# 2.2 Partial Differential Equations

## 2.2.2 One-Dimensional Burgers' Equation

Measurable Outcome 2.1, Measurable Outcome 2.2

Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895-1981). The one-dimensional Burgers' equation is given in differential form as:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0 \label{equ:burgers}\] | (2.17)

Which of the following is the correct mapping between the terms of the Burgers' equation and the terms of the canonical conservation equation?

Exercise 1
&nbsp; \(U=u, \vec{F}=u\) &nbsp;
&nbsp; \(U=u, \vec{F}=u^2\) &nbsp;
&nbsp; \(U=\frac{1}{2}u, \vec{F}=u^2\) &nbsp;
&nbsp; \(U=u,\vec{F}=\frac{1}{2}u^2\) &nbsp;

Answer: This solution can be verified by plugging in \(\vec{F} =\frac{1}{2} u^2\) and noting that \(\frac{\partial \vec{f}}{\partial x} = \frac{\partial \vec{F}}{\partial u}\frac{\partial u}{\partial x}\) and we arrive that the above differential form.