The Algebraic Method

In order to familiarize the reader with Prandtl s evaluation procedure [3-5] on which the algebraic method is based, let us examine the steady laminar motion of a fluid along a plate and denote by x the coordinate along the plate, by y the distance to the plate, by u and v the x and y components of the velocity, and by p the pressure. The equations of motion of an incompressible fluid of density p and kinematic viscosity v have the form 

As suggested by Prandtl, the entire zone of motion can be subdivided into two regions a boundary layer region near the plate of thickness 6h = 5,(x), in 

Denoting by u0 the value of u at the outer edge of the boundary layer and taking into account that 0 X 1 and 0 Y 1, one can write 

Replacing the terms of the equation of motion with their evaluations, one 

The exact solution of the problem leads to the same expression with a proportionality constant between 3 and 5, depending on the definition of the thickness of the boundary layer. In the following sections, the preceding evaluation procedure is applied to a large number of problems, particularly to complex cases for which limiting solutions can be obtained. As already noted in the introduction, the terms in the transport equations will be replaced by their evaluating expressions multiplied by constants. The undetermined constants will then be determined from solutions available for some asymptotic cases. 

If we have p projections of a picture and each projection contains r rays, we have a system of p-r equations in n2 unknowns, and a solution 

In order to have all equations in a compact form, the double index , k) of each ray is replaced by the single index (h) with the transformations 

In this way, all the projections of a picture are represented by a single column vector [g ], and the geometrical parameters form a matrix [afa], known as the weightingfactorsmatrix, which has p -r = t rows 

A reconstruction is a procedure which reverses the projection process, and the reconstruction equations can therefore be obtained from equation 3.1 with a matrix [bfa] which represents the inverse weighting factors matrix-. 

The values f of the reconstructed picture are obtained therefore by the following equations  

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The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. 

Algebraic methods - in these techniques calculation of grid coordinates is based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. 

These results are identical to those obtained using the gra]riiical and the algebraic methods. 

When the algebraic methods are used, care must be taken that the constraints are obeyed. This usually means following a boundary until the search leaves the vicinity of the constraints. This should be kept in mind while reading about the various procedures. 

The double degeneracy of the 0(2) case corresponds to the fact that the algebraic method describes in this case two Morse potentials related to each other by a reflection around x = 0. This is a peculiar feature of one-dimensional problems, and it does not appear in the general case of three dimensions. If one uses the 0(2) basis for calculations, this peculiarity can be simply dealt with by considering only the positive branch of M. 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

If one assumes that S/8h is independent of x, one arrives again at Eq. (58). If a particular form is chosen for the function F, as one proceeds in the method of polynomials, the calculation of the constants A, B, and E becomes possible. While in this particular problem one can follow a parallelism between the algebraic method and the method of polynomials, the same parallelism can no longer be identified in the other examples examined. It is worth emphasizing that the use of the boundary layer thickness concept in the algebraic method does not imply the existence of a similarity solution. In general, the algebraic method interpolates between the two similarity solutions which are valid in the two asymptotic cases. 

Let us compare the result obtained on the basis of the algebraic method with those obtained from an exact solution of Eq. (67). The solution of the latter equation for the boundary conditions (66) can be obtained through a similarity transformation. Indeed,... 

Fig. 1. Comparison of the predictions of the algebraic method (solid line) with the experimental data of Gryzagoridis and with the series solution of Szewczyk (broken line).

Let us note that Eqs. (107) can be obtained from scaling arguments alone, since near the wall u/u oc y/8h, where 8h is given by 8k oc (vx/u )112 [see Eq. (26)]. The algebraic method leads in this case to... 

It is important to note that the algebraic equation (131) probably holds even when the velocity of suction at the wall v0 x) assumes a form different from Eq. (114). The latter equation involves the existence of a similarity solution, but this is not required for applying the algebraic method of interpolation. 

Pleskov and Filinovski is about 6% below the exact results, the comparison indicates that the algebraic method is accurate within 5 to 10%. 

The operating point may be found as the intersection of plots of the pump and system heads as functions of the flow rate. Or an equation may be fitted to the pump characteristic and then solved simultaneously with Eq. (7.16). Figure 7.17 has such plots, and Example 7.2 employs the algebraic method. 

In this Sample Problem, you will see two different methods of solving the problem the algebraic method and the ratio method. Choose the method you prefer to solve this type of problem. 

In order to apply the algebraic methods based on so(4, 2) it is necessary to carry out a noncanonical and nonunitary transformation of Eq. (249). Thus, multiplying on the left by r and applying the scaling transformation (cf. Section V and Appendix B) to operators and functions... 

Strictly, the application of the algebraic method in non-linear perturbation theory requires the existence of a small parameter e in the equations, and this can be revealed by a non-dimensionalization procedure. However, even for such a simple set of equations, this is a complicated process which can be avoided by carrying out a numerical investigation of the time-scales present in the problem. By examining the eigenvalues of a linear approximation to the system as described in Section 4.7, it becomes clear that there are two negative fast modes in the above equations over all conditions tested. These are indicated by the presence of two large... 

These values are plotted on the equilibrium-distribution diagram, as shown in Figure 3.7. The resulting curve intersects the equilibrium curve to give the interface compositions xA. - 0.231 and yA = 0.494. This solution agrees with the solution obtained previously by the algebraic method. 

Thus we are left with the problem of fitting five independent parameters, while the algebraic expansion depends only on three parameters (four, including N). The root-mean-square (rms) error of the algebraic fit of Table I is 1.5 cm , while the error achieved with the Dunham expansion (3.56) is 1.1 cm . As pointed out earlier, it is difficult to highlight the advantages of the algebraic method in this simple case. However, we will soon see how the economy of parameters achieved here becomes substantial in more complex situations. 

The purported ease of calculating various quantities via the algebraic method has been cited throughout this article. As such, this ease of computation deserves further comment and explanation. Therefore, we outline in Section V.C the computer routines used most commonly in the algebraic model. 

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