Advection Equation with initial data $u(x,0)=\sin(x)$

Question

I am looking for analytic solution of the Advection Equation: $$ \frac{\partial{u(x,t)}}{\partial{t}} + c \frac{\partial{u(x,t)}}{\partial{x}}=0, $$ with initial condition $$ u_0(x)=u(x,0)=\sin(x). $$ I tried asking Wolfram|Alpha, but with no success.

Answer 1

The equation $u_t+cu_x=0$ can be solved using method of characteristic curves.
$dt=\dfrac{dx}{c}\implies ct=x+k$ and we can define $\zeta=x-ct$.
A general solution for the equation can be written as $u(t,x)=F(\zeta)=F(x-ct)$.
Imposing the boundary condition, we find $u(0,x)=F(x)=\sin(x)$, so the function $F$ acts on real numbers as $\mathbb R \ni\xi\mapsto\sin(\xi)$. In conclusion the solution is $u(t,x)=\boxed{\sin(x-ct)}$.

Given an equation of the form $\sum_{j=1}^n a^j(\textbf x) \frac{\partial u(\textbf x)}{\partial x^j}=0$ s.t. $a^j(\textbf x)\ne 0$, $\forall j=1,\dots,n$, $\forall \textbf x\in\mathbb R^n$ (in general you could consider the equation for $\textbf x\in\mathbb R^n\setminus\{\textbf x\in\mathbb R^n :a^j(\textbf x)=0\}_{j\in\{1,\dots,n\}}$), you can introduce a vector field $X$ defined as $$\mathbb R^n\ni\textbf x\mapsto (a^1(\textbf x),\dots, a^n(\textbf x))^t.$$ The integral curves $\sigma:I\subseteq\mathbb R\to\mathbb R^n$ of the vector field $X$ satisfy $X(\sigma(s))=\dfrac{d\sigma}{ds}$ and a function $u(\textbf x)$ satisfies the equation $\sum_{j=1}^n a^j(\textbf x) \frac{\partial u(\textbf x)}{\partial x^j}=0$ iff it's constant along the integral curves of the vector field since $\dfrac{du(\sigma(s))}{ds}=\dfrac{d\sigma^j}{ds}\dfrac{\partial u}{\partial x^j}=a^j(\textbf x)\dfrac{\partial u}{\partial x^j}$.