• Nonlinear Functional Analysis and Applications
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• v.29 no.3
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• pp.803-823
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• 2024

In this paper, we study the existence and uniqueness of solutions for the fractional time-varying delay integrodifferential equation with multi-point multi-term nonlocal and fractional integral boundary conditions by using fixed point theorems. The fractional derivative considered here is in the Caputo sense. Examples are provided to illustrate the results.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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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.

• Nuclear Engineering and Technology
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• v.52 no.9
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• pp.2017-2024
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• 2020

In this study, a fractional order proportional integral derivative (FOPID) controller is designed to create the reference power trajectory and to conquer the uncertainties and external disturbances. A fractional nonlinear model was utilized to describe the nuclear reactor dynamic behaviour considering thermal-hydraulic effects. The controller parameters were tuned using optimization method in Matlab/Simulink. The FOPID controller was simulated using Matlab/Simulink and the controller performance was evaluated for Hard variation of the reference power and compared with that of integer order a proportional integral derivative (IOPID) controller by two models of fractional neutron point kinetic (FNPK) and classical neutron point kinetic (CNPK). Also, the FOPID controller robustness was appraised against the external disturbance and uncertainties. Simulation results showed that the FOPID controller has the faster response of the control attempt signal and the smaller tracking error with respect to the IOPID in tracking the reference power trajectory. In addition, the results demonstrated the ability of FOPID controller in disturbance rejection and exhibited the good robustness of controller against uncertainty.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.497-502
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• 2022

Existence and uniqueness for fractional differential equations satisfying a general nonlocal initial or boundary condition are proven by means of Schauder's fixed point theorem. The nonlocal condition is given as an integral with respect to a signed measure, and includes the standard initial value condition and multi-point boundary value condition.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.161-184
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• 2020

In this article, we establish some new weighted Hardy-type inequalities involving some variants of extended Riemann-Liouville fractional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also prove the boundedness of the k-fractional integral operator on L_p[a, b].

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.601-617
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• 2017

We give a function associated with generalized Ostrowski type inequality and its integral representation for local fractional calculus. Then, using this function and its integral representation, we establish several inequalities of generalized Ostrowski type for twice local fractional differentiable functions. We also consider some special cases of the main results which are further applied to a concrete function to yield two interesting inequalities associated with two generalized means.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.167-181
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• 2021

In this paper, the Saigo's k-fractional integral and derivative operators involving k-hypergeometric function in the kernel are applied to the generalized k-Bessel function; results are expressed in term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Bessel functions are considered.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.537-555
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• 2023

In this paper, we consider two types of fractional boundary value problems, one of them is an implicit type and the other will be an integro-differential type with nonlocal integral multi-point boundary conditions in the frame of generalized Hilfer fractional derivatives. The existence and uniqueness results are acquired by applying Krasnoselskii's and Banach's fixed point theorems. Some various numerical examples are provided to illustrate and validate our results. Moreover, we get some results in the literature as a special case of our current results.

• Honam Mathematical Journal
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• v.39 no.3
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• pp.401-416
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• 2017

Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions of various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations using Prabhakar-type operators. The idea of present paper is motivated by Tomovski et al. [21].

• Proceedings of the Korean Statistical Society Conference
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• 2005.05a
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• pp.173-178
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• 2005

We derive an It${\hat{o}}$ formula for generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameter ${H_1},\;H_2$ ${\epsilon}$ (0,1). As an application, we consider a stochastic integral representation for the local time of the fractional Brownian sheet.