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In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. [ABSTRACT FROM AUTHOR]

Extensive research was conducted on the transmission dynamics of tuberculosis epidemics during its reemergence from the 1980s to the early 1990s, but this global problem of investigating tuberculosis spread dynamics remains of paramount importance. Our study utilized a fractional-order delay differential model to study tuberculosis transmission, where the time delay in the model was attributed to the disease's latent period. What is more, this model accounts for endogenous reactivation, exogenous reinfection, and treatment of tuberculosis. The model qualitative properties and the basic reproduction number were analyzed. The primary goal of the study was to recover the important dynamic parameters of tuberculosis. Our understanding of these complex processes leverages the efficacy of efforts for controlling the disease, forecasting future dynamics, and applying further appropriate strategies to prevent its spread.The calibration itself was carried out via minimization of a quadratic cost functional. Computational simulations demonstrated that the algorithm is capable of working with noisy real data. [ABSTRACT FROM AUTHOR]

In order to investigate the honeybee population dynamics, many differential equation models were proposed. Fractional derivatives incorporate the history of the honeybee population dynamics. We numerically study the inverse problem of parameter identification in models with Caputo and Caputo–Fabrizio differential operators. We use a gradient method of minimizing a quadratic cost functional. We analyze and compare results for the integer (classic) and fractional models. The present work also contains discussion on the efficiency of the numerical methods used. Computational tests with realistic data were performed and are discussed. [ABSTRACT FROM AUTHOR]