Affiliation:
1. Belarusian State University
Abstract
In this paper, we represent new examples of constructing model problems of the mechanics of a deformable solid using a fractional differentiation apparatus. The solutions to boundary problems of mechanics are found, in which the defining differential equations have a fractional order. In particular, such problems as a model of a “fractal” oscillator, a model problem on the dynamic of wave propagation in rock, model problems on the deformation of wave propagation in deformable viscoelastic media (a semi-infinite viscoelastic rod) for various viscoelasticity models are considered. When building the solutions, the Mainardi algorithm and the Laplace transformation are used. Model solutions for the considered problems are built. Asymptotic solutions of wave propagation equations in viscoelastic media under different viscoelasticity models are obtained.
Publisher
Publishing House Belorusskaya Nauka
Subject
Computational Theory and Mathematics,General Physics and Astronomy,General Mathematics
Reference11 articles.
1. Samko S. G., Kilbas A. А., Marichev A. I. Fractional Integrals and Derivatives and Some Applications. Minsk, Nauka i tekhnika Publ., 1987. 687 p. (in Russian)
2. Miller K., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, Wiley, 1993. 384 p.
3. Zhuravkov M., Romanova N. Review of methods and approaches for mechanical problem solutions based on fractional calculus. Mathematics and Mechanics of Solids, 2014, vol. 21, no. 5, pp. 595–620 https://doi.org/10.1177/1081286514532934
4. Bosiakov S. Fractional Calculus in Biomechanics. Encyclopedia of Continuum Mechanics, vol. 2. Berlin, Heidelberg, Springer, 2020, pp. 946–953. https://doi.org/10.1007/978-3-662-55771-6_76
5. Rossikhin Y. A., Shitikova M. V. Calculus Models in Dynamic Problems. Viscoelastivity. Handbook of Fractional Calculus with Applications, 2019, vol. 7, part A, pp. 139–158.