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Darboux problem for proportional partial fractional differential equations

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  • Ben Makhlouf, Abdellatif
  • Benjemaa, Mondher
  • Boucenna, Djalal
  • Hammami, Mohamed Ali

Abstract

In this paper, we introduce and study a novel fractional derivative called the generalized proportional Caputo fractional derivative. Under adequate assumptions on the source term, we prove existence and uniqueness aspects of the Darboux problems involving the generalized proportional fractional operator. A numerical scheme based on finite differences approximation is also investigated, and results of convergence are established. Several numerical tests are performed in illustration.

Suggested Citation

  • Ben Makhlouf, Abdellatif & Benjemaa, Mondher & Boucenna, Djalal & Hammami, Mohamed Ali, 2023. "Darboux problem for proportional partial fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:chsofr:v:166:y:2023:i:c:s0960077922010852
    DOI: 10.1016/j.chaos.2022.112906
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    References listed on IDEAS

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    1. Ravi Agarwal & Snezhana Hristova, 2022. "Impulsive Memristive Cohen–Grossberg Neural Networks Modeled by Short Term Generalized Proportional Caputo Fractional Derivative and Synchronization Analysis," Mathematics, MDPI, vol. 10(13), pages 1-12, July.
    2. Balamurali Ramakrishnan & Murat Erhan Cimen & Akif Akgul & Chunbiao Li & Karthikeyan Rajagopal & Hakan Kor & Abdellatif Ben Makhlouf, 2022. "Chaotic Oscillations in a Fractional-Order Circuit with a Josephson Junction Resonator and Its Synchronization Using Fuzzy Sliding Mode Control," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-11, August.
    3. Abdellatif Boutiara & Mohammed K. A. Kaabar & Zailan Siri & Mohammad Esmael Samei & Xiao-Guang Yue & Abdellatif Ben Makhlouf, 2022. "Investigation of the Generalized Proportional Langevin and Sturm–Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Appl," Mathematical Problems in Engineering, Hindawi, vol. 2022, pages 1-21, April.
    4. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
    5. He, Lin & Wang, Yangyang & Wei, Yujiang & Wang, Mingwei & Hu, Xiaosong & Shi, Qin, 2022. "An adaptive central difference Kalman filter approach for state of charge estimation by fractional order model of lithium-ion battery," Energy, Elsevier, vol. 244(PA).
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