© 2000, 2008
Clyde M. Davenport
Updated 5/29/03, 7/2/06, 8/10/08
|Introduction||Hypercomplex Solution||Elementary Properties||Conclusions|
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 commutative hypercomplex mathematics. This is a system of mathematics that obeys the axioms of the classical complex variables and behaves in all ways like the classical complex analysis, while treating a 4-D independent variable. In order to understand the following, the reader should first review the Hypercomplex Math page.
We shall present something new in the form of the general analytical solution for the Burgers equation in terms of commutative hypercomplex mathematics. We shall show that, unlike for linear partial differential equations, there is not a variety of eigenfunctions from which a solution may be constructed in a variety of ways, but only one characteristic function that satisfies the equation when analyticity is required. The solution will be presented as an analytic function of one 4-D variable and as a complementary pair of functions of a classical complex variable. The latter are further reduced to a complementary pair of real-valued functions that are verified as solutions of the Burgers equation. Top
We emphasize analytical solution because the solution that we shall develop is analytic in the classical complex variable sense (i.e., is continuous and single-valued in the region of interest) and is "conservative" in physics terminology. Other solutions might be available that do not preserve, for example, total energy or mass in the full four-space. The solution is general in that we shall first transform the PDE into an ODE and then perform two integrations to obtain a solution with two arbitrary constants of integration. It is also general in the sense that it is a 4-D function that implicitly contains several different solution forms in 2-D and in one dimension and time, as we shall show.
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 about a point. They are typically derived from a force-energy-momentum-mass balance on an infinitesimal element at a point. As an aside, we note that they usually represent approximations only to first or second order of the effects that they describe, in order to keep the equations tractable. We might obtain a closed-form solution for the PDE, only to find that it does not accurately describe the extremes of the physical behavior that we are attempting to model. Top
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 body-centered coordinates whose origin moves with the subject particle of mass. At any given instant of time and for given local conditions, the integrated result indicates how the particle of mass will move next within the body-centered frame. We continue to emphasize: The integrated result describes an immediate, localized reaction, and says nothing about the long-term motion of a given particle of mass. For that reason, we must do a finite-element-like numerical calculation in order to coordinate the motions and interactions of all the particles, thereby obtaining a view of the overall motion of the fluid. This view explains why there is not any analytic-function solution of the Burgers equation that models turbulent behavior in the large. Any "solution" is point-localized. Top
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 Z=1x+iy+ jz+kct. Refer to the Hypercomplex Math page for a derivation. These relations transform partial differentials into an ordinary derivative with respect to a 4-D independent variable, Z. Observe that the first two equalities on the left are identical to those for 2-D classical complex variables, and the two on the right are extensions for the 4-D case. In the rightmost relation, c is the characteristic speed of propagation in whatever medium that we are working. These relations, being direct consequences of the 4-D Cauchy-Riemann conditions, are therefore continuity conditions. By applying them to the Burgers equation, we force continuity upon any solution(s). One might argue that we are simultaneously solving the Burgers equation with the Cauchy-Riemann equations. However, that would be somewhat misleading because the Cauchy-Riemann equations are satisfied by any analytic function. Top
Making the partial derivative conversions, we obtain:
where k is a 4-D algebraic basis element. This equation is still nonlinear, but is solvable by simple, direct methods. A first integration yields:
where A is an arbitrary 4-D constant of integration. This is again integratable. We complete the square on the left, move like terms to separate sides of the equation, then integrate to obtain: Top
where B is another arbitrary 4-D constant of integration. Here, our result is dependent upon the condition 2A-c2 > (0,0,0,0). Otherwise, we would have a tanh-1() integrated form. We shall investigate this alternate solution, later below. Our above result can be solved for u(Z):
This is the general analytical solution for the Burgers equation in terms of the 4-D commutative hypercomplex variable Z. That means that it is the complete solution for the ODE form inasmuch as we have integrated twice and have a solution including two arbitrary constants of integration. Moreover, it is the complete solution (i.e., the most general characteristic function) of the PDE form of the Burgers equation when continuity conditions are imposed. Again, we remark: Continuity conditions are nearly always assumed for PDEs, but are seldom explicitly imposed. Top
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 in the absence of any other disturbance, reflection, interference, etc. in the medium. Of course, in a real problem all parts of the medium might be simultaneously receiving independent impulses, and the net external impulse on a specific increment of material at a given time is an integration over all the disturbances originating in all of the rest of the material increments, taking account of the fact that the speed of propagation may vary from point to point in the medium. Consequently, a solution for a given problem involving arbitrary boundary and initial conditions must be pieced together point by point, similar to what is done in a finite element method. It might seem that we have gained no advantage. However, the analytical solution can yield further theoretical insight and provide new avenues for numerical solution.
