Abstract
In this study, the optimal control problem for Cattaneo–Hristov heat diffusion, a partial differential equation including both fractional-order Caputo–Fabrizio and integer-order derivatives, is formulated for a rigid heat conductor with finite length. The state and control functions denoting temperature and heat source, respectively, are represented by eigenfunctions to eliminate the spatial coordinate. The necessary optimality conditions corresponding to a time-dependent dynamical system are derived via Hamilton’s principle. Because the optimality system contains both integer-order and fractional-order left- and right-side Caputo–Fabrizio derivatives, it cannot be solved analytically. Therefore, a numerical method based on the Volterra integral approach combined with the forward–backward finite difference schemes is applied to solve the system. Finally, the physical behaviours of the temperature state and heat control under the variation of the fractional parameter are depicted graphically.
Similar content being viewed by others
References
Kirk, D.E.: Optimal Control Theory: An Introduction. Courier Corporation, New York (2004)
Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890 (1996)
Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581 (1997)
Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002)
Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A: Math. Theor. 39(33), 10375 (2006)
Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor. 40(24), 6287 (2007)
Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(1–4), 323–337 (2004)
Agrawal, O.P.: A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14(9–10), 1291–1299 (2008)
Odzijewicz, T., Malinowska, A.B., Torres, D.F.: Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal-Theor. 75(3), 1507–1515 (2012)
Herzallah, M.A.: Variational calculus with fractional and classical derivatives. Rom. J. Phys. 57(9–10), 1261–1269 (2012)
Kumar, D., Singh, J., Tanwar, K., Baleanu, D.: A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. Int. J. Heat Mass Transf. 138, 1222–1227 (2019)
Kumar, D., Singh, J., Baleanu, D.: On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law. Math. Method. Appl. Sci. 43(1), 443–457 (2020)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 73–85 (2015)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)
Zhang, J., Ma, X., Li, L.: Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives. Adv. Differ. Equ. 2017(1), 1–14 (2017)
Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80(1), 11–27 (2017)
Bastos, N.R.: Calculus of variations involving Caputo-Fabrizio fractional differentiation. Stat. Optim. Inf. Comput. 6(1), 12–21 (2018)
Abdeljawad, T., Atangana, A., Gómez-Aguilar, J.F., Jarad, F.: On a more general fractional integration by parts formulae and applications. Physica A Stat. Mech. Appl. 536, (2019)
Yıldız, T.A., Jajarmi, A., Yıldız, B., Baleanu, D.: New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete Contin. Dyn. Syst. Ser. S 13(3), 407–428 (2020)
Mortezaee, M., Ghovatmand, M., Nazemi, A.: An application of generalized fuzzy hyperbolic model for solving fractional optimal control problems with Caputo-Fabrizio derivative. Neural Process. Lett. 52(3), 1997–2020 (2020)
Naik, P.A., Owolabi, K.M., Yavuz, M., Zu, J.: Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos Soliton. Fract. 140, (2020)
Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., Townley, S.: Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. Eur. Phys. J. Plus 135(10), 1–42 (2020)
Özdemir, N., Uçar, E.: Investigating of an immune system-cancer mathematical model with Mittag-Leffler kernel. AIMS Math. 5(2), 1519–1531 (2020)
Kumar, D., Singh, J. (eds.): Fractional Calculus in Medical and Health Science. CRC Press, Abingdon (2020)
Bonyah, E., Gomez-Aguilar, J.F., Adu, A.: Stability analysis and optimal control of a fractional human African trypanosomiasis model. Chaos Soliton. Fract. 117, 150–160 (2018)
Jajarmi, A., Ghanbari, B., Baleanu, D.: A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence. Chaos 29(9), (2019)
Baleanu, D., Jajarmi, A., Sajjadi, S.S., Mozyrska, D.: A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos 29(8), (2019)
Sweilam, N.H., Al Mekhlafi, S.M.: Optimal control for a nonlinear mathematical model of tumor under immune suppression: a numerical approach. Optim. Control. Appl. Metal 39(5), 1581–1596 (2018)
Özdemir, N., Povstenko, Y., Avcı, D., İskender, B.B.: Optimal boundary control of thermal stresses in a plate based on time-fractional heat conduction equation. J. Therm. Stresses 37(8), 969–980 (2014)
Povstenko, Y., Avcı, D., İskender Eroğlu, B.B., Özdemir, N.: Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain. Therm. Sci. 21(1 Part A), 19–28 (2017)
İskender Eroğlu, B.B., Avcı, D., Özdemir, N.: Optimal control problem for a conformable fractional heat conduction equation. Acta Phys. Pol., A 132(3), 658–662 (2017)
Lazo, J.M., Torres, D.F.M.: Variational calculus with conformable fractional derivatives. IEEE-CAA J. Autom. 4, 340–352 (2017)
İskender Eroğlu, B.B., Yapışkan, D.: Local generalization of transversality conditions for optimal control problem. Math. Model. Nat. Phenom. 14(3), 310 (2019)
İskender Eroğlu, B.B., Yapışkan, D.: Generalized conformable variational calculus and optimal control problems with variable terminal conditions. AIMS Math. 5(2), 1105–1126 (2020)
Hristov, J.: Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci. 20(2), 757–762 (2016)
Povstenko, Y.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stresses 34(2), 97–114 (2011)
Povstenko, Y., Ostoja-Starzewski, M.: Doppler effect described by the solutions of the Cattaneo telegraph equation. Acta Mech. 232, 725–740 (2021)
Hristov, J.: Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. In: Bhalekar, S. (ed.) Current Developments in Mathematical Sciences Volume: 1 Frontiers in Fractional Calculus, pp. 270–342. Bentham Science Publishers, Sharjah (2017)
Hristov, J.: Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative from Cattaneo concept with Jeffrey’s Kernel and analytical solutions. Therm. Sci. 21(2), 827–839 (2017)
Alkahtani, B.S.T., Atangana, A.: A note on Cattaneo–Hristov model with non–singular fading memory. Therm. Sci. 21(1 Part A), 1–7 (2017)
Koca, I., Atangana, A.: Solutions of Cattaneo–Hristov model of elastic heat diffusion with Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Therm. Sci. 21(6 Part A), 2299–2305 (2017)
Sene, N.: Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. Int. J. Anal. Appl. 17(2), 191–207 (2019)
İskender Eroğlu, B.B., Avcı, D.: Separable solutions of Cattaneo-Hristov heat diffusion equation in a line segment: Cauchy and source problems. Alex. Eng. J. 60(2), 2347–2353 (2021)
Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015)
Caputo, M., Fabrizio, M.: On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica 52, 3043–3052 (2017)
Atanackovic, T.M., Pilipovic, S., Zorica, D.: Properties of the Caputo-Fabrizio fractional derivative and its distributional settings. Frac. Calc. Appl. Anal. 21(1), 29–44 (2018)
Atanackovic, T.M., Janev, M., Pilipovic, S.: Wave equation in fractional Zener-type viscoelastic media involving Caputo-Fabrizio fractional derivatives. Meccanica 54, 155–167 (2019)
Hristov, J.: Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels: Pragmatic approach, memory kernel correspondence requirement and analyses. Eur. Phys. J. Plus 134, 283 (2019)
Cattaneo, C.: On the conduction of heat (in Italian). Atti Sem. Mat. Fis. Univ. Modena 3(1), 83–101 (1948)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Author contributions
All persons who meet authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, design, analysis, writing, or revision of the manuscript. Furthermore, each author certifies that this material or similar material has not been and will not be submitted to or published in any other publication.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Avcı, D., Eroğlu, B.B.İ. Optimal control of the Cattaneo–Hristov heat diffusion model. Acta Mech 232, 3529–3538 (2021). https://doi.org/10.1007/s00707-021-03019-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-021-03019-z