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). In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... The hydrogeology is study about of water-bearing formation. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... A gas-dynamic control system is one where the path of an object in flight is controlled by either the generation or redirection of gas flow out of an orifice rather than with the traditional movable control surfaces. ... Generalities The mathematical study of traffic flow, and in particular vehicular traffic flow, is done with the aim to get a better understanding of these phenomena and to hopefully avoid some problems of traffic congestion. ... Johannes Martinus Burgers (1895-1981) was a Dutch physicist. ...

For a given velocity u and viscosity coefficient ν, the general form of Burgers' equation is: The velocity of an object is simply its speed in a particular direction. ... The pitch drop experiment at the University of Queensland. ...

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When ν = 0, Burgers' equation becomes the inviscid Burgers' equation:

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which is a prototype for equations for which the solution can develop discontinuities (shock waves). Introduction The shock wave is one of several different ways in which a gas in a supersonic flow can be compressed. ...

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 In mathematics, the method of characteristics is a technique for solving partial differential equations. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

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

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 at EqWorld: The World of Mathematical Equations.