2.1.72 $$u_t + x u_x = 0$$ with $$u(x,0)=x^2$$. Math 5587

problem number 72

From Math 5587 midterm I, Fall 2016, practice exam, problem 3.

Solve for $$u(x,t)$$ \begin {align*} u_t + x u_x = 0 \end {align*}

With with $$u(x,0)=x^2$$

Mathematica

$\left \{\left \{u(x,t)\to e^{-2 t} x^2\right \}\right \}$

Maple

$u \left (x , t\right ) = x^{2} {\mathrm e}^{-2 t}$

Hand solution

Solve $$u_{t}+xu_{x}=0$$ with $$u\left ( x,0\right ) =x^{2}$$. Using the method of characteristics, the systems of characteristic lines are (from the PDE itself)\begin {align} \frac {dt}{ds} & =1\tag {1}\\ \frac {dx}{ds} & =x\tag {2}\\ \frac {du}{ds} & =0\tag {3} \end {align}

With initial conditions at $$s=0$$$t\left ( 0\right ) =0,x\left ( 0\right ) =\xi ,u\left ( 0\right ) =\xi ^{2}$ Equation (1) gives\begin {align} t & =s+t\left ( 0\right ) \nonumber \\ & =s\tag {5} \end {align}

Equation (2) gives\begin {align} \ln x & =s+x\left ( 0\right ) \nonumber \\ x & =\xi e^{s}\tag {6} \end {align}

From (5,6) solving for $$\xi$$ gives\begin {align} \xi & =xe^{-s}\nonumber \\ & =xe^{-t}\tag {7} \end {align}

Equation (3) gives\begin {align*} u & =u\left ( 0\right ) \\ & =\xi ^{2}\\ & =x^{2}e^{-2t} \end {align*}

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