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3 Fundamenta Informaticae

3 2017

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SM-Algorithms for Approximating the Variable-Order Fractional Derivative of High Order

Moghaddam B. P., Machado J. A. T.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

293--311

In this paper we discuss different definitions of variable-order derivatives of high order and we propose accurate and robust algorithms for their approximate calculation. The proposed algorithms are based on finite difference approximations and B-spline interpolation. We compare the performance of the algorithms by experimental convergence order. Numerical examples are presented demonstrating the efficiency and accuracy of the proposed algorithms.

Dynamics of Commodities Prices : Integer and Fractional Models

David S. A., Machado J. A. T., Trevisan L. R., Inácio Jr. C. M. C., Lopes A. M.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

389--408

This paper examines the time series of four important agricultural commodities, namely the soybean, corn, coffee and sugar prices. Time series can exhibit long-range dependence and persistence in their observation. The long memory feature of data is a documented fact and there has been an increasing interest in studying such concepts in the perspective of economics and finance. In this work, we start by analyzing the time series of the four commodities by means of the Fractional Fourier Transform (FrFT) to unveil time-frequency patterns in the data. In a second phase, we apply Auto Regressive Integrated Moving Average (ARIMA) and Auto Regressive Fractionally Integrated Moving Average (ARFIMA) models for obtaining the spot price composition and predict future price. The ARFIMA process is a known class of long memory model, representing a generalization of the ARIMA algorithm. We compare the performances of the ARIMA and the ARFIMA models and we show that the ARFIMA has a superior performance for future price forecasting.

A New Family of the Local Fractional PDEs

Yang X.-J., Machado J. A. T., Nieto J. J.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

63--75

A new family of the local fractional PDEs is investigated in this article. The linear, quasilinear, semilinear and nonlinear local fractional PDEs are presented. Furthermore, three types of the local fractional PDEs are discussed, namely, parabolic, hyperbolic and elliptic. Several examples illustrate the corresponding models in nonlinear mathematical physics.