• 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.

• 대한수학회논문집
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• 제33권1호
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• pp.171-177
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• 2018

In this research paper considering a differential equation with impulsive effect and dependent delay and applied Banach fixed point theorem using the impulsive condition to the impulsive fractional functional differential equation of an order ${\alpha}{\in}(1,2)$ to get an uniqueness solution. At last, theorem is verified by using a numerical example to illustrate the uniqueness solution.

• Korean Journal of Mathematics
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• 제29권4호
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• pp.749-763
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• 2021

This research article deals with the Mittag-Leffler-Hyers-Ulam stability of linear and impulsive fractional order differential equation which involves the Caputo derivative. The application of the generalized fractional Fourier transform method and fixed point theorem, evaluates the existence, uniqueness and stability of solution that are acquired for the proposed non-linear problems on Lizorkin space. Finally, examples are introduced to validate the outcomes of main result.

• Nonlinear Functional Analysis and Applications
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• 제28권2호
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• pp.449-471
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• 2023

In this paper, we study a linear system of homogeneous commensurate /incommensurate fractional-order differential equations by developing a new semi-analytical scheme. In particular, by decoupling the system into two fractional-order differential equations, so that the first equation of order (δ + γ), while the second equation depends on the solution for the first equation, we have solved the under consideration system, where 0 < δ, γ ≤ 1. With the help of using the Adomian decomposition method (ADM), we obtain the general solution. The efficiency of this method is verified by solving several numerical examples.

• 대한수학회보
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• 제47권6호
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• pp.1225-1234
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• 2010

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order $O({\Delta}x+{\Delta}t)$ is obtained in the discrete $L_2$ norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

• 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.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.215-228
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• 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.

• 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.

• 충청수학회지
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• 제28권4호
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• pp.583-590
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• 2015

This paper deals with linear impulsive fractional differential equations involving the Caputo derivative with non-integer order q. We provide exact solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions. Then we apply the exact solutions to improve impulsive integral inequalities with singularity.

• 대한수학회지
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• 제56권1호
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• pp.127-147
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• 2019

In this paper we establish some new explicit solutions for impulsive linear fractional differential equations with impulses at fixed times, which provides a handy tool in deriving singular integral-sum inequalities and an impulsive fractional comparison principle. Thus we study the Mittag-Leffler stability of impulsive differential equations with the Caputo fractional derivative by using the impulsive fractional comparison principle and piecewise continuous functions of Lyapunov's method. Also, we give some examples to illustrate our results.