# Burgers' equation

Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation[1] occurring in various areas of applied mathematics, such as fluid mechanics,[2] nonlinear acoustics,[3] gas dynamics, and traffic flow.[4] The equation was first introduced by Harry Bateman in 1915[5][6] and later studied by Johannes Martinus Burgers in 1948.[7] For a given field ${\displaystyle u(x,t)}$ and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) ${\displaystyle \nu }$, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}.}$

The term ${\displaystyle u\partial u/\partial x}$ can also rewritten as ${\displaystyle \partial (u^{2}/2)/\partial x}$. When the diffusion term is absent (i.e. ${\displaystyle \nu =0}$), Burgers' equation becomes the inviscid Burgers' equation:

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0,}$

which is a prototype for conservation equations that can develop discontinuities (shock waves).

The reason for the formation of sharp gradients for small values of ${\displaystyle \nu }$ becomes intuitevely clear when one examines the left-hand side of the equation. The term ${\displaystyle \partial /\partial t+u\partial /\partial x}$ is evidently a wave operator describing a wave propagating in the positive ${\displaystyle x}$-direction with a speed ${\displaystyle u}$. Since the wave speed is ${\displaystyle u}$, regions exhibiting large values of ${\displaystyle u}$ will be propagated rightwards quickly than regions exhibiting smaller values of ${\displaystyle u}$; in other words, if ${\displaystyle u}$ is decreasing in the ${\displaystyle x}$-direction, initially, then larger ${\displaystyle u}$'s that lie in the backside will catch up with smaller ${\displaystyle u}$'s that is on the front side. The role of the right-side diffusive term is essentially to stop the gradient becoming infinite.

## 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

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0,\quad u(x,0)=f(x)}$

can be constructed by the method of characteristics. Let ${\displaystyle t}$ be the parameter charcetising any given characteristics in the ${\displaystyle x}$-${\displaystyle t}$ plane, then the characteristic equations are given by

${\displaystyle {\frac {dx}{dt}}=u,\quad {\frac {du}{dt}}=0.}$

Integration of the second equation tells us that ${\displaystyle u}$ is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,

${\displaystyle u=c,\quad x=ut+\xi }$

where ${\displaystyle \xi }$ is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since ${\displaystyle u}$ at ${\displaystyle x}$-axis is known from the initial condition and the fact that ${\displaystyle u}$ is unchanged as we move along the characteristic emanating from each point ${\displaystyle x=\xi }$, we write ${\displaystyle u=c=f(\xi )}$ on each characteristic. Therefore, the family of trajectories of characteristics parametrized by ${\displaystyle \xi }$ is

${\displaystyle x=f(\xi )t+\xi .}$

Thus, the solution is given by

${\displaystyle u(x,t)=f(\xi )=f(x-ut),\quad \xi =x-f(\xi )t.}$

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]

${\displaystyle t_{b}={\frac {-1}{\inf _{x}\left(f^{\prime }(x)\right)}}.}$

### Complete integral of the inviscid Burgers' equation

The implicit solution described above containing an arbitrary function ${\displaystyle f}$ is called the general integral. However, the inviscid Burgers' equation, being a first-order partial differential equation, also has a complete integral which contains two arbitrary constants (for the two independent variables).[10][better source needed] Subrahmanyan Chandrasekhar provided the complete integral in 1943,[11] which is given by

${\displaystyle u(x,t)={\frac {ax+b}{at+1}}.}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are arbitrary constants. The complete integral satisfies a linear initial condition, i.e., ${\displaystyle f(x)=ax+b}$. One can also construct the geneal integral using the above complete integral.

## Viscous Burgers' equation

The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,[12][13][14]

${\displaystyle u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \varphi (x,t),}$

which turns it into the equation

${\displaystyle 2\nu {\frac {\partial }{\partial x}}\left[{\frac {1}{\varphi }}\left({\frac {\partial \varphi }{\partial t}}-\nu {\frac {\partial ^{2}\varphi }{\partial x^{2}}}\right)\right]=0,}$

which can be integrated with respect to ${\displaystyle x}$ to obtain

${\displaystyle {\frac {\partial \varphi }{\partial t}}-\nu {\frac {\partial ^{2}\varphi }{\partial x^{2}}}=\varphi {\frac {df(t)}{dt}},}$

where ${\displaystyle df/dt}$ is an arbitrary function of time. Introducing the transformation ${\displaystyle \varphi \to \varphi e^{f}}$ (which does not affect the function ${\displaystyle u(x,t)}$), the required equation reduces to that of the heat equation[15]

