Mathematical  Culture and Thought

Mathematical Culture and Thought

Fractional Calculus and Its Emerging Applications in Engineering

Document Type : Review

Authors
1 Faculty of Statistics, Mathematics, and Computer, Allameh Tabataba’i University, Iran
2 Department of Mechanical ‎Engineering‎‎, ‎University of Zanjan‎, ‎Iran
Abstract
In this article, the history of fractional calculus along with the disciplines and involved fields are introduced briefly. In the following, the basic formulas of fractional calculus are obtained. By presenting physical evidence, the existence of two types of absolute and cosmic time is proven and with their help, the geometrical and physical interpretations of fractional integration and derivation are presented. As a new application of concepts of fractional calculus, modeling of viscoelastic materials where these materials are widely used in mechanical, electrical, material, chemical and civil engineering, is offered. Classical mechanical models of viscoelastic materials including integer order models and fractional orders of materials are introduced. Finally, the applications of fractional calculus in control, dynamic systems and fractional variational calculus are referred.
Keywords

Subjects


[۱] اکرمی، محمدحسین، حسابان کسری از نظریه تا کاربرد، ریاضی و جامعه، ۴ (۱۳۹۶)، ۵۹-۶۹.
[۲] تواضعی، محمدصالح؛ توکلی کاخکی، مهسان، سیستم‌ها و کنترل‌کننده‌های مرتبه صحیح، انتشارات دانشگاه صنعتی خواجه نصیرالدین طوسی، تهران، ۱۳۹۵.
[۳] خاتمی، زهرا؛ علیزاده، یحیی، دیفرانسیل و انتگرال از مرتبه کسری، فرهنگ و اندیشۀ ریاضی، ۲۹ (۱۳۸۱)، ۳۰-۱۷.
[۴] رودین، و.، اصول آنالیز ریاضی، ترجمۀ علیاکبر عالم‌زاده، انتشارات علمی و فنی، تهران، ۱۴۰۱.
[۵] سیدصادقی، میرصادق؛ شفیعی دیزج، محمد، حسابان کسری تغییرات با مرتبه متغیر، جهاد دانشگاهی استان اردبیل، اردبیل، ۱۴۰۰.
[۶] کاشانی، سیدمحمدباقر، سخنرانی اینشتین: «ماهیت فضا» ارائه شده توسط مایکل عطیه»، فرهنگ و اندیشه ریاضی، ۴۰ (۱۳۸۷)، ۶۳-۷۰.
[۷] کبیری، رضا؛ چهارپاشلو، رضا، سیستمهای مرتبه کسری و کاربرد آنها در علم و صنعت، در همایش ملی الکترونیکی دستاوردهای نوین در علوم مهندسی و پایه، ۱۳۹۳.
[8] موسىزاده موسوى، سیف الله، ریشه‌ها، مبانى و سیر تکاملى نظریة استورم-لیوویل، فرهنک و اندیمشه ریاضى، ۶۰ (۱۳۹۶)، ۸۷-۶۷.
[9]    Almeida, R., Tavares, D., Torres, D. F. M., The Variable-Order Fractional Calculus of Variations, Springer Verlag, Switzerland, 2019.
[10]    Bagley, R. L., Torvik, P. J., On the fractional calculus model of viscoelastic behavior, J. Rheol., 30 (1986), 133-155.
[11]    Bingi, K., Ibrahim, R., Karsiti, M. N., Hassan, S. M., Harindran, V. R., Fractional-order set-point weighted controllers, in Fractional-order Systems and PID Controllers, Studies in Systems, Decision and Control, vol. 264. Springer, Cham, 2020.
[12]    Calcagni, G., Toward multifractional calculus, Front. Phys., 6 (2018), 58.
[13]    Caputo, M., Fabrizio, M., The kernel of the distributed order fractional derivatives with an application to complex materials. Fractal Fract., 1 (2017), no. 1, 13.
[14]    Chevalier, Y., Tuong, J. V., eds., Mechanics of Viscoelastic Materials and Wave Dispersion, John Wiley & Sons, London, , 2010.
[15]    Daigneault, A., Sangalli, A., Einstein’s static universe: an idea whose time has come back? Notices Amer. Math. Soc., 48 (2001), no.1, 9-16.
[16]    Diethelm, K., The Analysis of Fractional Differential Equations; An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin, 2010.
[17]    Ding, W., Patnaik, S., Sidhardh, S., Semperlotti, F., Applications of distributed-order fractional operators: A review, Entropy, 23 (2021), no.1, 110.
[18]    Faal, R. T., Using fractional derivatives for improved viscoelastic modeling of textile composites, Part I: Fabric yarns, J. Compos. Mater., 54 (2020), 3245-3260.
