• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.287-304
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• 2017

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

• Kyungpook Mathematical Journal
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• v.60 no.1
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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.

• 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.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.455-466
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• 2008

Current studies on products of analytic functionals have been based on applying convolution products in D' and the Fourier exchange formula. There are very few results directly computed from the ultradistribution space Z'. The goal of this paper is to introduce a definition for the product of analytic functionals and construct a new multiplier space $F(N_m)$ for $\delta^{(m)}(s)$ in a one or multiple dimension space, where Nm may contain functions without compact support. Several examples of the products are presented using the Cauchy integral formula and the multiplier space, including the fractional derivative of the delta function $\delta^{(\alpha)}(s)$ for $\alpha>0$.

• 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.31 no.1
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• pp.115-129
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• 2016

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. [9] evaluated the fractional integrals of the product of Appell function of the third kernel $F_3$ and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function $F_3$ and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, $Erd\acute{e}lyi$-Kober and Saigo operators, associated with multivariable H-function and Appell function $F_3$ are shown to be deduced as special cases of our findings.

• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1191-1204
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• 2001

The paper is devoted to the study of fractional integration and differentiation on a finite interval [a, b] of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int_{a}^{x}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation ${\delta}+{\mu}({\delta}=xD,D=d/dx)$ with real $\mu$, being taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X^{p}_{c}$(a, b) of Lebesgue measurable functions f on $R_{+}=(0,{\infty})$ such that for c${\in}R=(-{\infty}{\infty})$, in particular in the space $L^{p}(0,{\infty})\;(1{\le}{\le}{\infty})$. The existence almost every where is established for the coorresponding Hadamard-type fractional derivative for a function g(x) such that $x^{p}$g(x) have $\delta$ derivatives up to order n-1 on [a, b] and ${\delta}^{n-1}[x^{\mu}$g(x)] is absolutely continuous on [a, b]. Semigroup and reciprocal properties for the above operators are proved.

• Korean Journal of Mathematics
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• v.27 no.1
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• pp.119-129
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• 2019

Balance function is one of the joint factors to determine fall in risk theory. It helps to moderate the progression and riskiness of falls for detecting balance and fall risk factors. Nevertheless, the objective measures for balance function require expensive equipment with the assessment of any expertise. We establish the existence and uniqueness of a multi-order fractional differential equations based on ${\psi}$-Hilfer operator on time scales with balance function. This class describes the dynamic of time scales derivative. Our tool is based on the Schauder fixed point theorem. Here, sufficient conditions for Ulam-stability are given.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.269-284
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• 2016

In this paper, we firstly use Krasnosel'skii fixed point theorem to investigate positive solutions for the following three-point boundary value problems for p-Laplacian with a parameter $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+{\lambda}f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1), λ > 0 is a parameter. Then we use Leggett-Williams fixed point theorem to study the existence of three positive solutions for the fractional boundary value problem $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1).

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1241-1261
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• 2018

We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.