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Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

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  • Vasily E. Tarasov

    (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
    Faculty of Information Technologies and Applied Mathematics, Moscow Aviation Institute (National Research University), Moscow 125993, Russia)

Abstract

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

Suggested Citation

  • Vasily E. Tarasov, 2020. "Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients," Mathematics, MDPI, vol. 8(12), pages 1-11, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2231-:d:463056
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    References listed on IDEAS

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    1. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    2. West, Bruce J., 2015. "Exact solution to fractional logistic equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 429(C), pages 103-108.
    3. Vasily E. Tarasov, 2019. "Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models," Mathematics, MDPI, vol. 7(6), pages 1-50, June.
    4. Area, Iván & Losada, Jorge & Nieto, Juan J., 2016. "A note on the fractional logistic equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 182-187.
    5. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
    6. Vasily E. Tarasov, 2020. "Non-Linear Macroeconomic Models of Growth with Memory," Mathematics, MDPI, vol. 8(11), pages 1-22, November.
    7. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Logistic map with memory from economic model," Papers 1712.09092, arXiv.org.
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    Cited by:

    1. Fernando Alcántara-López & Carlos Fuentes & Carlos Chávez & Jesús López-Estrada & Fernando Brambila-Paz, 2022. "Fractional Growth Model with Delay for Recurrent Outbreaks Applied to COVID-19 Data," Mathematics, MDPI, vol. 10(5), pages 1-18, March.
    2. Ivana Eliašová & Michal Fečkan, 2022. "Poincaré Map for Discontinuous Fractional Differential Equations," Mathematics, MDPI, vol. 10(23), pages 1-16, November.

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