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Solution of Fractional Differential Boundary Value Problems with Arbitrary Values of Derivative Orders for Time Series Analysis

Author

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  • Dmitry Zhukov

    (Institute of Radio Electronics and Informatics, MIREA-Russian Technological University, 78 Vernadsky Avenue, 119454 Moscow, Russia)

  • Vadim Zhmud

    (Department of Laser Systems, Novosibirsk State Technical University, Prosp. K. Marksa 20, 630073 Novosibirsk, Russia)

  • Konstantin Otradnov

    (Department of Applied Informatics and Intelligent Systems in the Humanities, RUDN University, 6 Miklukho-Maklaya St., 117198 Moscow, Russia)

  • Vladimir Kalinin

    (Department of Applied Informatics and Intelligent Systems in the Humanities, RUDN University, 6 Miklukho-Maklaya St., 117198 Moscow, Russia)

Abstract

The paper considers the solution of a fractional differential boundary value problem, that is, a diffusion-type equation with arbitrary values of the derivative orders on an infinite axis. The difference between the obtained results and other authors’ ones is that these involve arbitrary values of the derivative orders. The solutions described in the literature, as a rule, are considered in the case when the fractional time derivative β lies in the range: 0 < β ≤ 1, and the fractional state derivative α (the variable describing the state of the process) is in the range: 1 < α ≤ 2. The solution presented in the article allows us to consider any ranges for α and β, if the inequality 0 < β/α ≤ 0.865 is satisfied in the range β/α. In order to solve the boundary value problem, the probability density function of the observed state x of a certain process (for example, the magnitude of the deviation of the levels of a time series) from time t (for example, the time interval for calculating the amplitudes of the deviation of the levels of a time series) can be captured.

Suggested Citation

  • Dmitry Zhukov & Vadim Zhmud & Konstantin Otradnov & Vladimir Kalinin, 2024. "Solution of Fractional Differential Boundary Value Problems with Arbitrary Values of Derivative Orders for Time Series Analysis," Mathematics, MDPI, vol. 12(24), pages 1-24, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:24:p:3905-:d:1541536
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    References listed on IDEAS

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    1. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
    2. Enrica Pirozzi, 2024. "Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications," Mathematics, MDPI, vol. 12(19), pages 1-20, October.
    3. Jahanshahi, Hadi & Sajjadi, Samaneh Sadat & Bekiros, Stelios & Aly, Ayman A., 2021. "On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
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