Smoothing coefficients

One more thing is worth noting here. It would be erroneous to think that every scheme, which is convergent in the case of smooth coefficients, is obliged to converge in the case of discontinuity coefficients. Further explorations are connected with a family of schemes converging in the class of discontinuity coefficients, thus expanding possibilities. Let us stress that in the sequel we will deal with such schemes only. 

The error of approximation in the class of smooth coefficients. The main... 

Remark It can be shown that the approximation in the class of smooth coefficients is necessary and sufficient for the convergence of the homogeneous scheme (2). 

The error of approximation in the class of smooth coefficients. The main point of the theory is the accurate account of the accuracy of the uniform scheme (16)—(17) in the class of continuous and discontinuous functions k(x), q(x) and f(x). In preparation for this, let u(x) be an exact solution of the original problem... 

A 3-band DWT for the spectrum x = (xq.xi,. .., xg) is shown in Fig. 1. There is one low pass and two high pass filters producing one set of scaling (or smoothed) coefficients and two sets of wavelet (or detailed) coefficients. As before, to go from one level to the next, only the scaling coefficients are filtered and the number of coefficients in each band is reduced by one third when moving from one level to the next. We have presented a transform with two levels (niev = 2)... 

Becau.se of the general utility of smoothing and its wide application, guidelines have been developed for its use, equations are available for calculating the smoothing coefficients, and it has been applied to two-dimensional data such as diode-array spectra. 

Figure 4.4 shows the variation of the discontinuous coefficient of friction, (4.1), and the smoothed coefficient of friction function, (4.21), for two values of the parameter r. As can be seen, the approximation of the smoothed function is considerably improved by increasing the value of r from 20 to 200. The stick-slip limit cycle of the discontinuous system is shown in Fig. 4.5. The stick boundary is shown in this figure by a horizontal dashed line at m = 1 /p (i.e., y = Vb). In this figure, two more limit cycles are shown which correspond to the system with...

The difference between the results obtained from the system with smoothed coefficient of friction, and the original system (discontinuous friction) is significant for r = 20. This, of course, is the same as the difference shown in Fig. 4.5. For r = 200, on the other hand, the differences among numerical simulation of the original equations, numerical simulation of the smoothed equations, and the averaging results are much smaller. 

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