Research article

A new local non-integer derivative and its application to optimal control problems

  • Received: 28 February 2022 Revised: 03 April 2022 Accepted: 14 April 2022 Published: 12 July 2022
  • MSC : 26A33, 34A08, 49M37, 49M25

  • Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.

    Citation: Xingfa Yang, Yin Yang, M. H. Noori Skandari, Emran Tohidi, Stanford Shateyi. A new local non-integer derivative and its application to optimal control problems[J]. AIMS Mathematics, 2022, 7(9): 16692-16705. doi: 10.3934/math.2022915

    Related Papers:

  • Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.



    加载中


    [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
    [2] Q. M. Al-Mdallal, H. Yusuf, A. Ali, A novel algorithm for time-fractional foam drainage equation, Alex. Eng. J., 59 (2020), 1607–1612. https://doi.org/10.1016/j.aej.2020.04.007 doi: 10.1016/j.aej.2020.04.007
    [3] M. Alqhtani, K. M. Saad, Numerical solutions of space-fractional diffusion equations via the exponential decay kernel, AIMS Math., 7 (2022), 6535–6549. https://doi.org/10.3934/math.2022364 doi: 10.3934/math.2022364
    [4] M. T. Darvishi, M. Najafi, A. M. Wazwaz, Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions, Chaos Solitons Fract., 150 (2021), 111187. https://doi.org/10.1016/j.chaos.2021.111187 doi: 10.1016/j.chaos.2021.111187
    [5] M. Eslami, H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53 (2016), 475–485. https://doi.org/10.1007/s10092-015-0158-8 doi: 10.1007/s10092-015-0158-8
    [6] F. Fahroo, I. M. Ross, Costate estimation by a Legendre pseudospectral method, J. Guid. Control Dyn., 24 (2001), 270–277. https://doi.org/10.2514/2.4709 doi: 10.2514/2.4709
    [7] M. Habibli, M. H. Noori Skandari, Fractional Chebyshev pseudospectral method for fractional optimal control problems, Optimal Control Appl. Methods, 40 (2919), 558–572. https://doi.org/10.1002/oca.2495 doi: 10.1002/oca.2495
    [8] Y. Huang, F. M. Zadeh, M. H. Noori Skandari, H. A. Tehrani, E. Tohidi, Space-time Chebyshev spectral collocation method for nonlinear time-fractional Burgers equations based on efficient basis functions, Math. Methods Appl. Sci., 44 (2021), 4117–4136. https://doi.org/10.1002/mma.7015 doi: 10.1002/mma.7015
    [9] R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math, 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
    [10] U. N. Katugampola, A new fractional derivative with classical properties, J. Amer. Math. Soc., 2014, 1–8. https://doi.org/10.48550/arXiv.1410.6535 doi: 10.48550/arXiv.1410.6535
    [11] N. I. Mahmudov, M. Aydin, Representation of solutions of nonhomogeneous conformable fractional delay differential equations, Chaos Soliton. Fract., 150 (2021), 111190. https://doi.org/10.1016/j.chaos.2021.111190 doi: 10.1016/j.chaos.2021.111190
    [12] M. H. Noori Skandari, M. Habibli, A. Nazemi, A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems, Math. Control Related Fields, 10 (2020), 171–187. https://doi.org/10.3934/mcrf.2019035 doi: 10.3934/mcrf.2019035
    [13] I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
    [14] X. B. Pang, X. F. Yang, M. H. Noori Skandari, E. Tohidi, S. Shateyi, A new high accurate approximate approach to solve optimal control problems of fractional order via efficient basis functions, Alex. Eng. J., 61 (2022), 5805–5818. https://doi.org/10.1016/j.aej.2021.11.007 doi: 10.1016/j.aej.2021.11.007
    [15] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Switzerland: Gordon and Breach Science Publishers, 1993.
    [16] K. Shah, F. Jarad, T. Abdeljawad, Stable numerical results to a class of time-space fractional partial differential equations via spectral method, J. Adv. Res., 25 (2020), 39–48. https://doi.org/10.1016/j.jare.2020.05.022 doi: 10.1016/j.jare.2020.05.022
    [17] J. Shen, T. Tang, L. L. Wang, Spectral methods: Algorithms, analysis and applications, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7
    [18] H. M. Srivastavaa, K. M. Saadd, M. M. Khader, An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus, Chaos Solitons Fract., 140 (2020), 110174. https://doi.org/10.1016/j.chaos.2020.110174 doi: 10.1016/j.chaos.2020.110174
    [19] D. Z. Zhao, M. K. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903–917. https://doi.org/10.1007/s10092-017-0213-8 doi: 10.1007/s10092-017-0213-8
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1282) PDF downloads(57) Cited by(3)

Article outline

Figures and Tables

Figures(5)  /  Tables(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog