Research article Special Issues

Existence of solutions and a numerical scheme for a generalized hybrid class of n-coupled modified ABC-fractional differential equations with an application

  • Received: 19 October 2022 Revised: 28 November 2022 Accepted: 12 December 2022 Published: 05 January 2023
  • MSC : 34A08, 49J15, 65P99

  • In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions; additionally, a numerical scheme based on the the Lagrange's interpolation polynomial is established and applied to a dynamical system for the applications. We also study the uniqueness and Hyers-Ulam stability for the solutions of the presumed mABC-FDEs system. Such a system has not been studied for the mentioned mABC-operator and this work generalizes most of the results studied for the ABC operator. This study will provide a base to a large number of dynamical problems for the existence, uniqueness and numerical simulations. The results are compared with the classical results graphically to check the accuracy and applicability of the scheme.

    Citation: Hasib Khan, Jehad Alzabut, Dumitru Baleanu, Ghada Alobaidi, Mutti-Ur Rehman. Existence of solutions and a numerical scheme for a generalized hybrid class of n-coupled modified ABC-fractional differential equations with an application[J]. AIMS Mathematics, 2023, 8(3): 6609-6625. doi: 10.3934/math.2023334

    Related Papers:

  • In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions; additionally, a numerical scheme based on the the Lagrange's interpolation polynomial is established and applied to a dynamical system for the applications. We also study the uniqueness and Hyers-Ulam stability for the solutions of the presumed mABC-FDEs system. Such a system has not been studied for the mentioned mABC-operator and this work generalizes most of the results studied for the ABC operator. This study will provide a base to a large number of dynamical problems for the existence, uniqueness and numerical simulations. The results are compared with the classical results graphically to check the accuracy and applicability of the scheme.



