#### 2.1.1 Transport equation $$u_t+ u_x = 0$$

problem number 1

Taken from Mathematica Symbolic PDE document

Solve for $$u(x,t)$$ $u_t+ u_x = 0$

Mathematica

$\{\{u(x,t)\to c_1(t-x)\}\}$

Maple

$u \left (x , t\right ) = \mathit {\_F1} \left (t -x \right )$

Hand solution

\begin {equation} u_{t}+u_{x}=0\tag {1} \end {equation}

Let $$u\equiv u\left ( x\left ( t\right ) ,t\right )$$. Then \begin {equation} \frac {du}{dt}=\frac {\partial u}{\partial x}\frac {dx}{dt}+\frac {\partial u}{\partial t}\tag {2} \end {equation}

Comparing (1) to (2) then we see that

\begin {align} \frac {du}{dt} & =0\tag {3}\\ \frac {dx}{dt} & =1\tag {4} \end {align}

(3) says that $$u$$ is constant. Since no initial conditions are given, let $$u=F\left ( x\left ( 0\right ) \right )$$ where $$F$$ is arbitrary function. To ﬁnd $$x\left ( 0\right )$$ we solve (4). The solution to (4) is $$x=x\left ( 0\right ) +t$$. Hence $$x\left ( 0\right ) =x-t$$.  Therefore

$u\left ( x,t\right ) =F\left ( x-t\right )$

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