Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 5: 7489-7506. doi: 10.3934/math.2022420
Previous Article Next Article
Research article

Nonlinear higher order fractional terminal value problems

  • 1.
    Cankaya University, Department of Mathematics, 06530 Balgat, Ankara, Turkey
  • 2.
    Institute of Space Sciences, Magurele-Bucharest, Romania
  • 3.
    Department of Medical Research, China Medical University Hospital, China Medical, University, Taichung, Taiwan
  • 4.
    Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China
  • Received: 07 December 2021 Revised: 16 January 2022 Accepted: 24 January 2022 Published: 14 February 2022

  • MSC : 34A08, 45G05

  • Terminal value problems for systems of fractional differential equations are studied with an especial focus on higher-order systems. Discretized piecewise polynomial collocation methods are used for approximating the exact solution. This leads to solving a system of nonlinear equations. For solving such a system an iterative method with a required tolerance is introduced and analyzed. The existence of a unique solution is guaranteed with the aid of the fixed point theorem. Order of convergence for the given numerical method is obtained. Numerical experiments are given to support theoretical results.

    Citation: Dumitru Baleanu, Babak Shiri. Nonlinear higher order fractional terminal value problems[J]. AIMS Mathematics, 2022, 7(5): 7489-7506. doi: 10.3934/math.2022420

    Related Papers:

  • Terminal value problems for systems of fractional differential equations are studied with an especial focus on higher-order systems. Discretized piecewise polynomial collocation methods are used for approximating the exact solution. This leads to solving a system of nonlinear equations. For solving such a system an iterative method with a required tolerance is introduced and analyzed. The existence of a unique solution is guaranteed with the aid of the fixed point theorem. Order of convergence for the given numerical method is obtained. Numerical experiments are given to support theoretical results.


    加载中


    [1] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
    [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amsterdam, 2006.
    [3] M. E. Ahmed, M. A. Khan, Modeling and analysis of the polluted lakes system with various fractional approaches, Chaos Soliton. Fract., 134 (2020), 109720. http://doi.org/10.1016/j.chaos.2020.109720 doi: 10.1016/j.chaos.2020.109720
    [4] L. Song, S. Xu, J. Yang, Dynamical models of happiness with fractional order, Commun. Nonlinear Sci., 15 (2010), 616–628. http://doi.org/10.1016/j.cnsns.2009.04.029 doi: 10.1016/j.cnsns.2009.04.029
    [5] S. Abbas, V. S. Erturk, S. Momani, Dynamical analysis of the Irving-Mullineux oscillator equation of fractional order, Signal Process., 102 (2014), 171–176. http://doi.org/10.1016/j.sigpro.2014.03.019 doi: 10.1016/j.sigpro.2014.03.019
    [6] M. K. Sadabad, A. J. Akbarfam, B. Shiri, A numerical study of eigenvalues and eigenfunctions of fractional Sturm-Liouville problems via Laplace transform, Indian J. Pure Appl. Math., 51 (2020), 857–868. http://doi.org/10.1007/s13226-020-0436-2 doi: 10.1007/s13226-020-0436-2
    [7] G. Yang, B. Shiri, H. Kong, G. C. Wu, Intermediate value problems for fractional differential equations, Comput. Appl. Math., 40 (2021), 1–20. http://doi.org/10.1007/s40314-021-01590-8 doi: 10.1007/s40314-021-01590-8
    [8] N. J. Ford, M. L. Morgado, M. Rebelo, High order numerical methods for fractional terminal value problems, Comput. Meth. Appl. Math., 14 (2014), 55–70. http://doi.org/10.1515/cmam-2013-0022 doi: 10.1515/cmam-2013-0022
    [9] N. J. Ford, M. L. Morgado, M. Rebelo, A nonpolynomial collocation method for fractional terminal value problems, J. Comput. Appl. Math., 275 (2015), 392–402. http://doi.org/10.1016/j.cam.2014.06.013 doi: 10.1016/j.cam.2014.06.013
    [10] M. A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357 (2019), 103–122. http://doi.org/10.1016/j.cam.2019.01.046 doi: 10.1016/j.cam.2019.01.046
    [11] M. A. Almalahi, M. S. Abdo, S. K. Panchal, On the theory of fractional terminal value problem with $\psi$-Hilfer fractional derivative, AIMS Math., 5 (2020), 4889–4908. http://doi.org/10.3934/math.2020312 doi: 10.3934/math.2020312
    [12] M. Benchohra, S. Bouriah, J. J. Nieto, Terminal value problem for differential equations with Hilfer-Katugampola fractional derivative, Symmetry, 11 (2019), 672. http://doi.org/10.3390/sym11050672 doi: 10.3390/sym11050672
    [13] B. Shiri, G. C. Wu, D. Baleanu, Collocation methods for terminal value problems of tempered fractional differential equations, Appl. Numer. Math., 156 (2020), 385–395. http://doi.org/10.1016/j.apnum.2020.05.007 doi: 10.1016/j.apnum.2020.05.007
    [14] B. Shiri, G. C. Wu, D. Baleanu, Terminal value problems for the nonlinear systems of fractional differential equations, Appl. Numer. Math., 170 (2021), 162–178. http://doi.org/10.1016/j.apnum.2021.06.015 doi: 10.1016/j.apnum.2021.06.015
    [15] K. Diethelm, On the separation of solutions of fractional differential equations, Fract. Calc. Appl. Anal., 11 (2008), 259–268.
    [16] K. Diethelm, N. J. Ford, Volterra integral equations and fractional calculus: Do neighboring solutions intersect? J. Integral Eq. Appl., 24 (2012), 25–37. http://doi.org/10.1216/JIE-2012-24-1-25 doi: 10.1216/JIE-2012-24-1-25
    [17] N. D. Cong, H. T. Tuan, Generation of non-local fractional dynamical systems by fractional differential equations, J. Integral Eq. Appl., 29 (2017), 585–608. http://doi.org/10.1216/JIE-2017-29-4-585 doi: 10.1216/JIE-2017-29-4-585
    [18] K. Diethelm, N. J. Ford, A note on the well-posedness of terminal value problems for fractional differential equations, J. Integral Eq. Appl., 30 (2018), 371–376. http://doi.org/10.1216/JIE-2018-30-3-371 doi: 10.1216/JIE-2018-30-3-371
    [19] S. H. Shah, M. ur Rehman, A note on terminal value problems for fractional differential equations on infinite interval, Appl. Math. Lett., 52 (2016), 118–125. http://dx.doi.org/10.1016/j.aml.2015.08.008 doi: 10.1016/j.aml.2015.08.008
    [20] D. Baleanu, B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos Soliton. Fract., 116 (2018), 136–145. http://doi.org/10.1016/j.chaos.2018.09.020 doi: 10.1016/j.chaos.2018.09.020
    [21] W. Rudin, Principles of mathematics analysis, McGraw-hill, London, 1964. http://doi.org/10.2307/3608793
  • 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
AIMS Mathematics
1.8 3.4

Metrics

Article views(1621) PDF downloads(112) Cited by(30)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(6)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog