Past exam of the applied partial differential equations course of the University of Oxford / 2023 / 2 / c / i

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- Solution i

Solution

The initial condition is a period-one square wave: height one on each interval

[k - 1/2, k]

and zero on each gap

(k, k + 1/2)

k

. Each period has mean

1/2

.

Figure 1.
A periodic train of pulses evolves into a travelling sawtooth of decreasing amplitude
. Original exact profiles at $t = 0$ , $t = 0.5$ , $t = 1$ , and $t = 3$ . The dashed line marks the conserved mean $1/2$ . Jump-point values can be chosen without changing the weak solution. Details.

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