Burgers' equation (original) (raw)

From Wikipedia, the free encyclopedia

Partial differential equation

Solutions of the Burgers equation starting from a Gaussian initial condition u ( x , 0 ) = e − x 2 / 2 {\displaystyle u(x,0)=e^{-x^{2}/2}} $u(x,0)=e^{-x^{2}/2}$ .

N-wave type solutions of the Burgers equation, starting from the initial condition u ( x , 0 ) = e − ( x − 1 ) 2 / 2 − e − ( x + 1 ) 2 / 2 {\displaystyle u(x,0)=e^{-(x-1)^{2}/2}-e^{-(x+1)^{2}/2}} $u(x,0)=e^{-(x-1)^{2}/2}-e^{-(x+1)^{2}/2}$ .

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 u ( x , t ) {\displaystyle u(x,t)} $u(x,t)$ and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) ν {\displaystyle \nu } $\nu$ , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

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

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

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

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

Inviscid Burgers' equation

[edit]

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

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

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

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

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

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

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

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

Thus, the solution is given by

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

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]

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

Complete integral of the inviscid Burgers' equation

[edit]

The implicit solution described above containing an arbitrary function f {\displaystyle f} $f$ 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

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

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

Viscous Burgers' equation

[edit]

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

u ( x , t ) = − 2 ν ∂ ∂ x ln ⁡ φ ( x , t ) , {\displaystyle u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \varphi (x,t),} $u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \varphi (x,t),$

which turns it into the equation

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

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

∂ φ ∂ t − ν ∂ 2 φ ∂ x 2 = φ d f ( t ) d t , {\displaystyle {\frac {\partial \varphi }{\partial t}}-\nu {\frac {\partial ^{2}\varphi }{\partial x^{2}}}=\varphi {\frac {df(t)}{dt}},} ${\frac {\partial \varphi }{\partial t}}-\nu {\frac {\partial ^{2}\varphi }{\partial x^{2}}}=\varphi {\frac {df(t)}{dt}},$

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

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

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

φ ( x , t ) = 1 4 π ν t ∫ − ∞ ∞ φ 0 ( x ′ ) exp ⁡ [ − ( x − x ′ ) 2 4 ν t ] d x ′ . {\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'.} $\varphi (x,t)={\frac {1}{\sqrt {4\pi \nu t}}}\int _{-\infty }^{\infty }\varphi _{0}(x')\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}\right]dx'.$

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

ln ⁡ φ 0 ( x ) = − 1 2 ν ∫ 0 x f ( x ′ ) d x ′ , {\displaystyle \ln \varphi _{0}(x)=-{\frac {1}{2\nu }}\int _{0}^{x}f(x')dx',} $\ln \varphi _{0}(x)=-{\frac {1}{2\nu }}\int _{0}^{x}f(x')dx',$

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

u ( x , t ) = − 2 ν ∂ ∂ x ln ⁡ { 1 4 π ν t ∫ − ∞ ∞ exp ⁡ [ − ( x − x ′ ) 2 4 ν t − 1 2 ν ∫ 0 x ′ f ( x ″ ) d x ″ ] d x ′ } {\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\}} $u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{{\frac {1}{\sqrt {4\pi \nu t}}}\int _{-\infty }^{\infty }\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}f(x'')dx''\right]dx'\right\}$

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

u ( x , t ) = − 2 ν ∂ ∂ x ln ⁡ { ∫ − ∞ ∞ exp ⁡ [ − ( x − x ′ ) 2 4 ν t − 1 2 ν ∫ 0 x ′ f ( x ″ ) d x ″ ] d x ′ } . {\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\}.} $u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{\int _{-\infty }^{\infty }\exp \left[-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}f(x'')dx''\right]dx'\right\}.$

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

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]

Steadily propagating traveling wave

[edit]

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

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

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 f + = 2 {\displaystyle f^{+}=2} $f^{+}=2$ and f − = 0 {\displaystyle f^{-}=0} $f^{-}=0$ to

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

with c = 1 {\displaystyle c=1} $c=1$ .

Delta function as an initial condition

[edit]

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

u ( x , t ) = ν π t [ ( e R e − 1 ) e − x 2 / 4 ν t 1 + ( e R e − 1 ) e r f c ( x / 4 ν t ) / 2 ] . {\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}})/2}}\right].} $u(x,t)={\sqrt {\frac {\nu }{\pi t}}}\left[{\frac {(e^{Re}-1)e^{-x^{2}/4\nu t}}{1+(e^{Re}-1)\mathrm {erfc} (x/{\sqrt {4\nu t}})/2}}\right].$

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

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

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

u ( x , t ) = { x t , 0 < x < 2 ν R e t , 0 , otherwise . {\displaystyle u(x,t)={\begin{cases}{\frac {x}{t}},\quad 0<x<{\sqrt {2\nu Re\,t}},\\0,\quad {\text{otherwise}}.\end{cases}}} $u(x,t)={\begin{cases}{\frac {x}{t}},\quad 0<x<{\sqrt {2\nu Re\,t}},\\0,\quad {\text{otherwise}}.\end{cases}}$

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

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

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

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

Multi-dimensional Burgers' equation

[edit]

In two or more dimensions, the Burgers' equation becomes

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

One can also extend the equation for the vector field u {\displaystyle \mathbf {u} } $\mathbf {u}$ , as in

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

Generalized Burgers' equation

[edit]

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

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

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

Stochastic Burgers' equation

[edit]

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

∂ u ∂ t + u ∂ u ∂ x = ν ∂ 2 u ∂ x 2 − λ ∂ η ∂ x . {\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}}.} ${\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}-\lambda {\frac {\partial \eta }{\partial x}}.$

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

^ 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.
^ 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
^ 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
^ 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.
^ 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.
^ Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.
^ 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.
^ 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.
^ 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.
^ Forsyth, A. R. (1903). A Treatise on Differential Equations. London: Macmillan.
^ Chandrasekhar, S. (1943). On the decay of plane shock waves (Report). Ballistic Research Laboratories. Report No. 423.
^ 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.
^ 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. ISBN 0-534-12216-7.
^ 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.
^ 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.
^ 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.

Burgers' Equation at EqWorld: The World of Mathematical Equations.
Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.