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Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Parabolic Partial Differential Equations > Burgers Equation
1.
∂w
=
∂2w
+w
∂w
. ∂t ∂x Burgers equation. It is used for describing wave processes in acoustics and hydrodynamics. ∂x2
1◦ . Solutions: 2 , x + λt + A 4x + 2A w(x, t) = 2 , x + Ax + 2t + B 6(x2 + 2t + A) w(x, t) = 3 , x + 6xt + 3Ax + B 2λ w(x, t) = , 1 + A exp(−λ2 t − λx) £ exp A(x − λt)] − B £ ¤ w(x, t) = −λ + A , exp A(x − λt) + B w(x, t) = λ +
where A, B, and λ are arbitrary constants. 2◦ . Other solutions can be obtained using the following formula (Hopf–Cole transformation): 2 ∂u , u ∂x
w(x, t) =
where u = u(x, t) is a solution of the linear heat equation, ut = uxx . 3◦ . The Cauchy problem with the initial condition: w = f (x)
at
t = 0,
−∞ < x < ∞.
Solution: ∂ w(x, t) = 2 ln F (x, t), ∂x
1 F (x, t) = √ 4πt
Z
· ¸ Z (x − ξ)2 1 ξ 0 0 − exp − f (ξ ) dξ dξ. 4t 2 0 −∞ ∞
References Hopf, E., The partial differential equation ut + uux = µuxx , Comm. Pure and Appl. Math., Vol. 3, pp. 201–230, 1950. Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., Vol. 9, No. 3, pp. 225–236, 1951. Ibragimov, N. H. (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, 1994. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations , Chapman & Hall/CRC, Boca Raton, 2004.
Burgers Equation c 2004 Andrei D. Polyanin Copyright °
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