• Nonlinear Functional Analysis and Applications
• /
• v.29 no.2
• /
• pp.435-450
• /
• 2024

This study depends on the modified conformable fractional derivative definition to generalize and proves some theorems of the classical power series into the fractional power series. Furthermore, a comprehensive formulation of the generalized Taylor's series is derived within this context. As a result, a new technique is introduced for the fractional power series. The efficacy of this new technique has been substantiated in solving some fractional differential equations.

• Kyungpook Mathematical Journal
• /
• v.62 no.3
• /
• pp.583-593
• /
• 2022

Lienard's equations are important nonlinear differential equations with application in many areas of applied mathematics. In the present article, a new approach known as the modified fractional Taylor series method (MFTSM) is proposed to solve the nonlinear fractional Lienard equations with Caputo fractional derivatives, and the convergence of this method is established. Numerical examples are given to verify our theoretical results and to illustrate the accuracy and effectiveness of the method. The results obtained show the reliability and efficiency of the MFTSM, suggesting that it can be used to solve other types of nonlinear fractional differential equations that arise in modeling different physical problems.

• Journal of applied mathematics & informatics
• /
• v.23 no.1_2
• /
• pp.215-228
• /
• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• Journal of applied mathematics & informatics
• /
• v.26 no.5_6
• /
• pp.1101-1121
• /
• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Journal of applied mathematics & informatics
• /
• v.19 no.1_2
• /
• pp.19-32
• /
• 2005

It is shown that the problem of minimizing (maximizing) a quadratic cost functional (quadratic gain functional) given the dynamics dx = (fx + gu)dt + hdb(t, a) where b(t, a) is a fractional Brownian motion of order a, 0 < 2a < 1, can be solved completely (and meaningfully!) by using the dynamical equations of the moments of x(t). The key is to use fractional Taylor's series to obtain a relation between differential and differential of fractional order.

• The Korean Journal of Applied Statistics
• /
• v.28 no.6
• /
• pp.1235-1243
• /
• 2015

Increased data volume in the ICT area has increased the importance of forecasting accuracy for internet traffic. Forecasting results may have paper plans for traffic management and control. In this paper, we propose combined forecasts based on several time series models such as Seasonal ARIMA and Taylor's adjusted Holt-Winters and Fractional ARIMA(FARIMA). In combined forecasting methods, we use simple-combined method, MSE based method (Armstrong, 2001), Ordinary Least Squares (OLS) method and Equality Restricted Least Squares (ERLS) method. The results show that the Seasonal ARIMA model outperforms in 3 hours ahead forecasts and that combined forecasts outperform in longer periods.

• Journal of Korea Water Resources Association
• /
• v.51 no.spc
• /
• pp.1079-1089
• /
• 2018

The majority of existing studies for quantifying uncertainties in climate change impact assessments suggest only the uncertainties of each stage, and not the total uncertainty and its propagation in the whole procedure. Therefore, this study has proposed a new method, the Uncertainty Delta Method (UDM), which can quantify uncertainties using the variances of projections (as the UDM is derived from the first-order Taylor series expansion), to allow for a comprehensive quantification of uncertainty at each stage and also to provide the levels of uncertainty propagation, as follows: total uncertainty, the level of uncertainty increase at each stage, and the percentage of uncertainty at each stage. For quantifying uncertainties at each stage as well as the total uncertainty, all the stages - two emission scenarios (ES), three Global Climate Models (GCMs), two downscaling techniques, and two hydrological models - of the climate change assessment for water resources are conducted. The total uncertainty took 5.45, and the ESs had the largest uncertainty (4.45). Additionally, uncertainties are propagated stage by stage because of their gradual increase: 5.45 in total uncertainty consisted of 4.45 in emission scenarios, 0.45 in climate models, 0.27 in downscaling techniques, and 0.28 in hydrological models. These results indicate the projection of future water resources can be very different depending on which emission scenarios are selected. Moreover, using Fractional Uncertainty Method (FUM) by Hawkins and Sutton (2009), the major uncertainty contributor (emission scenario: FUM uncertainty 0.52) matched with the results of UDM. Therefore, the UDM proposed by this study can support comprehension and appropriate analysis of the uncertainty surrounding the climate change impact assessment, and make possible a better understanding of the water resources projection for future climate change.