scholarly journals Estimation of Integral Inequalities Using the Generalized Fractional Derivative Operator in the Hilfer Sense

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Evaluation Process ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Practical Significance ◽  
Caputo Fractional Derivatives ◽  
Inequality Theory ◽  
Left And Right ◽  
Derivatives Of

With the great progress of fractional calculus, integral inequalities have been greatly enriched by fractional operators; users and researchers have formed a real-world phenomenon in the production of the evaluation process, which results in convexity. Monotonicity and inequality theory has a strong relationship, whichever we work on, and we can apply it to the other one due to the strong correlation produced between them, especially in the past few years. In this article, we introduce some estimations of left and right sides of the generalized Caputo fractional derivatives of a function for n th order differentiability via convex function, and related inequalities have been presented. Monotonicity and convexity of functions are used with some usual and straightforward inequalities. Moreover, we establish some new inequalities for C ⌣ eby s ⌣ ev and Gr u ¨ ss type involving the generalized Caputo fractional derivative operators. The finding provides the theoretical basis and practical significance for the establishment of fractional calculus in convexity. It also introduces new ways of thinking and methods for innovative scientific research.

scholarly journals Toward fractional gradient elasticity

2014 ◽  
Vol 23 (1-2) ◽  
pp. 41-46 ◽  
Author(s):  
Vasily E. Tarasov ◽  
Elias C. Aifantis
Keyword(s):  
Fractional Derivative ◽  
Gradient Elasticity ◽  
Point Load ◽  
Three Dimensions ◽  
Riesz Fractional Derivative ◽  
Caputo Fractional Derivatives ◽  
Stress Equilibrium ◽  
First Case ◽  
Elastic Bar ◽  
Derivatives Of

AbstractThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

Vectorial fractional integral inequalities with convexity

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
George Anastassiou
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
General Integral ◽  
Hardy Type ◽  
Wide Range ◽  
Left And Right ◽  
Multiple Integrals ◽  
P Type ◽  
Derivatives Of ◽  
Increasing Functions

AbstractHere we present vectorial general integral inequalities involving products of multivariate convex and increasing functions applied to vectors of functions. As specific applications we derive a wide range of vectorial fractional inequalities of Hardy type. These involve the left and right: Erdélyi-Kober fractional integrals, mixed Riemann-Liouville fractional multiple integrals. Next we produce multivariate Poincaré type vectorial fractional inequalities involving left fractional radial derivatives of Canavati type, Riemann-Liouville and Caputo types. The exposed inequalities are of L p type, p ≥ 1, and exponential type.

scholarly journals Estimates of Mean-Type Fractional Inequalities For Differentiable Functions

2021 ◽  
Author(s):  
Muhammad Samraiz ◽  
Zahida Perveen ◽  
Sajid Iqbal ◽  
Saima Naheed ◽  
Thabet Abdeljawad
Keyword(s):  
Fractional Derivative ◽  
Convex Functions ◽  
Fractional Derivatives ◽  
Integral Inequalities ◽  
Hilfer Fractional Derivative ◽  
Caputo Fractional Derivatives ◽  
Differentiable Functions ◽  
Wide Range ◽  
Type Integral ◽  
Special Case

In this article, we established a wide range of fractional mean-type integral inequalities for notable Hilfer fractional derivative using twice differentiable convex and $s$-convex functions for $s\in(0,1]$ with related identities. Also the results for Caputo fractional derivatives are derived as a special case of our general results.

On a Caputo-type fractional derivative

2019 ◽  
Vol 10 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Daniela S. Oliveira ◽  
Edmundo Capelas de Oliveira
Keyword(s):  
Differential Operator ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fundamental Theorem ◽  
Arbitrary Order ◽  
Fractional Operator ◽  
Derivatives Of ◽  
Generalized Fractional Derivative

Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

scholarly journals Approximate construction of new conservative physical magnitudes through the fractional derivative of polynomialtype functions: a particular case in semiconductors of type AlxGa1-xAs

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 874
Author(s):  
J. C. Campos-García ◽  
M. E. Molinar-Tabares ◽  
C. Figueroa-Navarro ◽  
L. Castro-Arce
Keyword(s):  
Fractional Calculus ◽  
Potential Energy ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Natural Sciences ◽  
Semiconductor Structures ◽  
Approximate Construction ◽  
Self Similar ◽  
The Many ◽  
Derivatives Of

The fractional calculus has a very large diversification as it relates to applications from physical interpretations to experimental facts to modeling of new problems in the natural sciences. Within the framework of a recently published article, we obtained the fractional derivative of the variable concentration x (z), the effective mass of the electron dependent on the position m (z) and the potential energy V (z), produced by the confinement of the electron in a semiconductor of type AlxGa1-xAs, with which we can intuit a possible geometric and physical interpretation. As a consequence, it is proposed the existence of three physical and geometric conservative quantities approximate character, associated with each of these parameters of the semiconductor, which add to the many physical magnitudes that already exist in the literature within the context of fractional variation rates. Likewise, we find that the fractional derivatives of these magnitudes, apart from having a common critical point, manifest self-similar behavior, which could characterize them as a type of fractal associated with the type of semiconductor structures under study.

Lagrangians With Linear Velocities Within Hilfer Fractional Derivative

2011 ◽  
Author(s):  
Dumitru Baleanu ◽  
Om P. Agrawal ◽  
Sami I. Muslih
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Major Area ◽  
Lagrange Equations ◽  
Hilfer Fractional Derivative ◽  
Caputo Fractional Derivatives ◽  
Generalized Fractional Derivative

Fractional variational principles started to be one of the major area in the field of fractional calculus. During the last few years the fractional variational principles were developed within several fractional derivatives. One of them is the Hilfer’s generalized fractional derivative which interpolates between Riemann-Liouville and Caputo fractional derivatives. In this paper the fractional Euler-Lagrange equations of the Lagrangians with linear velocities are obtained within the Hilfer fractional derivative.

scholarly journals Some trapezoid and midpoint type inequalities via fractional $(p,q)$-calculus

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pheak Neang ◽  
Kamsing Nonlaopon ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas ◽  
Praveen Agarwal
Keyword(s):  
Fractional Calculus ◽  
Mathematical Analysis ◽  
Arbitrary Order ◽  
Continuous Functions ◽  
Integral Inequalities ◽  
Research Subjects ◽  
Derivatives Of

AbstractFractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional $(p,q)$ ( p , q ) -calculus on finite intervals, particularly the fractional $(p,q)$ ( p , q ) -integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional $(p,q)$ ( p , q ) -integral on finite intervals. Then, the obtained results are used to derive some fractional $(p,q)$ ( p , q ) -trapezoid and $(p,q)$ ( p , q ) -midpoint type inequalities.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

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 Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

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