1

I'm completely stuck on the following problem:

Consider an example of unidirectional non-linear wave motion:

$u_t + uu_x = 0$, with $u(x, 0) = f(x)$.

i) Show that the characteristic variable $s$ is defined by the implicit equation $x = f(s)t + s$ and that the solution can also be expressed implicitly in the form $u(x, t) = f(x − tu)$.

ii) Using implicit differentiation, show that $u_x = \dfrac{f′′(x − tu)}{1 + t f''(x - tu)}$ and hence determine a condition when the solution will break down.

iii) With the example $f(x) = −x$,

a) Show that the solution breaks down at $t = 1$

b) Sketch the characteristic curves for this case and explain your sketch in relation to part (a)

c) Sketch the solution as a function of time between $0 ≤ t ≤ 1$.

I've spent hours on it, but I can't seem to make any progress. And, unfortunately, the book has no solutions for me to read and learn from.

I would greatly appreciate it if people could please take the time to demonstrate how this is solved, with explanations that justify their work so that I may learn what It is I'm not understanding. I will study any answers and reference it with my textbook, so that I can figure out what it is that I'm not understanding.

Community
  • 1
The Pointer
  • 4,686
  • It's supposed to be single derivatives $f'(x-tu)$ not $f''(x-tu)$ right? – Winther Aug 10 '18 at 01:18
  • Hi @Winther. I just checked: It's written as $f''(x-tu)$ – The Pointer Aug 10 '18 at 01:20
  • @Winther Why? Do you think this is a textbook error? The textbook does have a lot of errors. – The Pointer Aug 10 '18 at 01:20
  • 1
    Yes that is a typo. Try to take the $x$-derivative of $u = f(x-tu)$ – Winther Aug 10 '18 at 01:20
  • @Winther Thank you for that. Can you please explain why/how you know it's a typo? Thanks. – The Pointer Aug 10 '18 at 01:21
  • You'll need to show us what you have tried, so that we can give you a solution you can understand. – Matthew Cassell Aug 10 '18 at 02:40
  • @Mattos I’m completely unsure of what to do, so any basic (basic in the sense that it will allow me to understand what to do and why steps were taken) answer with explanations for each step would be best. If the textbook had solutions, then I would use that. I’d be very grateful for this, since there is no other way for me to understand what I’m doing wrong. – The Pointer Aug 10 '18 at 02:43
  • 1
    Did you compute the implicit solution $u=f(x−tu)$? If you don't know how to, look at the related problems to the on the right hand side of the screen, there are multiple examples available. To get part ii), doing what Winther alluded to, we have \begin{align} u &= f(x - ut) \ \implies u_{x} &= f'(x-ut) - tu_{x} \cdot f'(x-ut) \cdot \ \implies u_{x} (1 + tf') &= f' \ \implies u_{x} &= \frac{f'}{1+tf'} \end{align} This is called the breaking time. So you can see there is a typo in the question. I'm sure you can do part iii) onwards. – Matthew Cassell Aug 10 '18 at 05:03
  • @Mattos I've gone off and had a consultation with my Professor. I will now attempt the problem and post my solution for review, sound good? – The Pointer Aug 10 '18 at 05:46
  • @ThePointer Sounds good, just make sure you post your solution in your original post above and not a new post so it is clear to others who read your question/solution. – Matthew Cassell Aug 10 '18 at 06:00
  • @ThePointer This is the inviscid Burgers' equation. The first part is solved in this post using the method of characteristics. Implicit differentiation is used in this post to obtain the solution's breakdown condition. The case $f(x) = -x$ is tackled in this post. – EditPiAf Aug 10 '18 at 12:32

0 Answers0