For a given field and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) , 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. ), Burgers' equation becomes the inviscid Burgers' equation:
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
Integration of the second equation tells us that is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,
where 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 -axis is known from the initial condition and the fact that is unchanged as we move along the characteristic emanating from each point , we write on each characteristic. Therefore, the family of trajectories of characteristics parametrized by is
Thus, the solution is given by
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by[8][9]
Inviscid Burgers' equation for linear initial condition[edit]
Subrahmanyan Chandrasekhar provided the explicit solution in 1943 when the initial condition is linear, i.e., , where a and b are constants.[10] The explicit solution is
This solution is also the complete integral of the inviscid Burgers' equation because it contains as many arbitrary constants as the number of independent variables appearing in the equation.[11][better source needed] Using this complete integral, Chandrasekhar obtained the general solution described for arbitrary initial conditions from the envelope of the complete integral.
where is an arbitrary function of time. Introducing the transformation (which does not affect the function ), the required equation reduces to that of the heat equation[15]
The diffusion equation can be solved. That is, if , then
The initial function is related to the initial function by
where the lower limit is chosen arbitrarily. Inverting the Cole–Hopf transformation, we have
which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarthim, to
Some explicit solutions of the viscous Burgers' equation[edit]
Explicit expressions for the viscous Burgers' equation are available. Some of the physically relevant solutions are given below:[16]
In the limit , the limiting behaviour isa diffusional spreading of a source and therefore is given by
On the other hand, In the limit , the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by
The shock wave location and its speed are given by and
The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,
where is any arbitrary function of u. The inviscid equation is still a quasilinear hyperbolic equation for and its solution can be constructed using method of characteristics as before.[18]
^It relates to the Navier–Stokes momentum equation with the pressure term removed Burgers Equation(PDF): here the variable is the flow speedy=u
^It arises from Westervelt equation with an assumption of strictly forward propagating waves and the use of a coordinate transformation to a retarded time frame: here the variable is the pressure
^Cameron, Maria (February 29, 2024). "Notes on Burger's Equation"(PDF). University of Maryland Mathematics Department, Maria Cameron's personal website. Retrieved February 29, 2024.
^Eberhard Hopf (September 1950). "The partial differential equation ut + uux = μuxx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.
^Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution Techniques. Belmont: Wadsworth. pp. 31–35. ISBN0-534-12216-7.
^ abLandau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. Page 352-354.
^Salih, A. "Burgers’ Equation." Indian Institute of Space Science and Technology, Thiruvananthapuram (2016).
^Whitham, Gerald Beresford. Linear and nonlinear waves. John Wiley & Sons, 2011.
^Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.