• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.257-273
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• 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
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• 제26권5_6호
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• pp.1101-1121
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• 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.

• East Asian mathematical journal
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• 제36권1호
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• pp.81-90
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• 2020

Many compartment models assume a kinetically homogeneous amount of materials that have well-stirred compartments. However, based on observations from such processes, they have been heuristically fitted by exponential or gamma distributions even though biological media are inhomogeneous in real environments. Fractional differential equations using a specific kernel in Pharmacokinetic/Pharmacodynamic (PKPD) model are recently introduced to account for abnormal drug disposition. We discuss a tumor growth inhibition (TGI) model using fractional-order derivative from it. This represents a tumor growth delay by cytotoxic agents and additionally show variations in the equilibrium points by the change of fractional order. The result indicates that the equilibrium depends on the tumor size as well as a change of the fractional order. We find that the smaller the fractional order, the smaller the equilibrium value. However, a difference of them is the number of concavities and this indicates that TGI over time profile for fitting or prediction should be determined properly either fractional order or tumor sizes according to the number of concavities shown in experimental data.

• Journal of Power Electronics
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• 제13권6호
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• pp.1008-1015
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• 2013

By using fractional calculus and the circuit-averaging technique, the modeling and analysis of a Buck converter with fractional order inductor and fractional order capacitor in discontinuous conduction mode (DCM) operations is investigated in this study. The equivalent averaged circuit model of the fractional order Buck converter in DCM operations is established. DC analysis is conducted by using the derived DC equivalent circuit model. The transfer functions from the input voltage to the output voltage, the duty cycle to the output voltage, the input impedance, and the output impedance of the fractional order Buck converter in DCM operations are derived from the corresponding AC-equivalent circuit model. Results show that the DC equilibrium point, voltage ratio, and all derived transfer functions of the fractional order Buck converter in DCM operations are affected by the inductor order and/or capacitor order. The fractional order inductor and fractional order capacitor are designed, and PSIM simulations are performed to confirm the correctness of the derivations and theoretical analysis.

• 한국전자통신학회논문지
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• 제10권12호
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• pp.1389-1394
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• 2015

최근에 fractional calculus의 개념을 적용하여 fractional 미분 방정식으로 표현되는 기법이 제어공학, 수학, 물리학 등에 적용하고자 하는 노력이 나타나고 있다. 본 논문에서는 Duffing 방정식으로 표현되는 동적 방정식을 정수 차수가 아닌 fractional 차수로 표현하고 이 fractional 실수 차수에서 차수의 크기에 따라 카오스 거동이 존재함을 실수 차수의 값을 변화시켜가면서 시계열 데이터와 위상공간으로 확인하고자 한다.

• 한국전자통신학회논문지
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• 제11권1호
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• pp.81-86
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• 2016

300년 전에 발표한 fractional calculus의 개념인 fractional 미분 방정식을 제어공학, 수학, 물리학 등에 적용하고자 하는 노력이 지속되고 있다. 본 논문에서는 Van der Pol 방정식으로 표현되는 동적 방정식을 정수 차수와 실수 차수를 가진 fractional 차수로 표현하고 실수 차수의 값을 변화시켜 가면서 시계열 데이터와 위상공간으로 정수 차수와 실수 차수의 비교를 수행한다.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.229-241
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• 2007

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractional-order dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.695-703
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• 2002

The fractional order evolutionary integral equations have been considered by first author in [6], the existence, uniqueness and some other properties of the solution have been proved. Here we study the continuation of the solution and its fractional order derivative. Also we study the generality of this problem and prove that the fractional order diffusion problem, the fractional order wave problem and the initial value problem of the equation of evolution are special cases of it. The abstract diffusion-wave problem will be given also as an application.

• Kyungpook Mathematical Journal
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• 제60권1호
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• pp.73-116
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• 2020

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제11권1호
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• pp.436-450
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• 2017

Recently, the method based on fractional order partial differential equation has been used in image processing. Usually, the optional order of fractional differentiation is determined by a lot of experiments. In this paper, a denoising model is proposed based on adaptive fractional order anisotropic diffusion. In the proposed model, the complexity of the local image texture is reflected by the local variance, and the order of the fractional differentiation is determined adaptively. In the process of the adaptive fractional order model, the discrete Fourier transform is applied to compute the fractional order difference as well as the dynamic evolution process. Experimental results show that the peak signal-to-noise ratio (PSNR) and structural similarity index measurement (SSIM) of the proposed image denoising algorithm is better than that of other some algorithms. The proposed algorithm not only can keep the detailed image information and edge information, but also obtain a good visual effect.