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Burgers' equation

Burgers equation
Order2nd
SolvableExactly solvable
The Cauchy problemSolvable by the Cole-Hopf transform
HierarchyBurgers' hierarchy
First introducedForsyth, 1906

Contents

Definition

u t +αu u x =μ u xx

It includes nonlinearity and dissipation together in the simplest possible way and may be thought of as a nonlinear version of the heat equation.

History

The equation was known to Forsyth (1906) and had been discussed by Bateman (1915). Due to extensive works of Burgers (1948) it is known as Burgers’ equation.

The Cole-Hopf transformation

The Burgers' equation then can be linearized by the Cole-Hopf transformation

u( x,t )= 2μ α z x z ,

z=z( x,t ) .

Substituting it to the Burgers' equation one'll get the linear heat equation

z t =μ z xx .

The Cauchy problem

Let us suppose that function u( x,t ) satisfies the Burgers' equation

u t +αu u x =μ u xx

and that in the inital moment

u( x,t=0 )=φ( x ) .

Finally one will obtain the solution of the Cauchy problem for the Burgers' equation in the form

u( x,t )= + xθ t exp{ αG( θ,x,t ) 2μ }dθ + exp{ αG( θ,x,t ) 2μ }dθ

with

G( θ,x,t )= ( xθ ) 2 2t + 0 θ φ( ξ )dξ .

Solutions

Let us change the scale by introducing the new variables x'= μ α x,t'= μ α 2 t . Then the Burgers' equation will take the form (primes are omitted)

u t u u x = u xx .

Rational solutions

One can obtain the following rational solutions of the Burgers' equation

u( x,t )= Ax B+t ,

u( x,t )=λ+ 2 x+λt+A , u( x,t )= 4x+2A x 2 +Ax+2t+B , u( x,t )= 6( x 2 +2t+A ) x 3 +6xt+3Ax+B ,

where A,B and λ are arbitrary constants.

Other solutions

One can also obtain the following solutions of the Burgers' equation

u( x,t )= 2λ 1+Aexp( λ 2 tλx ) , u( x,t )=λ+A exp[ A( xλt ) ]B exp[ A( xλt ) ]+B ,

u( x,t )=λ+2Ath[ A( xλt )+B ],

u( x,t )= λ λ 2 t+A [ 2th( λx+B λ 2 t+A )λxB ],

u( x,t )=λ+2Atg[ A( λtx )+B ],

u( x,t )= 2λcos( λx+A ) Bexp( λ 2 t )+sin( λx+A ) ,

u( x,t )= 2 π( t+λ ) exp[ ( x+A ) 2 4( t+λ ) ] [ B+Aerf( x+A 2 t+λ ) ] 1 ,

where A,B and λ are arbitrary constants and erf(z) 2 π 0 z e t 2 dt is the "error function".

Other solutions of the Burgers' equation can be obtained using the Cole-Hopf transformation.

Applications and connections

The Burgers' equation is used as a model in fields as wide as
  • acoustics
  • continues stochastic processes
  • dispersive water waves
  • gas dynamics
  • heat conduction
  • longitudinal elastic waves in an isotropic solid
  • number theory
  • shock waves
  • turbulence
and so forth.

Generalizations

There are some nonlinear partial differential equations that may be thought of as a formal generalization of the Burgers' equation:

the Burgers-Huxley equation

u t +αu u x =μ u xx +μu+η u 2 δ u 3

the Kolmogorov-Petrovsky-Piskunov equation (Fisher equation)

u t =μ u xx +μu+η u 2

the Korteweg-de Vries-Burgers equation

u t +αu u x + u xxx =μ u xx

the Kuramoto-Sivashinsky equation

u t +u u x +α u xx +β u xxx +γ u xxxx =0

See also

Burgers' hierarchy

References

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  2. Barenblatt G.I. Similarity, self-similarity, and intermediate asymptotics (in Russian) // 2nd edition, Gidrometeoizdat, Leningrad, 1982. — 256 p.
  3. Bateman H. Some recent researches on the motion of fluids // Monthly Weather Review, 1915. 43:4. Pp.163–170. DOI:10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2
  4. Benton E.R. Some new exact, viscous, nonsteady solutions of Burgers' equation // Phys. Fluids, 1966. 9. Pp.1247-1248. DOI:10.1063/1.1761828
  5. Burgers J.M. A mathematical model illustrating the theory of turbulence // Adv. Appl. Mech., 1948. 1. Pp.171-199.
  6. Cole J.D. On a quasilinear parabolic equation occurring in aerodynamics // Quart. Appl. Math., 1951. 9. Pp.225-236.
  7. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C. Solitons and nonlinear wave equations // Academic Press, New York, 1982. — 630 p.
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  9. Forsyth A.R. Theory of differential equations. Part 4. Partial differential equations (Vol. 5-6) // 1906. — 1118 p.
  10. Goriely A. Integrability, partial integrability and nonintegrability for systems of ordinary differential equations // J. Math. Phys., 1996. 37:4. Pp.1871-1893. DOI:10.1063/1.531484
  11. Hopf E. The partial differential equation ut + uux = μuxx // Comm. Pure and Appl. Math., 1950. 3:3. Pp.201–230. DOI:10.1002/cpa.3160030302
  12. Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
  13. Kudryashov N.A. Methods of nonlinear mathematical physics (in Russian) // Moscow Engineering Physics Institute, Moscow, 2008. — 352 p.
  14. Kudryashov N.A. Self-similar solutions of the Burgers hierarchy // Applied mathematics and computation, 2009. 215:5. Pp.1990-1993. DOI:10.1016/j.amc.2009.07.048
  15. Kudryashov N.A., Sinelshchikov D.I. Exact solutions of equations for the Burgers hierarchy // Appl. Math. Comput., 2009. 215:3. Pp.1293-1300. DOI:10.1016/j.amc.2009.06.010
  16. Loskutov A.Yu., Mikhailov A.S. Introduction to synergetics (in Russian) // Nauka, Moscow, 1990. — 270 p.
  17. Malfliet W. Approximate solution of the damped Burgers equation // J. Phys. A: Math. Gen., 1993. 26. Ll.723-728. DOI:10.1088/0305-4470/26/16/003
  18. Oleinik O.A. Discontinuous solutions of non-linear differential equations (in Russian) // UMN, 1957. 12:3(75). Pp.3–73.
  19. Parker A. On the periodic solution of the Burgers equation: a unified approach // Proc. R. Soc. Lond. A, 1992. 438. Pp.113-132.
  20. Polyanin A.D., Zaitsev V.F. Handbook of nonlinear equations of mathematical physics. Exact solutions (in Russian) // Fizmatlit, Moscow, 2002. — 432 p.
  21. Rosenblatt M. Remarks on the Burgers equation // J. Math. Phys., 1968. 9. Pp.1129-1136. DOI:10.1063/1.1664687
  22. Weiss J., Tabor M., Carnevale G. The Painlevé property for partial differential equations // J. Math. Phys., 1983. 24:3. Pp.522–526. DOI:10.1063/1.525721

External links

  1. Burgers' equation in Wikipedia, the free encyclopedia
  2. Burgers' equation in Wolfram MathWorld
  3. Burgers' equation in EqWorld, the world of mathematical equations

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