[19]    Fang, C. Q., Sun, H. Y., GuJ. P., Application of fractional calculus methods to viscoelastic response of amorphous shape memory polymers, J. Mech., 31 (2010), 427-432.
[20]    Heuchel, M., Cui, J., Kratz, K., Relaxation based modeling of tunable shape recovery kinetics observed under isothermal conditions for amorphous shape-memory polymers, Polymer (Guildf), 51 (2010), 6212-6218.
[21]    Kontou, E., Katsourinis, S., Application of a fractional model for simulation of the viscoelastic functions of polymers, J. Appl. Polym. Sci., 133 (2016), 100-120.
[22]    Lorenzo, C .F., Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57-98.
[23]    Di Lorenzo, S., Di Paola, M., La Mantia, F. P., Pirrotta, A., Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model, Meccanica, 52 (2017), 1843-1850,
[24]    Machado, J. A. T., Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2019.
[25]    Magin, R. L., Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006.
[26]    Machado, J. A. T., Galhano, A. M. S. F., Trujillo, J. J., On development of fractional calculus during the last fifty years, Scientometrics, 98 (2014), 577-582.
[27]    Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, Singapore, 2022.
[28]    Nadzharyan, T. A., Kostrov, S. A., Stepanov, G. V., Kramarenko, E. Y., Fractional rheological models of dynamic mechanical behavior of magnetoactive elastomers in magnetic fields, Polymer (Guildf), 142 (2018), 316-329.
[29]    Pan, Z., Liu, Z., A novel fractional viscoelastic constitutive model for shape memory polymers, J. Polym. Sci., Part B, Polym. Phys., 56 (2018), no. 16, 1125-1134.
[30]    Patnaik, S., Hollkamp, J. P., Semperlotti, F., Applications of variable-order fractional operators: A review, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 476 (2020), 20190498.
[31]    Podlubny, I., Fractional Differential Equations, San Diego, Academic Press, 1999.
[32]    Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367-386.
[33]    Research Square, Adapting Taylor series to fractional differential equations (2017, March 17) [Video file], available at https://www.youtube.com/watch?v=Bt2Uf5RJ9hs.
[34]    Samko, S. G., Ross, B., Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct., 1 (1993), 277-300.
[35]    Sepehri-Amin S., Faal, R.T., Das, R., Analytical and numerical solutions for vibration of a functionally graded beam with multiple fractionally damped absorbers, Thin-Walled Struct., 157 (2020), 106-711.
[36]    Shin, Y., Darbon, J., Karniadakis, G. Em., A Caputo fractional derivative-based algorithm for optimization (2021), available at arXiv: 2104.02259.
[37]    Spanier, J., Oldham, K. B., The Fractional Calculus, Academic Press, New York, 1974.
[38]    Sun, H. G., Zhang, Y, Balean, D., Chen, W., Chen, Y. Q., A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231.
[39]    Sun, H., Chang, A., Zhang, Y., Chen, W., A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2019), 27-59.
[40]    Tenreiro MacHado, J. A., Silva, M. F., Barbosa, R. S., Jesus, I. S., Rris,C. M., Marcus, M. G., Some applications of fractional calculus in engineering, Math. Probl. Eng., (2010), article ID 639801.
[41]    Zaky, M., Machado, J. T., On the formulation and numerical simulation of distributed-order fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 177-189.
[42]    Zaky, M., A Legendre collocation method for distributed-order fractional optimal control problems, Nonlinear Dynam., 91 (2018), 2667-2681.
[43]    Zhang, Y., Xu, S., Zhang, Q., Zhou, Y., Experimental and theoretical research on the stressrelaxation behaviors of PTFE coated fabrics under different temperatures, Adv. Mater. Sci. Eng., (2015), article ID 19473.

  • Receive Date 23 December 2022
  • Revise Date 08 June 2023
  • Accept Date 08 June 2023
  • Publish Date 22 July 2024