Research article Special Issues

On a nonlinear mixed-order coupled fractional differential system with new integral boundary conditions

  • Received: 12 December 2020 Accepted: 22 March 2021 Published: 26 March 2021
  • MSC : 34A08, 34B15

  • We present the criteria for the existence of solutions for a nonlinear mixed-order coupled fractional differential system equipped with a new set of integral boundary conditions on an arbitrary domain. The modern tools of the fixed point theory are employed to obtain the desired results, which are well-illustrated by numerical examples. A variant problem dealing with the case of nonlinearities depending on the cross-variables (unknown functions) is also briefly described.

    Citation: Bashir Ahmad, Soha Hamdan, Ahmed Alsaedi, Sotiris K. Ntouyas. On a nonlinear mixed-order coupled fractional differential system with new integral boundary conditions[J]. AIMS Mathematics, 2021, 6(6): 5801-5816. doi: 10.3934/math.2021343

    Related Papers:

  • We present the criteria for the existence of solutions for a nonlinear mixed-order coupled fractional differential system equipped with a new set of integral boundary conditions on an arbitrary domain. The modern tools of the fixed point theory are employed to obtain the desired results, which are well-illustrated by numerical examples. A variant problem dealing with the case of nonlinearities depending on the cross-variables (unknown functions) is also briefly described.



