Burgers' equation
From Biocrawler, the free encyclopedia.
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:
.
Here μ > 0 is a viscosity coefficient. When μ = 0, Burgers' equation becomes the inviscid Burgers' equation:
,
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
- dX(t) / dt = 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
- dX / dt = U
- dU / dt = 0.
The solutions of this system are given in terms of the initial values by
- X(t) = X(0) + tU(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 + tu(η,0)
- U(t) = U(0).
Conclusion:
- u(η,0) = U(0) = U(t) = u(X(t),t) = u(η + tu(η,0),t).
This is an implicit relation that determines the solution of the inviscid Burgers' equation.
External link
- Burgers' Equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde1301.pdf) at EqWorld: The World of Mathematical Equations.
