Burgers equation
Recall our basic conservation law expressed as a PDE:
$$u_t + \varphi_x = f.$$
Burgers equation is a model of nonlinear advection that arises from the conservation law by assuming the source is zero and the flux is $\frac{1}{2}u^2$. Derive Burgers equation.
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This problem is kinda like problem 10 on page 38 of our text. Can you see what the flux is in that problem?
Comments
Recall our basic conservation law expressed as a PDE:
$$u_t + \varphi_x = f.$$
Here we assume the source $f = 0$ and the flux $\varphi = \frac{1}{2}u^2$. Taking the derivative of the flux with respect to $x$ by using the chain rule, we find $\varphi_x = uu_x$. Plugging these variables back into the original equation, we get
$$u_t + uu_x = 0.$$
The basic conservation law expressed as a PDE is:
$$u_t + \phi_x = f.$$
To find Burgers equation, first assume that the source $f = 0$ and the flux $\phi = \frac{1}{2}u^2$.
To find $\phi_x$ take the chain rule derivative of $\frac{1}{2} * u(x,t)^2$ with respect to $x$.
Therefore $\phi_x = u(x,t) * u_x(x,t)$
Inputting back into the basic conservation law gives:
$$u_t + uu_x = 0$$