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,^[1] nonlinear acoustics,^[2] gas dynamics, traffic flow. The equation was first introduced by Harry Bateman in 1915^[3]^[4] and later studied by Johannes Martinus Burgers in 1948.^[5]

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

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

When the diffusion term is absent (i.e. $\nu =0$ ), Burgers' equation becomes the inviscid Burgers' equation:

\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 previous equation is the advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration

{\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=0.

Solution of 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

{\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. The characteristic equations are

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

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

u=c,\quad x=ut+\xi

where $\xi$ 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 $u=c=f(\xi )$ on that characteristic. Therefore, the trajectory of that characteristic is

x=f(\xi )t+\xi .

Thus, the solution is given by

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 shock wave. In fact, the breaking time before a shock wave can be formed is given by

t_{b}={\frac {-1}{\min f'(x)}}.

Complete integral

Subrahmanyan Chandrasekhar provided the explicit solution in 1943 (On the decay of plane shock waves) when the initial condition is linear, i.e., $f(x)=ax+b$ , where a and b are constants.^[6] The explicit solution is

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

This solution is also the complete integral of the invisicd Burger's equation because it contains as many arbitrary constants as the number of independent variables appearing in the equation.^[7] Explicit solutions for other relevant initial conditions are, in general, not known.

Solution of viscous Burgers' equation

This is a numerical solution of the viscous two dimensional Burgers equation using an initial Gaussian profile. We see shock formation, and dissipation of the shock due to viscosity as it travels.

The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation ^[8]^[9]

u=-2\nu \frac{1}{\phi}\frac{\partial\phi}{\partial x},

which turns it into the equation

{\frac {\partial }{\partial x}}\left({\frac {1}{\phi }}{\frac {\partial \phi }{\partial t}}\right)=\nu {\frac {\partial }{\partial x}}\left({\frac {1}{\phi }}{\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)

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

{\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}+f(t)\phi

where $f(t)$ is a function that depends on boundary conditions. If $f(t)=0$ identically (e.g. if the problem is to be solved on a periodic domain), then we get the diffusion equation

\frac{\partial\phi}{\partial t}=\nu\frac{\partial^2\phi}{\partial x^2}.

The diffusion equation can be solved, and the Cole-Hopf transformation inverted, to obtain the solution to the Burgers' equation:

u(x,t)=-2\nu {\frac {\partial }{\partial x}}\ln \left\{(4\pi \nu t)^{-1/2}\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\}.

Other forms of Burger's equation

Generalized Burger's equation

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

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

where $c(u)$ is any arbitrary function of u. The inviscid $\nu =0$ equation is still a quasilinear hyperbolic equation for $c(u)>0$ and its solution can be constructed using method of characteristics as before.^[10]

Stochastic Burger's equation

Added space-time noise $\eta (x,t)$ forms a stochastic Burgers' equation^[11]

{\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 $h(x,t)$ upon substituting $u(x,t)=-\lambda \partial h/\partial x$ .

References

↑ 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
↑ Bateman, H. (1915). Some recent researches on the motion of fluids. Monthly Weather Review, 43(4), 163-170.
↑ 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. In Advances in applied mechanics (Vol. 1, pp. 171-199). Elsevier.
↑ Chandrasekhar, S. (1943). On the decay of plane shock waves (No. 423). Ballistic Research Laboratories.
↑ Forsyth, A. R. (1902). A treatise on differential equations.
↑ Julian Cole (1951). On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics, 9(3), 225-236.
↑ Eberhard Hopf (September 1950). "The partial differential equationy u_t + uu_x = μu_xx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.
↑ Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.
↑ W. Wang and A. J. Roberts. Diffusion approximation for self-similarity of stochastic advection in Burgers’ equation. Communications in Mathematical Physics, July 2014.

External links

Burgers' Equation at EqWorld: The World of Mathematical Equations.
Burgers' Equation at NEQwiki, the nonlinear equations encyclopedia.
Burgers' Equation Wolfram|Alpha search results.

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[1] 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

[2] 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

[3] Bateman, H. (1915). Some recent researches on the motion of fluids. Monthly Weather Review, 43(4), 163-170.

[4] Whitham, G. B. (2011). Linear and nonlinear waves (Vol. 42). John Wiley & Sons.

[5] Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. In Advances in applied mechanics (Vol. 1, pp. 171-199). Elsevier.

[6] Chandrasekhar, S. (1943). On the decay of plane shock waves (No. 423). Ballistic Research Laboratories.

[7] Forsyth, A. R. (1902). A treatise on differential equations.

[8] Julian Cole (1951). On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics, 9(3), 225-236.

[9] Eberhard Hopf (September 1950). "The partial differential equationy u_t + uu_x = μu_xx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.

[10] Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.

[11] W. Wang and A. J. Roberts. Diffusion approximation for self-similarity of stochastic advection in Burgers’ equation. Communications in Mathematical Physics, July 2014.