Burgers' equation#Viscous 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

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

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

:$\backslash frac\{dx\}\{dt\}\; =\; u,\; \backslash quad\; \backslash frac\{du\}\{dt\}=0.$

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

:$u=c,\; \backslash quad\; x\; =\; ut\; +\; \backslash xi$

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

:$x=f(\backslash xi)\; t+\; \backslash xi.$

Thus, the solution is given by

:$u(x,t)\; =\; f(\backslash xi)\; =\; f(x-ut),\; \backslash quad\; \backslash xi\; =\; x\; -\; f(\backslash xi)\; t.$

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{{Cite book|last=Olver|first=Peter J.|url=https://link.springer.com/book/10.1007/978-3-319-02099-0|title=Introduction to Partial Differential Equations|series=Undergraduate Texts in Mathematics | publisher=Springer| year=2013| isbn=978-3-319-02098-3|location=Online|pages=37|doi=10.1007/978-3-319-02099-0 |s2cid=220617008 }}{{Cite web |last=Cameron |first=Maria |date=February 29, 2024 |title=Notes on Burger's Equation |url=https://www.math.umd.edu/~mariakc/burgers.pdf |access-date=February 29, 2024 |website=University of Maryland Mathematics Department, Maria Cameron's personal website}}

:$t\_b\; =\; \backslash frac\{-1\}\{\backslash inf\_x\; \backslash left(f^\backslash prime(x)\backslash right)\}.$

The implicit solution described above containing an arbitrary function $f$ 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).{{cite book |last=Forsyth |first=A. R. |author-link=Andrew Forsyth |year=1903 |title=A Treatise on Differential Equations |location=London |publisher=Macmillan }}{{Better source needed|reason=Might be better to source this statement with a book less than 120 years old.|date=November 2020}} Subrahmanyan Chandrasekhar provided the complete integral in 1943,{{cite report |last=Chandrasekhar |first=S. |year=1943 |title=On the decay of plane shock waves |id=Report No. 423 |publisher=Ballistic Research Laboratories }} which is given by

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

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

The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,{{cite journal | first=Julian |last=Cole |author-link=Julian Cole |year=1951 |title=On a quasi-linear parabolic equation occurring in aerodynamics | journal=Quarterly of Applied Mathematics |volume=9 |issue=3 |pages=225–236 |doi=10.1090/qam/42889 |jstor=43633894 |doi-access=free }}{{cite journal | author=Eberhard Hopf|title=The partial differential equation u_t + uu_x = μu_xx | journal=Communications on Pure and Applied Mathematics|date=September 1950|volume=3|issue=3|pages=201–230 | doi=10.1002/cpa.3160030302 | author-link = Eberhard Hopf}}{{cite book |last=Kevorkian |first=J. |title=Partial Differential Equations: Analytical Solution Techniques |location=Belmont |publisher=Wadsworth |year=1990 |isbn=0-534-12216-7 | pages=31–35 }}

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

which turns it into the equation

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

which can be integrated with respect to $x$ to obtain

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

where $df/dt$ is an arbitrary function of time. Introducing the transformation $\backslash varphi\backslash to\; \backslash varphi\; e^f$ (which does not affect the function $u(x,t)$), the required equation reduces to that of the heat equationLandau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. Page 352-354.

:$\backslash frac\{\backslash partial\backslash varphi\}\{\backslash partial\; t\}=\; \backslash nu\; \backslash frac\{\backslash partial^2\backslash varphi\}\{\backslash partial\; x^2\}.$

The diffusion equation can be solved. That is, if $\backslash varphi(x,0)=\backslash varphi\_0(x)$, then

:$\backslash varphi(x,t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; \backslash nu\; t\}\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi\_0(x\text{'})\; \backslash exp\; \backslash left[-\backslash frac\{(x-x\text{'})^2\}\{4\backslash nu\; t\}\backslash right]dx\text{'}.$

The initial function $\backslash varphi\_0(x)$ is related to the initial function $u(x,0)=f(x)$ by

:$\backslash ln\; \backslash varphi\_0(x)\; =\; -\; \backslash frac\{1\}\{2\backslash nu\}\backslash int\_0^x\; f(x\text{'})\; dx\text{'},$

