The General Analytical Solution for the Burgers
Equation

© 2000, 2008

Clyde M. Davenport

cmdaven@comcast.net

Updated 5/29/03, 7/2/06, 8/10/08

Introduction | Hypercomplex Solution | Elementary Properties | Conclusions |

Introduction The Burgers equation is a nonlinear partial differential equation of second order, as follows: It is used in fluid dynamics teaching and in engineering as a simplified model for turbulence, boundary layer behavior, shock wave formation, and mass transport. It has been studied and applied for many decades. Many different closed-form, series approximation, and numerical solutions are known for particular sets of boundary conditions. Top In the following, we shall be applying We shall present something new in the form of the
We emphasize One might recall that partial differential equations, especially those used to model the behavior of some material substance, typically describe the behavior of some variable, parameter, or physical effect If we obtain a characteristic function of the Burgers equation by whatever means, then the function will embody the same description of physical effects as the PDE and must be viewed and applied in the same way; i.e., as describing the variation of effects about a point, and not necessarily the macro behavior over all space. The Burgers equation was developed to describe the immediate, localized
reaction of a tiny, incremental element of mass to given external forces and momentum and energy inputs. In physics terms, the integrated result is expressed in Hypercomplex Solution As mentioned earlier, our basic approach to solution is to first convert the (nonlinear) Burgers equation to an ODE, then solve it by means of classical methods. To convert partial differentials to ordinary derivatives, we shall use the following consequences of the 4-D Cauchy-Riemann equations: where Making the partial derivative conversions, we obtain: where where where u(Z):
This is the We remind the reader that this is not the one and only, final solution of the Burgers equation. Just as does the Burgers PDE, it merely represents the behavior about a point. Given an external force, impulse, etc. on a small increment of material at the point, the characteristic function that we have found indicates how the disturbance will be propagated away from that point The above solution is in terms of one 4-D independent variable, The eigenvalues are classical complex variables, and the above definition means that a differential operator that operates on the 4-D variable i.e., the eigenfunctions of When we are given a 4-D expression such as
All of the subscripted components are real or classical complex numbers. The A Recall that any 4-D commutative hypercomplex entity, such as an analytic function of a 4-D variable, can be represented in 4 X 4 real matrix form. If this were done for our In the above terms, the eigenfunctions are: We can carry this a step further and expand
This time, the individual vector components of
Elementary Properties We have obtained a solution Now, because we have established a classical analysis that is applied to 4-D functions [see the
Hypercomplex Math page], we can treat the 4-D constants u (x,ct) in the usual way,
The reader may verify that if we combine these equations by addition, then we get precisely the one-dimensional Burgers equation. Our reduced, one-dimensional solution
As before, A These represent wavetrains of infinite-amplitude shocks moving to the right and left, respectively. The following figure displays two successive shocks in either wavetrain, depicted at A way to visualize this in a real medium is as
follows. Consider an xyz-space completely filled with a Burgers medium, and let it be divided into two halves by a thin, rigid plate at the origin and lying in the yz-plane. Now let the plate experience a sharp, uniform impulse in the negative x-direction. On that side of the plate, a planar compression-rarefaction disturbance (compression leading) will move away in the negative x-direction.
This is the There are practical problems associated with these solutions. First, these idealized, propagating disturbances can occur only under well-prescribed circumstances. Basically, the wavetrain must propagate into a uniform, stationary medium. There must be no interferences, resistances, reflections, etc. If any of these are present, then we must use a numerical, finite-element method of solution. Top Secondly, these solutions reveal just how idealized the Burgers equation is, with respect to the physical behavior that it (approximately) models. In the Burgers solutions, the compression, rarefaction transition is infinite in amplitude, but in a real medium is always finite. Another way to look at it is, the Burgers model treats the medium as if it is an infinitely hard, dense substance that reacts instantaneously, resists impulse, and yet propagates a disturbance at mundane speeds. A shock wave in a real material will never execute a pressure change in such an abrupt fashion. There will always be a smooth transition of Thirdly, a single shock of the above nature will never be seen in any real medium, even though single shocks are easily produced in real media. The problem is in the precise width, the discontinuous initiation, the infinite amplitude, and the discontinuous drop from positive to negative, none of which are seen in a real medium. There must always be smooth, nondiscontinuous transitions, even though they might be small. Below, we present a way out of the dilemma. Recall that we must view any Burgers solution (characteristic function) as The (-) sign is necessary because each Although remarkable and as fitting of the experimental facts as they are, the above solutions (characteristic functions) are not the complete story. Consider the trigonometric identity Now we have entirely different qualitative behavior! Recall the shape of the tanh(..) functions represent a state change from one level to another, with a smooth s-shaped transition, that is propagated, or diffuses, throughout the medium. Clearly, we could manipulate the constants to adjust the sharpness of the transition. This could represent an incremental change in temperature, pressure, or concentration that diffuses away from the point of disturbance with a wavefront speed c. Moreover, the
first derivatives of these functions represent solitary, unchanging disturbances (solitons) moving away to infinity in either direction.
Top
The derivative of the Conclusions We have found the most general characteristic function for the 1-D Burgers equation when analyticity is imposed. It indicates that a planar disturbance will be propagated outward at the characteristic speed for the medium, as expected. We can have a single shock front moving in either direction, dual shocks moving in opposite directions, a permanent pressure change propagating in either or both directions, or a derivative soliton moving in either or both directions. While these are all possibilities in a real medium, we can see that the Burgers equation overstates the amplitude in a shock situation. We have presented a damped solution form that can be fitted to the experimental data from a specific fluid. I have now completed similar analyses for the Korteweg-de Vries equation (KdV page) and the Boussinesq equation (Boussinesq page). 6/20/03 © 2000, 2008 |