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I'm trying to solve $u_t + u^2u_x = 0$ with $u(x, 0) = 2 + x$.

I'm thinking to proceed by characteristics where we have above that $\frac{dx}{dt} = 1$ and $dy/dt = u^2$, but not sure if this will help. This is from shock waves idea.

Here's what I have

ut + u^2ux = 0

q +u^2 p =0

u^2 p +q =0

dx/ u^2 = dy/1

and u^-1/-1 + C = y

then u(x,t)^-1 + C = y

is this correct?

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1  
Please learn to $\LaTeX$ format your quesitons. There are tutorials on the web and you can see this. Presumably ut is $u_t=\frac {\partial u}{\partial t}$ but I am not sure how to parse u2ux. Is it $\frac {\partial^2 u}{\partial x^2}?$ –  Ross Millikan Oct 14 '12 at 0:19
    
Given that the method of characteristics gives $\frac{dy}{dt} = u^2$, I think she means $u^2u_x$. –  Michael Albanese Oct 14 '12 at 0:24
    
@Michael, that is correct –  mary Oct 14 '12 at 1:02
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As timur comments, the characteristic equations are wrong. They should be stated against a parameter, not the involved variables. You are mistaking $t$ with $y$. The construction of the characteristics is based on the supposition that if $x = x(\eta)$ and $t = t(\eta)$, then $$\frac{d}{d\eta}u\big(x(\eta),t(\eta)\big) = u_x x'(\eta) + u_t t'(\eta) = u^2 u_x + u_t = 0$$ and then one says $x'(\eta) = u^2$, $t'(\eta) = 1$, $u'(\eta) = 0$. See my answer for a full analysis. –  Pragabhava Oct 16 '12 at 8:31
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Also, if I understand your work, the method you are using is for solving fully nonlinear first order PDE's, and you are using it wrong. Your problem is quasilinear, and there is no need to introduce $p$ and $q$. This are only introduced in the case the derivatives of $u$ are involved nonlinearly in the equation. I strongly suggest you to study the first chapter of John's Partial Differential Equations, as I believe you are very confused. Any doubts, we can try to help. –  Pragabhava Oct 16 '12 at 8:40
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1 Answer 1

up vote 8 down vote accepted
+50

The quasilinear first order PDE $$ a\big(x,y,u(x,y)\big) u_x(x,y) + b\big(x,y,u(x,y)\big)u_y(x,y) = c\big(x,y,u(x,y)\big) $$ where $a,\,b,\,c \in C^1$ with data $\mathcal{C}(\xi) = \big(x(\xi), y(\xi), u(\xi)\big) \in C^1$ and with $$ \begin{vmatrix} \frac{dx}{d\xi} & a \\ \frac{dy}{d\xi} & b\end{vmatrix} \neq 0 $$ has a unique solution near $\mathcal{C}$ given by \begin{align} \frac{d x}{d \eta} &= a & x\big|_{\eta = 0}&= x(\xi)\\ \frac{d y}{d \eta} &= b & y\big|_{\eta = 0}&= y(\xi)\\ \frac{d u}{d \eta} &= c & u\big|_{\eta = 0}&= u(\xi)\\ \end{align}

For proof and geometrical interpretation, see F. John's Partial Differential Equations1.4)

In your case, $\mathcal{C}(\xi) = \big(\xi,0,\xi+2\big)$. Near $\eta \sim 0$ $$ \begin{vmatrix} \frac{dx}{d\xi} & a \\ \frac{dy}{d\xi} & b\end{vmatrix} = \begin{vmatrix} 1 & u^2 \\ 0 & 1\end{vmatrix} = 1 $$ and the solution is unique.

The system of ODE's is \begin{align} \frac{d x}{d \eta} &= u^2 & x\big|_{\eta = 0}&= \xi\\ \frac{d t}{d \eta} &= 1 & t\big|_{\eta = 0}&= 0\\ \frac{d u}{d \eta} &= 0 & u\big|_{\eta = 0}&= \xi + 2\\ \end{align} with solution $$ t = \eta, \quad u = \xi + 2, \quad x = (\xi + 2)^2 \eta + \xi. $$

The characteristics are $t = \frac{x - \xi}{(\xi + 2)^2}$ hence $\xi = -2$ is a special point. As $\xi \rightarrow \infty$, $t \rightarrow 0$. As $\xi \rightarrow -\infty$, $t \rightarrow 0$. As $\xi \rightarrow -2$, $t \rightarrow \infty$.

Characteristic curves

This of course, means that there is no solution when the characteristics meet.

A simple explanation for this is that the transformation $$ (x,t) \rightarrow (\xi,\eta) $$ is invertible iff $$ \begin{vmatrix} \partial_\xi x & \partial_\eta x \\ \partial_\xi t & \partial_\eta t \end{vmatrix} = 1 + 4\eta + 2\xi \eta \neq 0 $$ meaning there is no solution when $\xi = -\frac{1 + 4 \eta}{2\eta}$ or, inverting the transformation, when $$ t = - \frac{1}{4(x+2)} $$

$\hskip.75in$Existence

Lastly, inverting for $\xi$

$$ \xi = \frac{-(1 + 4t) \pm \sqrt{1 + 4t(2 + x)}}{2 t} $$

and

$$ u(x,t) = \frac{-1 \pm \sqrt{1 + 4t(2 + x)}}{2 t}. $$

In order to determine the correct sign, we must look at the initial condition. For the minus sign $\lim_{t \rightarrow 0} u(x,t) = -\infty$, while the plus sign gives the correct answer. Hence

$$ u(x,t) = \frac{-1 + \sqrt{1 + 4t(2 + x)}}{2 t}. $$

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Thank you so much for your explanation. –  mary Oct 16 '12 at 9:55
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