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

• Published in: Mathematics (Basel) 2024-12, Vol.12 (24), p.3905
• Main Authors: Zhukov, Dmitry, Zhmud, Vadim, Otradnov, Konstantin, Kalinin, Vladimir
• Format: Article
• Language: English
• Subjects: Boundary value problems; Derivatives; Deviation; electoral processes; fractional differential equations; Numerical analysis; Probability density functions; Time series; time series analysis
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12243905

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.