• Nonlinear Functional Analysis and Applications
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• 제28권1호
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• pp.237-249
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• 2023

In this paper, by means of the Schauder fixed point theorem and Arzela-Ascoli theorem, the existence and uniqueness of solutions for a class of not instantaneous impulsive problems of nonlinear fractional functional Volterra-Fredholm integro-differential equations are investigated. An example is given to illustrate the main results.

• 호남수학학술지
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• 제45권1호
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• pp.160-183
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• 2023

In the present paper, we prove that our main inequality reduces to some trapezoid and Newton type inequalities for differentiable s-convex functions. These inequalities are established by using the well-known Riemann-Liouville fractional integrals. With the help of special cases of our main results, we also present some new and previously obtained trapezoid and Newton type inequalities.

• 호남수학학술지
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• 제45권2호
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• pp.285-299
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• 2023

In the present paper, we establish some perturbed Newton-type inequalities in the case of twice differentiable convex functions. These inequalities are established by using the well-known Riemann-Liouville fractional integrals. With the aid of special cases of our main results, we also give some previously obtained Newton-type inequalities.

• Kyungpook Mathematical Journal
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• 제64권1호
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• pp.31-46
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• 2024

The aim of this paper is to present an approach to improve reverse Minkowski and Hölder-type inequalities using k-weighted fractional integral operators _a⁺𝔍^𝜇_w with respect to a strictly increasing continuous function 𝜇, by introducing two parameters of integrability, p and q. For various choices of 𝜇 we get interesting special cases.

• 디지털콘텐츠학회 논문지
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• 제9권1호
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• pp.109-116
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• 2008

정칙적인 곡선이나 곡면에 대해서만 적용되고 있는 전통적인 개념의 미적분을 복잡하고 비 정칙적인 대상에도 적용할 수 있는 방법들이 다양하게 시도되고 있다. 이에 본 논문에서는 비 정수 차수의 도함수를 적분의 한 형태로 변환하여 표현하는 방법을 알아보고 이를 효과적으로 구현함으로 실제적인 응용을 할 수 있게 하였다.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.1-14
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• 2008

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and $Gr\ddot{u}nwald$-Letnikov(G-L) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

• 대한수학회보
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• 제55권6호
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• pp.1639-1657
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• 2018

In this paper, we initiate the study of boundary value problems involving Hilfer fractional derivatives. Several new existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating our results are also presented.

• Korean Journal of Mathematics
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• 제23권4호
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• pp.537-555
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• 2015

This paper is concerned with the existence of solutions to the following fractional differential inclusion $$\{-{\frac{d}{dx}}\(p_0D^{-{\beta}}_x(u^{\prime}(x)))+q_xD^{-{\beta}}_1(u^{\prime}(x))\){\in}{\partial}F_u(x,u),\;x{\in}(0,1),\\u(0)=u(1)=0,$$ where $_0D^{-{\beta}}_x$ and $_xD^{-{\beta}}_1$ are left and right Riemann-Liouville fractional integrals of order ${\beta}{\in}(0,1)$ respectively, 0 < p = 1 - q < 1 and $F:[0,1]{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is locally Lipschitz with respect to the second variable. Due to the general assumption on the constants p and q, the problem does not have a variational structure. Despite that, here we study it combining with an iterative technique and nonsmooth critical point theory, we obtain an existence result for the above problem under suitable assumptions. The result extends some corresponding results in the literatures.

• 대한수학회지
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• 제53권4호
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• pp.929-967
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• 2016

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.

• 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.