Super-fast Computer Implementations of the Classical Fourier Series | Chapter 9| Mathematics and Computer Science: Contemporary Developments Vol. 9

In recent series of publications by the author, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system on a finite segment has proven unusually effective. On its basis, it was first discovered that knowledge of a finite number of Fourier coefficients of the function f from an infinite dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Then the corresponding parametric biorthogonal systems for trigonometric Fourier series were constructed and the corresponding over-convergence phenomena were confirmed. As a result, some very fast algorithms for reconstructing a function from its truncated series are proposed. These algorithms for accelerating the convergence of classical Fourier series are convenient for implementation in computer systems that allow extensive symbolic computations. These results are presented below as a general scheme for all types of classical Fourier series, with an emphasis on the details of computer implementation.The presented numerical experiments confirm the high efficiency of these convergence accelerations for sufficiently smooth functions. In conclusion, the main results are summarized, and some prospects for developing and generalizing the proposed approaches are discussed.

 

Author (s) Details

Anry Nersessian
Institute of Mathematics, National Academy of Sciences of Armenia. Yerevan, Republic of Armenia.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v9/2784

The Use of Spectral Methods for Assessing Durability at Post-processing Stage Requires Justification | Chapter 2 | Advanced Aspects of Engineering Research Vol. 6

 It can be difficult or impossible to record the loading process in some enormous structures, such as off-shore platforms, over many years of operation. Knowing the characteristics of the input random processes, it may appear promising to estimate the output characteristics using the structural dynamic equation. The spectral densities are used in the estimation of longevity using this method. Many methods for estimating fatigue damage using spectral densities are currently in the works. Is it reasonable to apply the spectral technique in the post-processing stage of machines when the prototype exists and the loading process realisation might potentially be recorded? It's worth noting that many of the developers of those approaches' ultimate goal is to produce outcomes that are similar to those that can be acquired straight from the RAINFLOW approach. The genuine loading procedure is examined in this research. The purpose of this work is to demonstrate some of the spectral approach's inconsistencies. The short review with an examination of the spectrum method's accuracy is offered in this work. The method of the digitally modelled process is introduced to demonstrate that several of the assumptions of spectral approaches are dubious. Most importantly, the author wants to know why spectral techniques experts recommend such a contentious approach of estimate.

Author(s) Details

I. Gadolina
Mechanical Engineering Research Institute of the Russian Academy of Sciences (IMASH RAN), Russia.

View Book :- https://stm.bookpi.org/AAER-V6/article/view/995