• Journal of applied mathematics & informatics
• /
• v.28 no.1_2
• /
• pp.257-273
• /
• 2010

One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.

• 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.24 no.1_2
• /
• pp.167-178
• /
• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

• Journal of applied mathematics & informatics
• /
• v.25 no.1_2
• /
• pp.339-344
• /
• 2007

The relationships between binding number and fractional edge (vertex)-deletability or fractional k-extendability of graphs are studied. Furthermore, we show that the result about fractional vertex-deletability are best possible.

• Journal of the Korea Institute of Information and Communication Engineering
• /
• v.15 no.11
• /
• pp.2444-2450
• /
• 2011

In this paper, we propose the low fractional spur phase-locked loop(PLL) with multiple phase-frequency detector(PFD). The fractional spurs are suppressed by using a new PFD. The new PFD architecture with two different edge detection methods is used to suppress the fractional spur by limiting a maximum width of the output signals of PFD. The proposed PLL was simulated by HSPICE using a 0.35m CMOS parameters. The simulation results show that the proposed PLL is able to suppress fractional spurs with fast locking.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.6
• /
• pp.1423-1433
• /
• 2019

Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.583-586
• /
• 2011

A fractional Gagliardo-Nirenberg inequality is established. A sharp fractional Sobolev inequality is discussed as a direct consequence.

• Journal of applied mathematics & informatics
• /
• v.31 no.1_2
• /
• pp.299-309
• /
• 2013

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The fractional derivatives are presented in terms of Caputo sense. The application of the method to fractional BVPs leads to algebraic systems which can be solved by an appropriate method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

• 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.41 no.4
• /
• pp.811-819
• /
• 2023

In this research we have used the definition of standard fractional vector cross product to obtain fractional curl and fractional field of a standing wave, a travelling wave, a transverse wave, a vector field in xy plane, a complex vector field and an electric field. Fractional curl and fractional field for a complex order are also discussed. We have supported the study with calculation of impedance at γ = 0, 0 < γ < 1, γ = 1. The formula discussed in this paper are useful for study of polarization, reflection, impedance, boundary conditions where fractional solutions have applications.