Convex Functions on Discrete Time Domains

2016 ◽  
Vol 59 (2) ◽  
pp. 225-233 ◽  
Author(s):  
Ferhan M. Atıcı ◽  
Hatice Yaldız
Keyword(s):  
Fractional Calculus ◽  
Discrete Time ◽  
Convex Functions ◽  
Time Scale ◽  
Discrete Fractional Calculus ◽  
Hadamard Inequality ◽  
Time Domains ◽  
Definition Of

AbstractIn this paper, we introduce the definition of a convex real valued function f defined on the set of integers, ℤ. We prove that f is convex on Z if and only if Δ2 f ≥ 0 on ℤ. As a first application of this new concept, we state and prove discrete Hermite–Hadamard inequality using the basics of discrete calculus (i.e., the calculus on Z). Second, we state and prove the discrete fractional Hermite–Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

scholarly journals On Sumudu Transform Method in Discrete Fractional Calculus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Fahd Jarad ◽  
Kenan Taş
Keyword(s):  
Fractional Calculus ◽  
Time Scale ◽  
Initial Value Problems ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Sumudu Transform ◽  
Definition Of ◽  
General Time

In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the generalized discrete Sumudu transform and present some of its basic properties. We obtain the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences. We apply this transform to solve some fractional difference initial value problems.

s-convex functions on discrete time domains

Analysis ◽  
2017 ◽  
Vol 37 (4) ◽  
Author(s):  
Hatice Yaldız ◽  
Praveen Agarwal
Keyword(s):  
Discrete Time ◽  
Convex Functions ◽  
Time Domains ◽  
Definition Of

AbstractIn the present work, we give the definition of an

Discrete Fractional Diffusion Equation of Chaotic Order

2016 ◽  
Vol 26 (01) ◽  
pp. 1650013 ◽  
Author(s):  
Guo-Cheng Wu ◽  
Dumitru Baleanu ◽  
He-Ping Xie ◽  
Sheng-Da Zeng
Keyword(s):  
Porous Media ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Discrete Time ◽  
Chaotic Map ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Diffusion Modeling ◽  
Discrete Fractional Calculus

Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.

scholarly journals Some Inequalities Using Generalized Convex Functions in Quantum Analysis

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1402 ◽  
Author(s):  
Miguel J. Vivas-Cortez ◽  
Artion Kashuri ◽  
Rozana Liko ◽  
Jorge E. Hernández Hernández
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Generalized Convex Functions ◽  
Hadamard Inequality ◽  
Quantum Calculus ◽  
Quantum Analysis ◽  
Real Functions ◽  
Definition Of

In the present work, the Hermite–Hadamard inequality is established in the setting of quantum calculus for a generalized class of convex functions depending on three parameters: a number in ( 0 , 1 ] and two arbitrary real functions defined on [ 0 , 1 ] . From the proven results, various inequalities of the same type are deduced for other types of generalized convex functions and the methodology used reveals, in a sense, a symmetric mathematical phenomenon. In addition, the definition of dominated convex functions with respect to the generalized class of convex functions aforementioned is introduced, and some integral inequalities are established.

scholarly journals Discrete fractional diffusion model with two memory terms

2015 ◽  
Vol 19 (4) ◽  
pp. 1177-1181
Author(s):  
Yan-Mei Qin ◽  
Hua Kong ◽  
Kai-Teng Wu ◽  
Xiao-Ming Zhu
Keyword(s):  
Numerical Simulation ◽  
Fractional Calculus ◽  
Diffusion Model ◽  
Discrete Time ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Difference ◽  
Linear Evolution ◽  
Time Domains ◽  
Diffusion Behaviors

Fractional calculus can always exactly describe anomalous diffusion. Recently the discrete fractional difference is becoming popular due to the depiction of non-linear evolution on discrete time domains. This paper proposes a diffusion model with two terms of discrete fractional order. The numerical simulation is given to reveal various diffusion behaviors.

scholarly journals SOME HARDY-TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA DELTA FRACTIONAL INTEGRALS

Fractals ◽  
2021 ◽  
pp. 2240004
Author(s):  
FUZHANG WANG ◽  
USAMA HANIF ◽  
AMMARA NOSHEEN ◽  
KHURAM ALI KHAN ◽  
HIJAZ AHMAD ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Convex Functions ◽  
Type Inequality ◽  
Fractional Integrals ◽  
Discrete Fractional Calculus ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Hilbert Type

In this paper, some Jensen- and Hardy-type inequalities for convex functions are extended by using Riemann–Liouville delta fractional integrals. Further, some Pólya–Knopp-type inequalities and Hardy–Hilbert-type inequality for convex functions are also proved. Moreover, some related inequalities are proved by using special kernels. Particular cases of resulting inequalities provide the results on fractional calculus, time scales calculus, quantum fractional calculus and discrete fractional calculus.

scholarly journals Inequalities for a Unified Integral Operator for Strongly α , m -Convex Function and Related Results in Fractional Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chahn Yong Jung ◽  
Ghulam Farid ◽  
Kahkashan Mahreen ◽  
Soo Hak Shim
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Special Cases ◽  
Definition Of

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and α , m -convex functions. A new definition of function, namely, strongly α , m -convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

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