    加载中


    [1] K. Deimling, Nonlinear functional analysis, Berlin: Springer-Verlag, 1985. http://dx.doi.org/10.1007/978-3-662-00547-7
    [2] B. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equat. Appl., 2 (2010), 465–486. http://dx.doi.org/10.7153/dea-02-28 doi: 10.7153/dea-02-28
    [3] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), 1312–1324. http://dx.doi.org/10.1016/j.camwa.2011.03.041 doi: 10.1016/j.camwa.2011.03.041
    [4] S. Sitho, S. Ntouyas, J. Tariboon, Existence results for hybrid fractional integro-differential equations, Bound. Value Probl., 2015 (2015), 113. http://dx.doi.org/10.1186/s13661-015-0376-7 doi: 10.1186/s13661-015-0376-7
    [5] M. Awadalla, K. Abuasbeh, On system of nonlinear sequential hybrid fractional differential equations, Math. Probl. Eng., 2022 (2022), 8556578. http://dx.doi.org/10.1155/2022/8556578 doi: 10.1155/2022/8556578
    [6] S. Gul, R. Khan, H. Khan, R. George, S. Etemad, S. Rezapour, Analysis on a coupled system of two sequential hybrid BVPs with numerical simulations to a model of typhoid treatment, Alex. Eng. J., 61 (2022), 10085–10098. http://dx.doi.org/10.1016/j.aej.2022.03.020 doi: 10.1016/j.aej.2022.03.020
    [7] A. Khan, Z. Khan, T. Abdeljawad, H. Khan, Analytical analysis of fractional-order sequential hybrid system with numerical application, Adv. Cont. Discr. Mod., 2022 (2022), 12. http://dx.doi.org/10.1186/s13662-022-03685-w doi: 10.1186/s13662-022-03685-w
    [8] J. Losada, J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. http://dx.doi.org/10.12785/pfda/010202 doi: 10.12785/pfda/010202
    [9] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
    [10] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. http://dx.doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
    [11] M. Al-Refai, D. Baleanu, On an extension of the operator with Mittag-Leffler kernel, Fractals, 30 (2022), 2240129. http://dx.doi.org/10.1142/S0218348X22401296 doi: 10.1142/S0218348X22401296
    [12] B. Dhage, G. Khurape, A. Shete, J. Salunkhe, Existence and approximate solutions for nonlinear hybrid fractional integro-differential equations, Int. J. Anal. Appl., 11 (2016), 157–167.
    [13] B. Dhage, Existence and attractivity theorems for nonlinear hybrid fractional differential equations with anticipation and retardation, J. Nonlinear Funct. Anal., 2020 (2020), 47. http://dx.doi.org/10.23952/jnfa.2020.47 doi: 10.23952/jnfa.2020.47
    [14] M. Al-Refai, Proper inverse operators of fractional derivatives with nonsingular kernels, Rend. Circ. Mat. Palermo. II Ser., 71 (2022), 525–535. http://dx.doi.org/10.1007/s12215-021-00638-2 doi: 10.1007/s12215-021-00638-2
    [15] Z. Khan, A. Khan, T. Abdeljawad, H. Khan, Computational analysis of fractional order imperfect testing infection disease model, Fractals, 30 (2022), 2240169. http://dx.doi.org/10.1142/S0218348X22401697 doi: 10.1142/S0218348X22401697
    [16] R. Shi, Y. Cui, Global analysis of a mathematical model for Hepatitis C virus transmissions, Virus Res., 217 (2016), 8–17. http://dx.doi.org/10.1016/j.virusres.2016.02.006 doi: 10.1016/j.virusres.2016.02.006
    [17] M. Subramanian, M. Manigandan, C. Tunç, T. Gopal, J. Alzabut, On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order, J. Taibah Uni. Sci., 16 (2022), 1–23. http://dx.doi.org/10.1080/16583655.2021.2010984 doi: 10.1080/16583655.2021.2010984
    [18] M. Subramanian, J. Alzabut, M. Abbas, C. Thaiprayoon, W. Sudsutad, Existence of solutions for coupled higher-order fractional integro-differential equations with nonlocal integral and multi-point boundary conditions depending on lower-order fractional derivatives and integrals, Mathematics, 10 (2022), 1823. http://dx.doi.org/10.3390/math10111823 doi: 10.3390/math10111823
    [19] M. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 2014 (2014), 389386. http://dx.doi.org/10.1155/2014/389386 doi: 10.1155/2014/389386
    [20] N. Mahmudov, M. Matar, Existence of mild solution for hybrid differential equations with arbitrary fractional order, TWMS J. Pure Appl. Math., 8 (2017), 160–169.
    [21] H. Jafari, D. Baleanu, H. Khan, R. Khan, A. Khan, Existence criterion for the solutions of fractional order p-Laplacian boundary value problems, Bound. Value Probl., 2015 (2015), 164. http://dx.doi.org/10.1186/s13661-015-0425-2 doi: 10.1186/s13661-015-0425-2
