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Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws

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  • Solís-Pérez, J.E.
  • Gómez-Aguilar, J.F.
  • Atangana, A.

Abstract

Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

Suggested Citation

  • Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
    • Handle: RePEc:eee:chsofr:v:114:y:2018:i:c:p:175-185
    DOI: 10.1016/j.chaos.2018.06.032
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    6. Kumar, Devendra & Singh, Jagdev & Baleanu, Dumitru & Sushila,, 2018. "Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 155-167.
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    3. Hasib Khan & Jehad Alzabut & Haseena Gulzar & Osman Tunç & Sandra Pinelas, 2023. "On System of Variable Order Nonlinear p-Laplacian Fractional Differential Equations with Biological Application," Mathematics, MDPI, vol. 11(8), pages 1-17, April.
    4. Yadav, Swati & Pandey, Rajesh K. & Shukla, Anil K., 2019. "Numerical approximations of Atangana–Baleanu Caputo derivative and its application," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 58-64.
    5. Yadav, Swati & Pandey, Rajesh K., 2020. "Numerical approximation of fractional burgers equation with Atangana–Baleanu derivative in Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
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    7. Heydari, M.H. & Atangana, A., 2019. "A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 339-348.
    8. Zheng, Xiangcheng & Wang, Hong & Fu, Hongfei, 2020. "Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    9. Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2019. "A novel method to solve variable-order fractional delay differential equations based in lagrange interpolations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 266-282.
    10. Avcı, Derya & Yetim, Aylin, 2019. "Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 361-365.
    11. Ávalos-Ruiz, L.F. & Gómez-Aguilar, J.F. & Atangana, A. & Owolabi, Kolade M., 2019. "On the dynamics of fractional maps with power-law, exponential decay and Mittag–Leffler memory," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 364-388.
    12. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    13. Andrade, Dana I. & Specchia, Stefania & Fuziki, Maria E.K. & Oliveira, Jessica R.P. & Tusset, Angelo M. & Lenzi, Giane G., 2024. "Dynamic analysis and SDRE control applied in a mutating autocatalyst with chaotic behavior," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    14. Pho, Kim-Hung & Heydari, M.H. & Tuan, Bui Anh & Mahmoudi, Mohammad Reza, 2020. "Numerical study of nonlinear 2D optimal control problems with multi-term variable-order fractional derivatives in the Atangana-Baleanu-Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    15. Heydari, M. H. & Atangana, A., 2020. "An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    16. Soradi-Zeid, Samaneh & Jahanshahi, Hadi & Yousefpour, Amin & Bekiros, Stelios, 2020. "King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    17. Ghanbari, Behzad & Gómez-Aguilar, J.F., 2018. "Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 114-120.

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