Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} \end{cases}$$

By d´Alembert's formula we know that

$u(x,t)=\frac{f(x+\alpha t)+f(x-\alpha t)}{2} +\frac{1}{2\alpha}\int^{x+\alpha t}_{x-\alpha t}g(s)ds$

is only solution of the above problem. But in the proof, the uniqueness of $u$, second my book reference, is given by the uniqueness of the functions $f(x)\in C^{2}(\mathbb{R})$ and $g(x)\in C^{1}(\mathbb{R}).$

I do not understand this statement! Can anyone help me? thank you very much.

share|cite|improve this question
up vote 3 down vote accepted

Note that $f(x)\in C^{2}(\mathbb{R})$ and $g(x)\in C^{1}(\mathbb{R})$ imply $u(x,t)\in C^{2}$. The solution at point $(x, t)$ is uniquely determined by values of $f$ and $g$ ​in the interval $[x-\alpha t,x+\alpha t]$. This interval is the Domain of Dependency of solution at point $(x,t)$.

share|cite|improve this answer
Thank you @Anderson Lima. – Nathan Marke Feb 13 '13 at 19:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.