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I am currently trying to answer a problem but I'm not quite sure I understand what the actual question is asking. The question reads:

Fix a time $t_*>0$. Assume $\phi_i, \psi_i$ where $i=1,2$ are bounded functions on the real line. Let $u_i$ denote the solution to:$$\frac{\partial^2u_i}{\partial t^2}-c^2\cdot\frac{\partial^2u_i}{\partial x^2}=0, \>\>\>\>u_i(x,0)=\phi_i(x), \>\>\>\> \frac{\partial u_i(x,0)}{\partial t} = \psi_i(x)$$ For any function $f(x)$, with $||f(x)||_{\infty}$ denoting the supremum of $|f(x)|$ over $x\in\mathbb R$. Consider the statement: For every $\epsilon>0$ we can find $\delta>0$ such that:$$\text{if } ||\phi_1(x)-\phi_2(x)||_{\infty}<\delta\text{ and }||\psi_1(x)-\psi_2(x)||_{\infty}<\delta, \textbf{ then } ||u_1(x, t_*)-u_2(x,t_*)||_{\infty}<\epsilon$$ a) Explain in words what this statement means in terms of solving the IVP for the wave equation b)Prove the statement

Now for a), I'm assuming the way to describe it with words is that if the initial values have an error that is delta-fine, then the solutions that comes with the two separate IVP has to have error that is epsilon-fine, or in other words, the solution is unique.

But for b), I dont even know where to start. How can I prove this statement using epsilon delta definitions? I am not that knowledgeable when it comes to epsilon-delta definitions, and require some help here. If anyone could help it would be greatly appreciated. Thank you!

Felicio Grande
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  • For a, you're more or less correct. Another way to say it is "arbitrarily small changes in the initial values result in arbitrarily small changes of the solutions.". Note that these small changes are in the supremum of the function. To prove the statement, try using the fact that we know how u is a solution to our original problem, what does that say about $u_1 - u_2$? We also want to use bounded as well. – DaveNine Sep 27 '17 at 01:18
  • I'm assuming that if we know that $u$ is a solution, then it is unique, thus the error between any two solutions will have an error that gets $\epsilon$-fine? But I'm not sure how to directly write this as a proof? @DaveNine – Felicio Grande Sep 27 '17 at 02:45

1 Answers1

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To get you started, suppose $w(x, t) = u_1(x,t) - u_2(x, t)$. Use this to write down what the wave equation is going to be with $w(x, t)$ as the solution.

Then write down the solution of the problem using D'Lambert's Solution for $w(x, t)$, and find a bound for $\|w(x, t)\|_{\infty}$. Since you are fixing $t=t_{*}$, it will be okay that our expression involves these terms. This should give you the result.

In general, this kind of result is called a "stability" result, because if we had an equation and made a small change in the IC's and the solutions were wildly different, then it would be really difficult to produce numerical results.

DaveNine
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