Burgers' equation

Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field u ( x , t ) {displaystyle u(x,t)} and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) ν {displaystyle u } , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system: When the diffusion term is absent (i.e. ν = 0 {displaystyle u =0} ), Burgers' equation becomes the inviscid Burgers' equation: which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition can be constructed by the method of characteristics. The characteristic equations are Integration of the second equation tells us that u {displaystyle u} is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e., where ξ {displaystyle xi } is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since at the point, the velocity is known from the initial condition and the fact that this value is unchanged as we move along the characteristic emanating from that point, we write u = c = f ( ξ ) {displaystyle u=c=f(xi )} on that characteristic. Therefore, the trajectory of that characteristic is