    [22] A. Khan, Z. Khan, T. Abdeljawad, H. Khan, Analytical analysis of fractional-order sequential hybrid system with numerical application, Adv. Cont. Discr. Mod., 2022 (2022), 12. http://dx.doi.org/10.1186/s13662-022-03685-w doi: 10.1186/s13662-022-03685-w
    [23] G. Wang, B. Ahmad, L. Zhang, J. Nieto, Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci., 19 (2014), 401–403. http://dx.doi.org/10.1016/j.cnsns.2013.04.003 doi: 10.1016/j.cnsns.2013.04.003
    [24] W. Al-Sadi, Z. Huang, A. Alkhazzan, Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity, J. Taibah Uni. Sci., 13 (2019), 951–960. http://dx.doi.org/10.1080/16583655.2019.1663783 doi: 10.1080/16583655.2019.1663783
    [25] A. Shah, R. Khan, H. Khan, A fractional‐order hybrid system of differential equations: existence theory and numerical solutions, Math. Method. Appl. Sci., 45 (2022), 4024–4034. http://dx.doi.org/10.1002/mma.8029 doi: 10.1002/mma.8029
    [26] S. Aljoudi, B. Ahmad, J. Nieto, A. Alsaedi, A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos, Solition. Fract., 91 (2016) 39–46. http://dx.doi.org/10.1016/j.chaos.2016.05.005
    [27] S. Jose, R. Ramachandran, J. Cao, J. Alzabut, M. Niezabitowski, V. Balas, Stability analysis and comparative study on different eco‐epidemiological models: stage structure for prey and predator concerning impulsive control, Optim. Contr. Appl. Method., 43 (2022), 842–866. http://dx.doi.org/10.1002/oca.2856 doi: 10.1002/oca.2856
    [28] S. Etemad, B. Tellab, J. Alzabut, S. Rezapour, M. Abbas, Approximate solutions and Hyers-Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform, Adv. Differ. Equ., 2021 (2021), 428. http://dx.doi.org/10.1186/s13662-021-03563-x doi: 10.1186/s13662-021-03563-x
    [29] A. Selvam, D. Baleanu, J. Alzabut, D. Vignesh, S. Abbas, On Hyers-Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum, Adv. Differ. Equ., 2020 (2020), 456. http://dx.doi.org/10.1186/s13662-020-02920-6 doi: 10.1186/s13662-020-02920-6
    [30] A. Zada, J. Alzabut, H. Waheed, I. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions, Adv. Differ. Equ., 2020 (2020), 64. http://dx.doi.org/10.1186/s13662-020-2534-1 doi: 10.1186/s13662-020-2534-1
    [31] A. Akgul, M. Modanli, Crank-Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana-Baleanu Caputo derivative, Chaos Soliton. Fract., 127 (2019), 10–16. http://dx.doi.org/10.1016/j.chaos.2019.06.011 doi: 10.1016/j.chaos.2019.06.011
    [32] J. Solis-Perez, J. Gomez-Aguilar, A. Atangana, Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws, Chaos Soliton. Fract., 114 (2018), 175–185. http://dx.doi.org/10.1016/j.chaos.2018.06.032 doi: 10.1016/j.chaos.2018.06.032
    [33] M. Akinlar, F. Tchier, M. Inc, Chaos control and solutions of fractional-order Malkus waterwheel model, Chaos Soliton. Fract., 135 (2020), 109746. http://dx.doi.org/10.1016/j.chaos.2020.109746 doi: 10.1016/j.chaos.2020.109746
    [34] H. Khan, J. Gomez-Aguilar, A. Khan, T. Khan, Stability analysis for fractional order advection-reaction diffusion system, Physica A, 521 (2019), 737–751. http://dx.doi.org/10.1016/j.physa.2019.01.102 doi: 10.1016/j.physa.2019.01.102
    [35] S. Gul, R. Khan, H. Khan, R. George, S. Etemad, S. Rezapour, Analysis on a coupled system of two sequential hybrid BVPs with numerical simulations to a model of typhoid treatment, Alex. Eng. J., 61 (2022), 10085–10098. http://dx.doi.org/10.1016/j.aej.2022.03.020 doi: 10.1016/j.aej.2022.03.020
    [36] J. Singh, A. Alshehri, S. Momani, S. Hadid, D. Kumar, Computational analysis of fractional diffusion equations occurring in oil pollution, Mathematics, 10 (2022), 3827. http://dx.doi.org/10.3390/math10203827 doi: 10.3390/math10203827
    [37] J. Singh, D. Kumar, S. Purohit, A. Mishra, M. Bohra, An efficient numerical approach for fractional multidimensional diffusion equations with exponential memory, Numer. Meth. Part. D. E., 37 (2021), 1631–1651. http://dx.doi.org/10.1002/num.22601 doi: 10.1002/num.22601
    [38] V. Dubey, J. Singh, A. Alshehri, S. Dubey, D. Kumar, Analysis of local fractional coupled Helmholtz and coupled Burgers' equations in fractal media, AIMS Mathematics, 7 (2022), 8080–8111. http://dx.doi.org/10.3934/math.2022450 doi: 10.3934/math.2022450
  • Reader Comments
  • © 2023 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(1409) PDF downloads(199) Cited by(2)

Article outline

Figures and Tables

Figures(8)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog