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An operator method for composite fractional partial differential equations

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  • Wang, Huiwen
  • Li, Fang

Abstract

A composite fractional-order partial differential equation is a type of partial differential equation that combines integer-order and fractional-order derivatives, enabling more accurate characterization of the dynamical behaviors of complex systems. We study this kind of equation, which incorporates the Caputo fractional derivatives of orders 1<α<2 and 0≤β<1 in a Banach space. Utilizing the theory of (a,k)-regularized resolvent families of bounded and linear operators, we delineate the solution pertinent to the abstract form of these equations. In addition, we establish results pertaining to the existence and uniqueness of solutions. Specifically, we obtain the existence and uniqueness of solutions for the fractional oscillation equation with initial value conditions. Furthermore, applying our results, we solve a multi-term composite abstract fractional differential equation, a Sobolev-type composite fractional differential equation, a Bagley–Torvik equation and a fractional control system.

Suggested Citation

  • Wang, Huiwen & Li, Fang, 2025. "An operator method for composite fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 198(C).
    • Handle: RePEc:eee:chsofr:v:198:y:2025:i:c:s0960077925005156
    DOI: 10.1016/j.chaos.2025.116502
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    References listed on IDEAS

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    1. Carlos Lizama & Felipe Poblete, 2012. "On a Functional Equation Associated with (a, k)‐Regularized Resolvent Families," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Najafi, Alireza & Taleghani, Rahman, 2022. "Fractional Liu uncertain differential equation and its application to finance," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    3. Aljethi, Reem Abdullah & Kılıçman, Adem, 2023. "Analysis of fractional differential equation and its application to realistic data," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    4. Boutaous, Fatiha, 2024. "Study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    5. Z. Sharifi & B. P. Moghaddam & M. Ilie, 2024. "Efficient numerical simulation of fractional-order Van der Pol impulsive system," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 35(03), pages 1-15, March.
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