    加载中


    [1] A. V. Bicadze, A. A. Samarskiĭ, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 185 (1969), 739–740.
    [2] J. Andres, A four-point boundary value problem for the second-order ordinary differential equations, Arch. Math., 53 (1989), 384–389. doi: 10.1007/BF01195218
    [3] P. W. Eloe, B. Ahmad, Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions, Appl. Math. Lett., 18 (2005), 521–527. doi: 10.1016/j.aml.2004.05.009
    [4] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. Lond. Math. Soc., 74 (2006), 673–693. doi: 10.1112/S0024610706023179
    [5] J. R. Graef, J. R. L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal. Theor., 71 (2009), 1542–1551. doi: 10.1016/j.na.2008.12.047
    [6] M. Feng, X. Zhang, W. Ge, Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications, J. Appl. Math. Comput., 33 (2010), 137–153. doi: 10.1007/s12190-009-0278-x
    [7] L. Zheng, X. Zhang, Modeling and analysis of modern fluid problems, Academic Press, 2017.
    [8] J. R. Cannon, The solution of the heat equation subject to the specification of energy, Q. Appl. Math., 21 (1963), 155–160. doi: 10.1090/qam/160437
    [9] N. I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differentsial'nye Uravneniya, 13 (1977), 294–304.
    [10] R. Yu. Chegis, Numerical solution of a heat conduction problem with an integral condition, Liet. Mat. Rink., 24 (1984), 209–215.
    [11] C. Taylor, T. Hughes, C. Zarins, Finite element modeling of blood flow in arteries, Comput. Method. Appl. M., 158 (1998), 155–196. doi: 10.1016/S0045-7825(98)80008-X
    [12] J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), 553–563. doi: 10.1113/jphysiol.1955.sp005276
    [13] F. Nicoud, T. Schfonfeld, Integral boundary conditions for unsteady biomedical CFD applications, Int. J. Numer. Meth. Fl., 40 (2002), 457–465. doi: 10.1002/fld.299
    [14] R. Čiegis, A. Bugajev, Numerical approximation of one model of the bacterial self-organization, Nonlinear Anal. Model., 17 (2012), 253–270. doi: 10.15388/NA.17.3.14054
    [15] B. Ahmad, A. Alsaedi, Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions, Nonlinear Anal. Real, 10 (2009), 358–367. doi: 10.1016/j.nonrwa.2007.09.004
    [16] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. Theor., 70 (2009), 364–371. doi: 10.1016/j.na.2007.12.007
    [17] M. Boukrouche, D. A. Tarzia, A family of singular ordinary differential equations of the third order with an integral boundary condition, Bound. Value Probl., 2018 (2018), 1–11. doi: 10.1186/s13661-017-0918-2
    [18] J. Henderson, Smoothness of solutions with respect to multi-strip integral boundary conditions for $n$th order ordinary differential equations, Nonlinear Anal. Model., 19 (2014), 396–412. doi: 10.15388/NA.2014.3.6
    [19] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA, Nonlinear Diff., 15 (2008), 45–67. doi: 10.1007/s00030-007-4067-7
    [20] G. S. Wang, A. F. Blom, A strip model for fatigue crack growth predictions under general load conditions, Eng. Fract. Mech., 40 (1991), 507–533. doi: 10.1016/0013-7944(91)90148-T
    [21] B. Ahmad, T. Hayat, Diffraction of a plane wave by an elastic knife-edge adjacent to a rigid strip, Canad. Appl. Math. Quart., 9 (2001), 303–316.
    [22] T. V. Renterghem, D. Botteldooren, K. Verheyen, Road traffic noise shielding by vegetation belts of limited depth, J. Sound Vib., 331 (2012), 2404–2425. doi: 10.1016/j.jsv.2012.01.006
    [23] E. Yusufoglu, I. Turhan, A mixed boundary value problem in orthotropic strip containing a crack, J. Franklin I., 349 (2012), 2750–2769. doi: 10.1016/j.jfranklin.2012.09.001
    [24] M. S. Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays, Commun. Nonlinear Sci., 83 (2020), 105088. doi: 10.1016/j.cnsns.2019.105088
    [25] Y. Xu, Y. Li, W. Li, Adaptive finite-time synchronization control for fractional-order complex-valued dynamical networks with multiple weights, Commun. Nonlinear Sci., 85 (2020), 105239. doi: 10.1016/j.cnsns.2020.105239
    [26] Y. Ding, Z. Wang, H. Ye, Optimal control of a fractional-order HIV-immune system with memory, IEEE T. Contr. Syst. T., 20 (2012), 763–769. doi: 10.1109/TCST.2011.2153203
    [27] Y. Xu, W. Li, Finite-time synchronization of fractional-order complex-valued coupled systems, Physica A, 549 (2020), 123903. doi: 10.1016/j.physa.2019.123903
    [28] F. Zhang, G. Chen C. Li, J. Kurths, Chaos synchronization in fractional differential systems, Philos. T. R. Soc. A, 371 (2013), 201201553.
    [29] L. Xu, X. Chu, H. Hu, Quasi-synchronization analysis for fractional-order delayed complex dynamical networks, Math. Comput. Simulat., 185 (2021), 594–613. doi: 10.1016/j.matcom.2021.01.016
    [30] X. Chu, L. Xu, H. Hu, Exponential quasi-synchronization of conformable fractional-order complex dynamical networks, Chaos Soliton. Fract., 140 (2020), 110268. doi: 10.1016/j.chaos.2020.110268
    [31] V. J. Ervin, N. Heuer, J. P. Roop, Regularity of the solution to $1-$D fractional order diffusion equations, Math. Comput., 87 (2018), 2273–2294. doi: 10.1090/mcom/3295
    [32] H.Wang, X. Zheng, Well posedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475 (2019), 1778–1802. doi: 10.1016/j.jmaa.2019.03.052
    [33] X. Zheng, H. Wang, An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492–2514. doi: 10.1137/20M132420X
    [34] H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics, Theory and Application, Academic Press, 2017.
    [35] M. Javidi, B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system, Ecol. Model., 318 (2015), 8–18. doi: 10.1016/j.ecolmodel.2015.06.016
    [36] Z. Ming, G. Zhang, H. Li, Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions, Electron. J. Qual. Theo., 2019 (2019), 1–15.
    [37] Y. Wang, S. Liang, Q. Wang, Existence results for fractional differential equations with integral and multi-point boundary conditions, Bound. Value Probl., 2018 (2018), 1–11. doi: 10.1186/s13661-017-0918-2
    [38] Z. Cen, L. B. Liu, J. Huang, A posteriori error estimation in maximum norm for a two-point boundary value problem with a Riemann-Liouville fractional derivative, Appl. Math. Lett., 102 (2020), 106086. doi: 10.1016/j.aml.2019.106086
    [39] G. Iskenderoglu, D. Kaya, Symmetry analysis of initial and boundary value problems for fractional differential equations in Caputo sense, Chaos Soliton. Fract., 134 (2020), 109684. doi: 10.1016/j.chaos.2020.109684
    [40] C. S. Goodrich, Coercive nonlocal elements in fractional differential equations, Positivity, 21 (2017), 377–394. doi: 10.1007/s11117-016-0427-z
    [41] B. Ahmad, Y. Alruwaily, A. Alsaedi, J. J. Nieto, Fractional integro-differential equations with dual anti-periodic boundary conditions, Differ. Integral Equ., 33 (2020), 181–206.
    [42] D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, 2000.
    [43] A. S. Deshpande, V. Daftardar-Gejji, On disappearance of chaos in fractional systems, Chaos Soliton. Fract., 102 (2017), 119–126. doi: 10.1016/j.chaos.2017.04.046
    [44] S. Wang, M. Xu, Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Anal. Real, 10 (2009), 1087–1096. doi: 10.1016/j.nonrwa.2007.11.027
    [45] L. Xu, X. Chu, H. Hu, Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses, Appl. Math. Lett., 99 (2020), 106000. doi: 10.1016/j.aml.2019.106000
    [46] D. He, L. Xu, Exponential stability of impulsive fractional switched systems with time delays, IEEE T. Circuits-II, 2020, doi: 10.1109/TCSII.2020.3037654.
    [47] B. Ahmad, N. Alghamdi, A. Alsaedi, S. K. Ntouyas, A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions, Fract. Calc. Appl. Anal., 22 (2019), 601–618. doi: 10.1515/fca-2019-0034
    [48] B. Ahmad, S. K. Ntouyas, A. Alsaedi, Fractional order differential systems involving right Caputo and left Riemann-Liouville fractional derivatives with nonlocal coupled conditions, Bound. Value Probl., 2019 (2019), 1–12. doi: 10.1186/s13661-018-1115-7
    [49] B. Ahmad, A. Alsaedi, S. K. Ntouyas, Fractional order nonlinear mixed coupled systems with coupled integro-differential boundary conditions, J. Appl. Anal. Comput., 10 (2020), 892–903.
    [50] B. Ahmad, A. Alsaedi, Y. Alruwaily, S. K. Ntouyas, Nonlinear multi-term fractional differential equations with Riemann-Stieltjes integro-multipoint boundary conditions, AIMS Mathematics, 5 (2020), 1446–1461. doi: 10.3934/math.2020099
    [51] J. Henderson, R. Luca, A. Tudorache, On a system of fractional differential equations with coupled integral boundary conditions, Fract. Calc. Appl. Anal., 18 (2015), 361–386.
    [52] S. K. Ntouyas, H. H. Al-Sulami, A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions, Adv. Differ. Equ., 2020 (2020), 1–21.
    [53] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
    [54] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  • Reader Comments
  • © 2021 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(1388) PDF downloads(222) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog