overlay the characteristics. Note how the function is constant (same color) along the characteristic since the ODE along the characteristics is $$\frac{du}{ds} = 0 .$$

Next, we plot the graph of $$u(x,t)$$ in 3 dimensions. Note that Sage currently (unless this has been fixed already) labels the axes a little confusingly, so the $$t$$ axis is labeled as $$y .$$ You should be able to rotate the graph by clicking and dragging.

## Plotting Example 1.9.3

Let us work through $$xu_x + u_t + 2u = 0,$$ $$-\infty < x < \infty ,$$ $$t \geq 0,$$ with the initial condition $$u(x,0) = \cos(x) .$$

First, we plot the characteristic curves $$\xi = \text{constant}.$$ In this example, the characteristic coordinates $$(\xi,s)$$ were given by $$x=\xi e^s,$$ $$t=s .$$ Solving for $$\xi$$ in terms of $$x$$ and $$t$$ we find $$\xi=xe^{-t}.$$ We will only plot $$x$$ in the range $$-5$$ to $$5$$ and the $$t$$ in the range $$0$$ to $$1.$$ So to avoid a narrow graph we set the aspect ratio to 8 to stretch the $$t$$ axis by 8 times.

In the book, we solved the equation to find

$$\displaystyle u(x,t) = e^{-2t} \cos(xe^{-t}) .$$

We again make a contour plot of $$u$$ with the characteristics overlaid. In this case, the ODE along the characteristic is $$\frac{du}{ds}+2u=0 ,$$ which has the general solution $$u = Ce^{-2s} .$$ So the solution decays towards zero exponentially as we move along the characteristic.

And finally, the graph of $$u(x,t)$$ in 3 dimensions. Here we also stretch the vertical axis 3 times.