scholarly journals Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1069
Author(s):  
Asmaa M. Elsayed ◽  
Viktor N. Orlov

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method.

PRIMUS ◽  
1992 ◽  
Vol 2 (2) ◽  
pp. 165-172 ◽  
Author(s):  
Tilak de Alwis

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Chernet Tuge Deressa ◽  
Gemechis File Duressa

AbstractWe consider a SEAIR epidemic model with Atangana–Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic.


Author(s):  
Duangui Wang

Formulation of the problem. In the chamber-vocal genre, the composer exists in two images: he is both the interpreter of the poetic composition and the author of a new synthetic music and poetic composition. The experience of the style analysis of one of the best examples of Ukrainian vocal lyrics of the first third of the 20th century shows that the cycle op. 20 characterizes the mature style of the composer, which was formed, on the one hand, under the influence of European Romanticism. On the other hand, the essence of the Ukrainian “branch” of the Western European song-romance (“solo-singing”) is revealed by the prominent national song-romance intonation, filled with not only a romantic worldview, but also with some personal sincerity, chastity, intimate involvement with the great in depth and simplicity poetry line, read from the individual position of the musician. The paradox is as follows. Although Pushkin’s poetry is embodied in a “holistic adequacy” (A. Khutorskaya), and the composer found the fullest semantic analogue of the poetic source, however, in terms of translating the text into the Ukrainian language, the musical semantics changes its intonation immanence, which naturally leads to inconsistency of the listeners’ position and ideas about the style of Russian romance. We are dealing with inter-specific literary translation: Pushkin’s discourse creates the Ukrainian romance style and system of figurative thinking. The purpose of the article is to reveal the principle of re-semantization of the intonation-figurative concept of the vocal composition by V. Kosenko (in the context of translating Pushkin’s poetry into the Ukrainian language) in light of the theory of interspecific art translation. Analysis of recent publications on the topic. Among the most recent studies of Ukrainian musicology, one should point out the dissertation by G. Khafizova (Kyiv, 2017), in which the theory of modelling of the stylistic system of the vocal composition as an expression of Pushkin’s discourse is described. The basis for the further stylistic analysis of V. Kosenko’s compositions is the points from A. Hutorska’s candidate’s thesis; she develops the theory of interspecific art translation. The types of translation of poetry into music are classified according to two parameters. The exact translation creates integral adequacy, which involves the composer’s finding a maximally full semantic analogue of the poetic source. The free translation is characterized by compensatory, fragmentary, generalized-genre adequacy. Presenting the main material. The Zhitomir period for Viktor Kosenko was the time of the formation of his creative style. Alongside the lyrical imagery line, the composer acquired one more – dramatic, after his mother’s death. It is possible that the romances on the poems of A. Pushkin are more late reflection of this tragic experience (op. 20 was created in 1930). “I Loved You” opens the vocal cycle and has been dedicated by A. V. Kosenko. The short piano introduction contains the intonation emblem of the love-feeling wave. The form of the composition is a two part reprising (А А1) with the piano Introduction and Postlude. The semantic culmination is emphasized by the change of metro-rhythmic organization 5/4 (instead of 4) and the plastic phrase “as I wish, that the other will love you” sounding in the text. Due to these melodies (with national segments in melo-types, rhythm formulas and harmony) V. Kosenko should be considered as “Ukrainian Glinka”, the composer who introduced new forms and “figures” of the love language into the romantic “intonation dictionary”. In general, V. Kosenko’s solo-singing represents the Ukrainian analogue of Pushkin’s discourse – the theme of love. The melos of vocal piece “I Lived through My Desires” is remembered by the broad breath, bright expression of the syntactic deployment of emotion. On the background of bass ostinato, the song intonation acquires a noble courage. This solo-singing most intermediately appeal to the typical examples of the urban romance of Russian culture of the 19th century. “The Raven to the Raven” – a Scottish folk ballad in the translation by A. Pushkin. V. Kosenko as a profound psychologist, delicately transmits the techniques of versification, following each movement of a poetic phrase, builds stages of the musical drama by purely intonation means. The semantics of a death is embodied through the sound imaging of a black bird: a marching-like tempo and rhythm of the accompaniment, with a characteristic dotted pattern in a descending motion (like a raven is beating its wings). The middle section is dominated by a slow-motion perception of time space (Andante), meditative “freeze” (size 6/4). The melody contrasts with the previous section, its profile is built on the principle of descending move: from “h1” to “h” of the small octave (with a stop on S-harmony), which creates a psychologically immersed state, filled by premonition of an unexpected tragedy. In general, the Ukrainian melodic intonation intensified the tragic content of the ballad by Pushkin. The musical semantics of V. Kosenko’s romances is marked by the dependence on the romantic “musical vocabulary”, however, it is possible to indicate and national characteristics (ascending little-sixth and fifth intervals, which is filled with a gradual anti-movement; syllabic tonic versification, and other). Conclusion. The romances (“solo-singings”) by V. Kosenko belongs to the type of a free art translation with generalized-genre adequacy. There is a re-semantization of poetic images due to the national-mental intonation. Melos, rhythm, textural presentation (repetitions), stylization of different genre formulas testify to the rare beauty of Kosenko’s vocal style, spiritual strength and maturity of the master of Ukrainian vocal culture. Entering the “Slavic song area”, the style of Ukrainian romance, however, is differenced from the Russian and common European style system of figurative and intonation thinking (the picture of the world).


Author(s):  
Dali Chen ◽  
Dingyu Xue ◽  
YangQuan Chen

Firstly the one-dimension digital fractional order Savitzky-Golay differentiator (1-D DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is proposed to estimate the fractional order derivative of the noisy signal. The polynomial least square fitting technology and the Riemann-Liouville fractional order derivative definition are used to ensure robust and accuracy. Experiments demonstrate that 1-D DFOSGD can estimate the fractional order derivatives of both ideal signal and noisy signal accurately. Secondly, the two-dimension DFOSGD is obtained from 1-D DFOSGD by defining a group of direction operators, and a new image enhancing method based on 2-D DFOSGD is presented. Experiments demonstrate that 2-D DFOSGD has very good performance on image enhancement.


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