${\displaystyle {\frac {\partial \varphi }{\partial t}}=\nu {\frac {\partial ^{2}\varphi }{\partial x^{2}}}.}$

The diffusion equation can be solved. That is, if ${\displaystyle \varphi (x,0)=\varphi _{0}(x)}$, then

${\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi \nu t}}}\int _{-\infty }^{\infty }\varphi _{0}(x')\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}\right]dx'.}$

The initial function ${\displaystyle \varphi _{0}(x)}$ is related to the initial function ${\displaystyle u(x,0)=f(x)}$ by

${\displaystyle \ln \varphi _{0}(x)=-{\frac {1}{2\nu }}\int _{0}^{x}f(x')dx',}$

where the lower limit is chosen arbitrarily. Inverting the Cole–Hopf transformation, we have

${\displaystyle u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{{\frac {1}{\sqrt {4\pi \nu t}}}\int _{-\infty }^{\infty }\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}f(x'')dx''\right]dx'\right\}}$

which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarthim, to

${\displaystyle u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{\int _{-\infty }^{\infty }\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}f(x'')dx''\right]dx'\right\}.}$

This solution is derived from the solution of the heat equation for ${\displaystyle \varphi }$ that decays to zero as ${\displaystyle x\to \pm \infty }$; other solutions for ${\displaystyle u}$ can be obtained starting from solutions of ${\displaystyle \varphi }$ that satisfies different boundary conditions.

## Some explicit solutions of the viscous Burgers' equation

Explicit expressions for the viscous Burgers' equation are available. Some of the physically relevant solutions are given below:[16]

If ${\displaystyle u(x,0)=f(x)}$ is such that ${\displaystyle f(-\infty )=f^{-}}$ and ${\displaystyle f(+\infty )=f^{+}}$ and ${\displaystyle f'(x)<0}$, then we have a traveling-wave solution (with a constant speed ${\displaystyle c=(f^{+}+f^{-})/2}$) given by

${\displaystyle u(x,t)=c-{\frac {f^{+}-f^{-}}{2}}\tanh \left[{\frac {f^{+}-f^{-}}{4\nu }}(x-ct)\right].}$

This solution, that was originally derived by Harry Bateman in 1915,[5] is used to describe the variation of pressure across a weak shock wave[15]. When ${\displaystyle f^{+}=2}$ and ${\displaystyle f^{-}=0}$ to

${\displaystyle u(x,t)={\frac {2}{1+e^{x-t}}}}$

with ${\displaystyle c=1}$.

### Delta function as an initial condition

If ${\displaystyle u(x,0)=2\nu Re\delta (x)}$, where ${\displaystyle Re}$ (say, the Reynolds number) is a constant, then we have[17]

${\displaystyle u(x,t)={\sqrt {\frac {\nu }{\pi t}}}\left[{\frac {(e^{Re}-1)e^{-x^{2}/4\nu t}}{1+(e^{Re}-1)\mathrm {erfc} (x/{\sqrt {4\nu t}})/{\sqrt {2}}}}\right].}$

In the limit ${\displaystyle Re\to 0}$, the limiting behaviour isa diffusional spreading of a source and therefore is given by

${\displaystyle u(x,t)={\frac {2\nu Re}{\sqrt {4\pi \nu t}}}\exp \left(-{\frac {x^{2}}{4\nu t}}\right).}$

On the other hand, In the limit ${\displaystyle Re\to \infty }$, the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by

${\displaystyle u(x,t)={\begin{cases}{\frac {x}{t}},\quad 0

The shock wave location and its speed are given by ${\displaystyle x={\sqrt {2\nu Re\,t}}}$ and ${\displaystyle {\sqrt {\nu Re/t}}.}$

### N-wave solution

The N-wave solution comprises a compression wave followed by a rarafaction wave. A solution of this type is given by

${\displaystyle u(x,t)={\frac {x}{t}}\left[1+{\frac {1}{e^{Re_{0}-1}}}{\sqrt {\frac {t}{t_{0}}}}\exp \left(-{\frac {Re(t)x^{2}}{4\nu Re_{0}t}}\right)\right]^{-1}}$

where ${\displaystyle R_{0}}$ may be regarded as an initial Reynolds number at time ${\displaystyle t=t_{0}}$ and ${\displaystyle Re(t)=(1/2\nu )\int _{0}^{\infty }udx=\ln(1+{\sqrt {\tau /t}})}$ with ${\displaystyle \tau =t_{0}{\sqrt {e^{Re_{0}}-1}}}$, may be regarded as the time-varying Reynold number.

