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Variable-order fractal-fractional time delay equations with power, exponential and Mittag-Leffler laws and their numerical solutions

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Abstract

In this paper, a numerical method based on the Lagrangian piece-wise interpolation is proposed to solve variable-order fractal-fractional time delay equations with power law, exponential decay and Mittag-Leffler memories. These operators permit to describe physical phenomena with variable memory and fractal variable dimension. Numerical methods were applied to simulate the variable-order time delay Mackey–Glass and synaptically coupled FitzHugh–Nagumo models. Our numerical simulations display several new attractors.

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References

  1. Caputo M, Mainardi F (1971) A new dissipation model based on memory mechanism. Pure Appl Geophys 91:134–147

    MATH  Google Scholar 

  2. Caputo M, Fabricio M (2015) A new definition of fractional derivative without singular Kernel. Progr Fract Differ Appl 1(2):73–85

    Google Scholar 

  3. Kumar D, Singh J, Tanwar K, Baleanu D (2019) 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

    Google Scholar 

  4. Kumar D, Singh J, Baleanu D (2020) On the analysis of vibration equation involving a fractional derivative with Mittag–Leffler law. Math Methods Appl Sci 43(1):443–457

    MathSciNet  MATH  Google Scholar 

  5. Kumar D, Singh J, Al Qurashi M, Baleanu D (2019) A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Adva Differ Equ 2019(1):1–17

    MathSciNet  Google Scholar 

  6. Kumar D, Singh J, Purohit SD, Swroop R (2019) A hybrid analytical algorithm for nonlinear fractional wave-like equations. Math Model Nat Phenom 14(3):1–14

    MathSciNet  MATH  Google Scholar 

  7. Veeresha P, Prakasha DG, Kumar D, Baleanu D, Singh J (2020) An efficient computational technique for fractional model of generalized Hirota-Satsuma-coupled Korteweg-de Vries and coupled modified Korteweg-de Vries equations. J Comput Nonlinear Dyn 15(7):1–16

    Google Scholar 

  8. Goswami A, Singh J, Kumar D (2019) An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A Stat Mech Appl 524:563–575

    MathSciNet  Google Scholar 

  9. Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular Kernel. Theory and application to heat transfer model. Therm Sci 20(2):763–769

    Google Scholar 

  10. Samko SG (1995) Fractional integration and differentiation of variable order. Anal Math 21(3):213–236

    MathSciNet  MATH  Google Scholar 

  11. Jia J, Zheng X, Fu H, Dai P, Wang H (2020) A fast method for variable-order space-fractional diffusion equations. Numer Algorithms 1:1–22

    MathSciNet  MATH  Google Scholar 

  12. Chen C, Liu H, Zheng X, Wang H (2020) A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection-diffusion equations. Comput Math Appl 79(9):2771–2783

    MathSciNet  MATH  Google Scholar 

  13. Heydari MH, Atangana A, Avazzadeh Z, Mahmoudi MR (2020) An operational matrix method for nonlinear variable-order time fractional reaction-diffusion equation involving Mittag–Leffler kernel. Eur Phys J Plus 135(2):1–19

    Google Scholar 

  14. Patnaik S, Hollkamp JP, Semperlotti F (2020) Applications of variable-order fractional operators: a review. Proc R Soc A 476(2234):1–20

    MathSciNet  MATH  Google Scholar 

  15. Owolabi KM, Atangana A, Akgul A (2020) Modelling and analysis of fractal-fractional partial differential equations: application to reaction–diffusion model. Alex Eng J 1:1–17

    Google Scholar 

  16. Wang Y, Chen Y (2020) Shifted legendre polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model. Appl Math Model 81:159–176

    MathSciNet  MATH  Google Scholar 

  17. Avazzadeh Z, Heydari MH, Mahmoudi MR (2020) An approximate approach for the generalized variable-order fractional pantograph equation. Alex Eng J 1:1–13

    Google Scholar 

  18. Heydari MH, Avazzadeh Z (2020) New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq–Burger’s equations. Eng Comput 1:1–9

    Google Scholar 

  19. Ganji RM, Jafari H, Baleanu D (2020) A new approach for solving multi variable orders differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 130:1–10

    MathSciNet  Google Scholar 

  20. Gu Y, Sun H (2020) A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives. Appl Math Model 78:539–549

    MathSciNet  MATH  Google Scholar 

  21. Sweilam NH, AL-Mekhlafi SM, Alshomrani AS, Baleanu D (2020) Comparative study for optimal control nonlinear variable-order fractional tumor model. Chaos Solitons Fractals 136:1–10

    MathSciNet  Google Scholar 

  22. Patnaik S, Semperlotti F (2020) Variable-order particle dynamics: formulation and application to the simulation of edge dislocations. Philos Trans R Soc A 378(2172):1–20

    MathSciNet  MATH  Google Scholar 

  23. Tolba MF, Saleh H, Mohammad B, Al-Qutayri M, Elwakil AS, Radwan AG (2020) Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system. Nonlinear Dyn 1:1–12

    Google Scholar 

  24. Ganji RM, Jafari H, Nemati S (2020) A new approach for solving integro-differential equations of variable-order. J Comput Appl Math 1:1–11

    MathSciNet  MATH  Google Scholar 

  25. Keshi FK, Moghaddam BP, Aghili A (2019) A numerical technique for variable-order fractional functional nonlinear dynamic systems. Int J Dyn Control 1:1–8

    MathSciNet  Google Scholar 

  26. Hajipour M, Jajarmi A, Baleanu D, Sun H (2019) On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun Nonlinear Sci Numer Simul 69:119–133

    MathSciNet  MATH  Google Scholar 

  27. Chen R, Liu F, Anh V (2019) Numerical methods and analysis for a multi-term time-space variable-order fractional advection–diffusion equations and applications. J Comput Appl Math 352:437–452

    MathSciNet  MATH  Google Scholar 

  28. Doha EH, Abdelkawy MA, Amin AZ, Baleanu D (2019) Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations. Nonlinear Anal Model Control 24(2):176–188

    MathSciNet  MATH  Google Scholar 

  29. Zhou C, Li Z, Xie F (2019) Coexisting attractors, crisis route to chaos in a novel 4D fractional-order system and variable-order circuit implementation. Eur Phys J Plus 134(2):1–13

    Google Scholar 

  30. Owolabi KM, Hammouch Z (2019) Mathematical modeling and analysis of two-variable system with noninteger-order derivative. Chaos Interdiscip J Nonlinear Sci 29(1):1–10

    MathSciNet  MATH  Google Scholar 

  31. El-Sayed AA, Agarwal P (2019) Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math Methods Appl Sci 42(11):3978–3991

    MathSciNet  MATH  Google Scholar 

  32. Haq S, Ghafoor A, Hussain M (2019) Numerical solutions of variable order time fractional (1+1)-and (1+2)-dimensional advection dispersion and diffusion models. Appl Math Comput 360:107–121

    MathSciNet  MATH  Google Scholar 

  33. Heydari MH, Avazzadeh Z, Yang Y (2019) A computational method for solving variable-order fractional nonlinear diffusion-wave equation. Appl Math Comput 352:235–248

    MathSciNet  MATH  Google Scholar 

  34. Lu X, Li H, Chen N (2019) An indicator for the electrode aging of lithium-ion batteries using a fractional variable order model. Electrochim Acta 299:378–387

    Google Scholar 

  35. Atangana A (2017) Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 102:396–406

    MathSciNet  MATH  Google Scholar 

  36. Atangana A, Qureshi S (2019) Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos Solitons Fractals 123:320–337

    MathSciNet  MATH  Google Scholar 

  37. Heydari MH (2020) Numerical solution of nonlinear 2D optimal control problems generated by Atangana–Riemann–Liouville fractal-fractional derivative. Appl Math Comput 150:507–518

    MathSciNet  MATH  Google Scholar 

  38. Goufo EFD (2020) Fractal and fractional dynamics for a 3D autonomous and two-wing smooth chaotic system. Alex Eng J 1:1–9

    Google Scholar 

  39. Qureshi S, Atangana A (2020) Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fractals 136:1–10

    MathSciNet  Google Scholar 

  40. Abro KA, Atangana A (2020) Mathematical analysis of memristor through fractal-fractional differential operators: a numerical study. Math Methods Appl Sci 1:1–14

    MathSciNet  MATH  Google Scholar 

  41. Abro KA, Atangana A (2020) A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal-fractional differentiations. Eur Phys J Plus 135(2):1–16

    Google Scholar 

  42. Li Z, Liu Z, Khan MA (2020) Fractional investigation of bank data with fractal-fractional Caputo derivative. Chaos Solitons Fractals 131:1–10

    MathSciNet  Google Scholar 

  43. Heydari MH, Hosseininia M, Avazzadeh Z (2020) An efficient wavelet-based approximation method for the coupled nonlinear fractal-fractional 2D Schrödinger equations. Eng Comput 1:1–16

    MATH  Google Scholar 

  44. Atangana A, Shafiq A (2019) Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 127:226–243

    MathSciNet  MATH  Google Scholar 

  45. Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197(4300):287–289

    MATH  Google Scholar 

  46. Wang Q, Lu Q, Chen G, Duan L (2009) Bifurcation and synchronization of synaptically coupled FHN models with time delay. Chaos Solitons Fractals 39(2):918–925

    Google Scholar 

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Acknowledgements

Jesús Emmanuel Solís Pérez acknowledges the support provided by CONACyT through the assignment doctoral fellowship. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

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  1. Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México

    J. E. Solís-Pérez & J. F. Gómez-Aguilar

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  1. J. E. Solís-Pérez
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  2. J. F. Gómez-Aguilar
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Correspondence to J. F. Gómez-Aguilar.

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Solís-Pérez, J.E., Gómez-Aguilar, J.F. Variable-order fractal-fractional time delay equations with power, exponential and Mittag-Leffler laws and their numerical solutions. Engineering with Computers 38, 555–577 (2022). https://doi.org/10.1007/s00366-020-01065-0

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  • DOI: https://doi.org/10.1007/s00366-020-01065-0

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