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Operator-Based Approach for the Construction of Solutions to ( C D (1/ n ) ) k -Type Fractional-Order Differential Equations

Author

Listed:
  • Inga Telksniene

    (Mathematical Modelling Department, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania)

  • Zenonas Navickas

    (Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania)

  • Romas Marcinkevičius

    (Department of Software Engineering, Kaunas University of Technology, Studentu 50-415, LT-51368 Kaunas, Lithuania)

  • Tadas Telksnys

    (Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania)

  • Raimondas Čiegis

    (Mathematical Modelling Department, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania)

  • Minvydas Ragulskis

    (Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania)

Abstract

A novel methodology for solving Caputo D ( 1 / n ) C k -type fractional differential equations (FDEs), where the fractional differentiation order is k / n , is proposed. This approach uniquely utilizes fractional power series expansions to transform the original FDE into a higher-order FDE of type D ( 1 / n ) C k n . Significantly, this perfect FDE is then reduced to a k -th-order ordinary differential equation (ODE) of a special form, thereby allowing the problem to be addressed using established ODE techniques rather than direct fractional calculus methods. The effectiveness and applicability of this framework are demonstrated by its application to the fractional Riccati-type differential equation.

Suggested Citation

  • Inga Telksniene & Zenonas Navickas & Romas Marcinkevičius & Tadas Telksnys & Raimondas Čiegis & Minvydas Ragulskis, 2025. "Operator-Based Approach for the Construction of Solutions to ( C D (1/ n ) ) k -Type Fractional-Order Differential Equations," Mathematics, MDPI, vol. 13(7), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1169-:d:1626424
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    References listed on IDEAS

    as
    1. Farman, Muhammad & Xu, Changjin & Shehzad, Aamir & Akgul, Ali, 2024. "Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 461-488.
    2. Graef, John R. & Kong, Lingju & Ledoan, Andrew & Wang, Min, 2020. "Stability analysis of a fractional online social network model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 625-645.
    3. Zhang, Xugang & Gao, Xiyuan & Duan, Linchao & Gong, Qingshan & Wang, Yan & Ao, Xiuyi, 2025. "A novel method for state of health estimation of lithium-ion batteries based on fractional-order differential voltage-capacity curve," Applied Energy, Elsevier, vol. 377(PA).
    4. Sivalingam, S M & Kumar, Pushpendra & Trinh, Hieu & Govindaraj, V., 2024. "A novel L1-Predictor-Corrector method for the numerical solution of the generalized-Caputo type fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 462-480.
    5. Navickas, Z. & Telksnys, T. & Marcinkevicius, R. & Ragulskis, M., 2017. "Operator-based approach for the construction of analytical soliton solutions to nonlinear fractional-order differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 625-634.
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