## Other forms

### Multi-dimensional Burgers' equation

In two or more dimensions, the Burgers' equation becomes

${\displaystyle {\frac {\partial u}{\partial t}}+u\cdot \nabla u=\nu \nabla ^{2}u.}$

One can also extend the equation for the vector field ${\displaystyle \mathbf {u} }$, albeit it is not very useful, as in

${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} =\nu \nabla ^{2}\mathbf {u} .}$

### Generalized Burgers' equation

The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,

${\displaystyle {\frac {\partial u}{\partial t}}+c(u){\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}.}$

where ${\displaystyle c(u)}$ is any arbitrary function of u. The inviscid ${\displaystyle \nu =0}$ equation is still a quasilinear hyperbolic equation for ${\displaystyle c(u)>0}$ and its solution can be constructed using method of characteristics as before.[18]

### Stochastic Burgers' equation

Added space-time noise ${\displaystyle \eta (x,t)={\dot {W}}(x,t)}$, where ${\displaystyle W}$ is an ${\displaystyle L^{2}(\mathbb {R} )}$ Wiener process, forms a stochastic Burgers' equation[19]

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}-\lambda {\frac {\partial \eta }{\partial x}}.}$

This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field ${\displaystyle h(x,t)}$ upon substituting ${\displaystyle u(x,t)=-\lambda \partial h/\partial x}$.

## References

1. ^ Misra, Souren; Raghurama Rao, S. V.; Bobba, Manoj Kumar (2010-09-01). "Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation". International Journal of Computational Fluid Dynamics. 24 (8): 303–315. Bibcode:2010IJCFD..24..303M. doi:10.1080/10618562.2010.523518. ISSN 1061-8562. S2CID 123001189.
2. ^ It relates to the Navier–Stokes momentum equation with the pressure term removed Burgers Equation (PDF): here the variable is the flow speed y=u
3. ^ 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
4. ^ Musha, Toshimitsu; Higuchi, Hideyo (1978-05-01). "Traffic Current Fluctuation and the Burgers Equation". Japanese Journal of Applied Physics. 17 (5): 811. Bibcode:1978JaJAP..17..811M. doi:10.1143/JJAP.17.811. ISSN 1347-4065. S2CID 121252757.
5. ^ a b Bateman, H. (1915). "Some recent researches on the motion of fluids". Monthly Weather Review. 43 (4): 163–170. Bibcode:1915MWRv...43..163B. doi:10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2.
6. ^ Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.
7. ^ Burgers, J. M. (1948). "A Mathematical Model Illustrating the Theory of Turbulence". Advances in Applied Mechanics. 1: 171–199. doi:10.1016/S0065-2156(08)70100-5. ISBN 9780123745798.
8. ^ Olver, Peter J. (2013). Introduction to Partial Differential Equations. Undergraduate Texts in Mathematics. Online: Springer. p. 37. doi:10.1007/978-3-319-02099-0. ISBN 978-3-319-02098-3. S2CID 220617008.
9. ^ 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.
10. ^ Forsyth, A. R. (1903). A Treatise on Differential Equations. London: Macmillan.
11. ^ Chandrasekhar, S. (1943). On the decay of plane shock waves (Report). Ballistic Research Laboratories. Report No. 423.
12. ^ Cole, Julian (1951). "On a quasi-linear parabolic equation occurring in aerodynamics". Quarterly of Applied Mathematics. 9 (3): 225–236. doi:10.1090/qam/42889. JSTOR 43633894.
13. ^ 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.
14. ^ Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution Techniques. Belmont: Wadsworth. pp. 31–35. ISBN 0-534-12216-7.
15. ^ a b Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. Page 352-354.
16. ^ Salih, A. "Burgers’ Equation." Indian Institute of Space Science and Technology, Thiruvananthapuram (2016).
17. ^ Whitham, Gerald Beresford. Linear and nonlinear waves. John Wiley & Sons, 2011.
18. ^ Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.
19. ^ Wang, W.; Roberts, A. J. (2015). "Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation". Communications in Mathematical Physics. 333 (3): 1287–1316. arXiv:1203.0463. Bibcode:2015CMaPh.333.1287W. doi:10.1007/s00220-014-2117-7. S2CID 119650369.