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

:$u(x,t)=-2\backslash nu\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash ln\backslash left\backslash \{\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; \backslash nu\; t\}\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash exp\backslash left[-\backslash frac\{(x-x\text{'})^2\}\{4\backslash nu\; t\}\; -\; \backslash frac\{1\}\{2\backslash nu\}\backslash int\_0^\{x\text{'}\}f(x$)dx\right]dx'\right\}

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

:$u(x,t)=-2\backslash nu\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash ln\backslash left\backslash \{\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash exp\backslash left[-\backslash frac\{(x-x\text{'})^2\}\{4\backslash nu\; t\}\; -\; \backslash frac\{1\}\{2\backslash nu\}\backslash int\_0^\{x\text{'}\}f(x$)dx\right]dx'\right\}.

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

Explicit expressions for the viscous Burgers' equation are available. Some of the physically relevant solutions are given below:Salih, A. "Burgers’ Equation." Indian Institute of Space Science and Technology, Thiruvananthapuram (2016).

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

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

This solution, that was originally derived by Harry Bateman in 1915,{{r|bateman}} is used to describe the variation of pressure across a weak shock wave{{r|landau}}. When $f^+=2$ and $f^-=0$ this simplifies to

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

with $c=1$.

If $u(x,0)\; =\; 2\backslash nu\; Re\; \backslash delta(x)$, where $Re$ (say, the Reynolds number) is a constant, then we haveWhitham, Gerald Beresford. Linear and nonlinear waves. John Wiley & Sons, 2011.

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

In the limit $Re\backslash to\; 0$, the limiting behaviour is a diffusional spreading of a source and therefore is given by

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

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

:

0, \quad \text{otherwise}.\end{cases}

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

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

:$u(x,t)\; =\; \backslash frac\{x\}\{t\}\backslash left[1\; +\; \backslash frac\{1\}\{e^\{Re\_0-1\}\}\backslash sqrt\{\backslash frac\{t\}\{t\_0\}\}\backslash exp\backslash left(-\backslash frac\{Re(t)x^2\}\{4\backslash nu\; Re\_0\; t\}\backslash right)\backslash right]^\{-1\}$

where $Re\_0$ may be regarded as an initial Reynolds number at time $t=t\_0$ and $Re(t)\; =\; (1/2\backslash nu)\; \backslash int\_0^\backslash infty\; udx=\backslash ln\; (1+\backslash sqrt\{\backslash tau/t\})$ with $\backslash tau\; =\; t\_0\; \backslash sqrt\{e^\{Re\_0\}-1\}$, may be regarded as the time-varying Reynold number.

In two or more dimensions, the Burgers' equation becomes

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

One can also extend the equation for the vector field $\backslash mathbf\; u$, as in

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

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

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

where $c(u)$ is any arbitrary function of u. The inviscid $\backslash nu=0$ equation is still a quasilinear hyperbolic equation for $c(u)>0$ and its solution can be constructed using method of characteristics as before.Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.

Added space-time noise $\backslash eta(x,t)\; =\; \backslash dot\; W(x,t)$, where $W$ is an $L^2(\backslash mathbb\; R)$ Wiener process, forms a stochastic Burgers' equation{{cite journal |first1=W. |last1=Wang |first2=A. J. |last2=Roberts | title=Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation |journal=Communications in Mathematical Physics |volume=333 |pages=1287–1316 |year=2015 |issue=3 |doi=10.1007/s00220-014-2117-7 |arxiv=1203.0463 |bibcode=2015CMaPh.333.1287W |s2cid=119650369 }}

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

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

• Chaplygin's equation
• Conservation equation
• Euler–Tricomi equation
• Fokker–Planck equation
• KdV-Burgers equation

{{reflist|30em}}

• [http://eqworld.ipmnet.ru/en/solutions/npde/npde1301.pdf Burgers' Equation] at EqWorld: The World of Mathematical Equations.
• [http://www.primat.mephi.ru/wiki/ow.asp?Burgers%27_equation Burgers' Equation] at NEQwiki, the nonlinear equations encyclopedia.

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