Chemistry - Burgers' equation

Burgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modelling of gas dynamics and traffic flow . It is named for Johannes Martinus Burgers (1895-1981).

The general form of Burgers' equation is:

$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2}$ .

Here $μ > 0$ is a viscosity coefficient. When $μ = 0$ , Burgers' equation becomes the inviscid Burgers' equation:

$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0$ ,

which is a prototype for equations for which the solution can develop discontinuities (shock waves).

Solution

The inviscid Burgers' equation is a first order partial differential equation. Its solution can be constructed by the method of characteristics. This method yields that if $X (t)$ is a solution of the ordinary differential equation

d X (t) / d t = u (X (t), t)

then $U (t): = u (X (t), t)$ is constant as a function of $t$ . Hence $(X (t), U (t))$ is a solution of the system of ordinary equations

d X / d t = U

d U / d t = 0.

The solutions of this system are given in terms of the initial values by

X (t) = X (0) + t U (0)

U (t) = U (0).

Substitute $X (0) = η$ , then $U (0) = u (X (0),0) = u (η,0)$ . Now the system becomes

X (t) = η + t u (η,0)

U (t) = U (0).

Conclusion:

u (η,0) = U (0) = U (t) = u (X (t), t) = u (η + t u (η,0), t).

This is an implicit relation that determines the solution of the inviscid Burgers' equation.

External link

Burgers' Equation at EqWorld: The World of Mathematical Equations.

Categories: Partial differential equations | Fluid dynamics | Equations