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.
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