The above solution is in terms of one 4-D independent variable, Z. We can break this down into two separate solutions in terms of one classical complex variable, each, by writing it in the canonical form, as follows. Recall the definition of a commutative hypercomplex function or operator: Top
The eigenvalues are classical complex variables, and the above definition means that a differential operator that operates on the 4-D variable Z also operates separately on the two eigenvalues of Z. Similarly for functions, the 4-D solution u (Z) is the sum of two 2-D solutions u () and u ():
i.e., the eigenfunctions of u (Z) in the canonical form are also solutions of the original PDE. Top
When we are given a 4-D expression such as u (Z) and we wish to expand it into canonical form, it is not so simple as writing two identical copies with just different independent variables. We must write every component of the u (Z) expression, including all constants, in canonical form and then reduce the resulting expression into simplest canonical form, as follows. We first expand the arbitrary 4-D constants:
All of the subscripted components are real or classical complex numbers. The Ai , Bi are arbitrary real or complex values, and in the present case can be zero without causing trouble. However, for some other PDE problems they might need to be invertible, hence in those cases none should be zero. Top
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 u (Z) solution, then the eigenfunction components and their complex conjugates would be the eigenvalues of the matrix form of u (Z). As stated earlier, each of the eigenfunctions of u (Z) is a solution of a KdV equation which is stated in terms of the associated 2-D independent variable or . Top
In the above terms, the eigenfunctions are:
We can carry this a step further and expand u (Z) into 4-D vector form:
This time, the individual vector components of u (Z) definitely are not solutions of a Burgers equation, because the Burgers equation is nonlinear.
We have obtained a solution u (Z) in terms of one variable having three space dimensions and time. By use of our 4-D constants A,B, we can position our center of action and the orientation of the line of action to any direction in space. One of the first questions that must be answered is, "Does it reduce to a solution of the original, one-dimensional Burgers equation when the y,z components are set to zero?" This is easy enough to check. Setting y=z=0, we get: Top
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 A,B and the base element k as constants with respect to differentiation, much as we would the classical imaginary, i. Taking the requisite partial derivatives of 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 u (x,ct), above, is not immediately illuminating about what we know of the physical situation that it is used to model. We see the true behavior when we examine the eigenfunctions u1() and u2() with the y,z coordinates set to zero: Top
As before, A1,A2,B1,B2 are real or classical complex numbers, all arbitrary. We can choose meaningful real values for the constants such that we have purely real expressions that are each solutions of the Burgers PDE. For purposes of illustration and interpretation, consider the case wherein B1=B2=0 and the A-values are such that the two constant expressions containing radicals reduce as follows:
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 t=0 and with the vertical lines representing the actual shock fronts. The first formula has the wavetrain moving to the right, with the rarefaction spike leading. The second formula has the wavetrain moving to the left, with the compression spike leading. Top
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 u2(x,ct) solution. On the other side of the plate, a rarefaction-compression disturbance (rarefaction leading), will move away in the positive x-direction. This is the u1(x,ct) solution. If, instead of a thin plate, we had a solid body that filled either half space, then we would see only one of the two possible solutions.
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 some width, even if small. Moreover, this idealized treatment takes no account of the fact that real materials undergo phase changes when they experience large pressure increases, let alone infinite increases. Top
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 approximating the behavior about a point in the medium. The Burgers equation, itself, is an approximation. The characteristic functions that we have found have obvious, real-world deficiencies that we can easily correct, as follows. We can temper our solutions by doing orthogonal-function approximations of the tan() terms with the same arguments, for example, (x-ct), in such a way that the same general shape is maintained, but with finite amplitude and only near-discontinuous transition. Indeed, we can match the shape to what is observed in real wave experiments for given fluids. This would tailor the general solutions to specific fluids. We could then use the tailored solutions in numerical calculations. For example, one such approximation for the tan-form u1(x,ct) eigenfunction is: Top
The (-) sign is necessary because each x exp(-x^2n) term has the same broad, general form as the tan(x) function, except with opposite amplitude peaks and with finite amplitude. Each succeeding term is superimposed upon the others, but has a narrower width than the previous ones. Each term has skirts that taper off smoothly to zero on either side, instead of coming down abruptly to zero at a specific width. The factors are arbitrary weighting constants that control the aggregate shape and amplitude of the approximating function. They can be determined by a least-squares fit to experimental data, thereby tailoring the solution to a specific fluid or other Burgers medium. By this means we can model a single shock with real-world properties. Following is an example: Top
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 tan(i z)=i tanh(z) from classical complex variables. In our u1(), u2() functions, suppose that we choose the A1, A2, B1, and B2 (complex) constants such that: Top
Now we have entirely different qualitative behavior! Recall the shape of the tanh(x) function, below. The horizontal lines at top and bottom are at ±1, and the tanh-curve executes a smooth transition between the two.
In the present context, 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 tan(..) solution represents an infinite-amplitude soliton spike that travels away from the site of the original disturbance at a speed c. Likewise, the derivative of the tanh(..) solution represents a finite-amplitude soliton disturbance that moves away from the site of the original disturbance at a speed c. By incrementally manipulating the components of the 4-D constant A, we can observe a gradual transition between the two cases.
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.
© 2000, 2008