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.

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 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.

scholarly journals Hydrated metal(II) complexes of N-(6-amino-3,4-dihydro-3-methyl-5-nitroso-4-oxopyrimidin-2-yl) derivatives of glycine, glycylglycine, threonine, serine, valine and methionine: a monomeric complex and coordination polymers in one, two and three dimensions linked by hydrogen bonding. Corrigendum

2006 ◽  
Vol 62 (1) ◽  
pp. 165-165
Author(s):  
M. Luz Godino Salido ◽  
Paloma Arranz Mascarós ◽  
Rafaél López Garzón ◽  
M. Dolores Gutiérrez Valero ◽  
John N. Low ◽  
...  
Keyword(s):  
Hydrogen Bonding ◽  
Synchrotron Radiation ◽  
Data Collection ◽  
Coordination Polymers ◽  
Data Reduction ◽  
Three Dimensions ◽  
Compound Viii ◽  
Monomeric Complex ◽  
Cell Refinement ◽  
Derivatives Of

Some of the data collection details for compound (VIII) were incorrectly given in Table 1 of Godino Salido et al. (2004). The data for compound VIII in this paper were collected using synchrotron radiation at the Daresbury SRS station 9.8, λ = 0.6935 Å (Cernik et al., 1997; Clegg, 2000). The data were collected using a Bruker SMART 1K CCD diffractometer using ω rotation with narrow frames. The computer program used in the data collection was SMART (Bruker, 2001) and for cell refinement and data reduction SAINT (Bruker, 2001).

scholarly journals Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zaid Laadjal ◽  
Qasem M. Al-Mdallal ◽  
Fahd Jarad
Keyword(s):  
Boundary Conditions ◽  
Coupled System ◽  
Fractional Integrals ◽  
Langevin Equations ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Nonseparated Boundary Conditions ◽  
New Type ◽  
Predictor Corrector ◽  
Derivatives Of

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

scholarly journals Integral transforms of the κ-Hilfer fractional derivative

2021 ◽  
Author(s):  
Felix Costa ◽  
Junior Cesar Alves Soares ◽  
Stefânia Jarosz
Keyword(s):  
Fractional Derivative ◽  
Gamma Function ◽  
Fundamental Theorem ◽  
Beta Function ◽  
Integral Transforms ◽  
Fractional Operators ◽  
Riesz Fractional Derivative ◽  
Hilfer Fractional Derivative ◽  
Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

Navier Stokes Derivative Estimates in Three Dimensions with Boundary Values and Body Forces

1991 ◽  
Vol 43 (6) ◽  
pp. 1161-1212 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Stokes Equations ◽  
Finite Energy ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Boundary Values ◽  
Vector Solution ◽  
Navier Stokes Equations ◽  
Body Forces ◽  
Derivatives Of

AbstractFor a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

Riesz fractional derivative Elite-guided sine cosine algorithm

2019 ◽  
Vol 81 ◽  
pp. 105481 ◽  
Author(s):  
Wenyan Guo ◽  
Yuan Wang ◽  
Fengqun Zhao ◽  
Fang Dai
Keyword(s):  
Fractional Derivative ◽  
Riesz Fractional Derivative ◽  
Sine Cosine Algorithm
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