FRACTIONAL CALCULUS OF THERMOELASTIC p-WAVES REFLECTION UNDER INFLUENCE OF GRAVITY AND ELECTROMAGNETIC FIELDS

In this paper, we discussed the longitudinal harmonic waves reflection from a solid elastic half-space with electromagnetic and gravity fields influence, considering a fractional order via fractional exponential function method. The clarifications are required for the reflection amplitudes ratios (i.e. the ratios between the reflected waves amplitude and the incident waves amplitude). The results obtained were calculated analytically and displayed by graphs to show the physical meaning of the phenomenon. A comparison has been made between the fractional and integer derivatives. The results of this paper demonstrate the rigor and effectiveness of the considered fractional technique.

Present problems of vibration isolation in heavy mining machines at long-term cyclic loads

On the basis of the developed simulation model and the Boltzmann-Volterra integral relations with kernels of relaxation and aftereffect, an equation was worked out, which made it possible to take into account the rubber viscoelastic properties in full volume; in this equation, stiffness operator of elastic suspension in the machine is written by using fractional exponential function of the Yu. Rabotnov’s type; on the basis of the mathematical model, the basic parameters of the machine under the study were calculated; in particular, for the vortex mixer, the time dependences of amplitude of the mixer housing vibrations and coefficient of vibration isolation efficiency were calculated with taking into account aging of elastic link material in the machines; the calculation results were compared with the results of industrial tests of the vortex mixer operation lasting for 16 years. The theory and method for calculating vibration isolation systems with rubber elastic links for heavy mining machines were developed with taking into account material structure changes due to the effects of aging.

A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer

In this paper, we consider a general fractional-order derivataive of the Liouville-Caputo type with the non-singular kernel of the Rabotnov fractional-exponential function for the first time. A new general fractional-order derivataive heat transfer model is discussed in detail. The general fractional-order derivataive formula is a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

A new general fractional-order derivative with Rabotnov fractional-exponential kernel

In this article, a general fractional-order derivative of the Riemann-Liouville type with the non-singular kernel involving the Rabotnov fractional-exponential function is addressed for the first time. A new general fractional-order derivative model for the anomalous diffusion is discussed in detail. The general fractional-order derivative operator formula is as a novel and mathematical approach proposed to give the generalized presentation of the physical models in complex phenomena with power law.

Equilibrium of an elastic medium with after-effect

Editorial Note: The original version of the paper is usually cited asYu.N. Rabotnov, Equilibrium of an elastic medium with after-effect (in Russian). Prikladnaya Matematika i Mekhanika (J. Appl. Math. Mech.) 12, No 1 (1948), 53–62.This is a re-printed version of the paper, translated from Russian into English by Marina Shitikova, under the kind permission by the Editorial Board of the journal “Prikladnaya Matematika i Mekhanika”, Institute of Mechanics — Russian Academy of Sciences, http://pmm.ipmnet.ru/ru/, a journal published with English translation since 1958, as “Journal of Applied Mathematics and Mechanics” by Elsevierhttp://www.journals.elsevier.com/journal-of-applied-mathematics-and-mechanics/.Note that nowadays, the fractional exponential function ∋α (β, t) introduced by Rabotnov as (2.5), is known also as the Rabotnov function and as a special case of the Mittag-Leffler function widely used in fractional calculus. Rabotnov spoke also about differential equations with fractional derivatives (Sect. 3) but preferred to work with integral equations methods.Yury Nikolaevich Rabotnov (24 February 1914–15 May 1985) was a great Russian scientist in the field of Mechanics. More about his life and contributions, the readers can find in this journal (Fract. Calc. Appl. Anal.), in the articles: D. Valério, J. Tenreiro Machado, V. Kiryakova, 17, No 2 (2014), 552–578; DOI: 10.2478/s13540-014-0185-1; http://link.springer.com/article/10.2478/s13540-014-0185-1 Yu. A. Rossikhin and M. V. Shitikova, 10, No 2 (2007), 111–122; http://www.math.bas.bf/~fcaa Yu.A. Rossikhin, M.V. Shitikova, 17, No 3 (2014) — this same issue, 674–683; DOI: 10.2478/s13540-014-0192-2.