scholarly journals On Chebyshev and Riemann-Liouville Fractional Inequalities in q-Calculus

2019 ◽  
pp. 1-10
Author(s):  
Stephen. N. Ajega-Akem ◽  
Mohammed M. Iddrisu ◽  
Kwara Nantomah
Keyword(s):  
Fractional Calculus ◽  
Arbitrary Order ◽  
High Demand ◽  
Quantum Calculus ◽  
Integer Order ◽  
Model Complex ◽  
Derivatives Of ◽  
Increasing Functions ◽  
Anomalous Dynamics ◽  
New Inequalities

This paper presents some new inequalities on Fractional calculus in the context of q-calculus. Fractional calculus generalizes the integer order differentiation and integration to derivatives and integrals of arbitrary order. In other words, Fractional calculus explores integrals and derivatives of functions that involve non-integer order(s). Quantum calculus (q-Calculus) on the other hand focuses on investigations related to calculus without limits and in recent times, it has attracted the interest of many researchers due to its high demand of mathematics to model complex systems in nature with anomalous dynamics. This paper thus establishes some new extensions of Chebyshev and Riemann-Liouville fractional integral inequalities for positive and increasing functions via q-Calculus.

scholarly journals General Fractional Integrals and Derivatives of Arbitrary Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 755
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Basic Properties ◽  
Suitable Generalization ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Derivatives Of

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

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 Simulation of Chaotic Oscillators of Fractional Order

2019 ◽  
pp. 11-17
Author(s):  
Alejandro Silva-Juárez ◽  
Miguel De Jesús Salazar-Pedraza ◽  
Juan Jorge Ponce-Mellado ◽  
Gustavo Herrera-Sánchez
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Arbitrary Order ◽  
Chaotic Behavior ◽  
Pid Controllers ◽  
Chaotic Oscillators ◽  
Non Linear ◽  
Derivatives Of ◽  
Corrective Method

In 1695 the theory of fractional calculus was introduced, but it only developed as a pure mathematical branch. Currently several research groups have focused on the control, the implementation of filters, PID controllers, synchronization, the implementation of circuits of chaotic systems of fractional order, etc. Currently, the number of applications of fractional calculus is increasing rapidly, these mathematical phenomena have allowed us to describe and model a real object more accurately than the classical "integer" methods. Along with the development of the fractional calculation, it was shown that many fractional-order nonlinear dynamic systems behave in a chaotic manner. This is the type of non-linear systems that are addressed in this research topic with the focus on derivatives of arbitrary order, where numerical simulations of chaotic behavior are presented in non-linear, fractional-order autonomous models. The case studies are six chaotic oscillators of fractional order; The systems of Lorenz, Rӧssler, Financiero, Lui, Chen and Lü, whose attractors are obtained by applying the definitions of the Grünwald-Letnikov definitions and the predictive corrective method of Adams-Bashforth-Moulton.

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.

scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals CONSTRAINT ALGEBRAS IN GAUGE-INVARIANT SYSTEMS

1995 ◽  
Vol 10 (28) ◽  
pp. 4087-4105 ◽  
Author(s):  
KH. S. NIROV
Keyword(s):  
Wide Class ◽  
Arbitrary Function ◽  
Arbitrary Order ◽  
Local Symmetry ◽  
Poisson Brackets ◽  
Hamiltonian Description ◽  
Gauge Invariant ◽  
Symmetry Transformations ◽  
Constraint Algebra ◽  
Derivatives Of

A Hamiltonian description is constructed for a wide class of mechanical systems having local symmetry transformations depending on time derivatives of the gauge parameters of arbitrary order. The Poisson brackets of the Hamiltonian and constraints with each other and with an arbitrary function are explicitly obtained. The constraint algebra is proved to be of the first class.

Properties of Spatial-Rotation Transformations of Fractional Derivatives

2020 ◽  
pp. 198-223
Author(s):  
Ehab Malkawi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Scalar Fields ◽  
Two Dimensional ◽  
Spatial Rotation ◽  
Essential Step ◽  
Derivatives Of

The transformation properties of the fractional derivatives under spatial rotation in two-dimensional space and for both the Riemann-Liouville and Caputo definitions are investigated and derived in their general form. In particular, the transformation properties of the fractional derivatives acting on scalar fields are studied and discussed. The study of the transformation properties of fractional derivatives is an essential step for the formulation of fractional calculus in multi-dimensional space. The inclusion of fractional calculus in the Lagrangian and Hamiltonian dynamical formulation relies on such transformation. Specific examples on the transformation of the fractional derivatives of scalar fields are discussed.

A New Insight Into the Grünwald–Letnikov Discrete Fractional Calculus

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Yiheng Wei ◽  
Weidi Yin ◽  
Yanting Zhao ◽  
Yong Wang
Keyword(s):  
Fractional Calculus ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Integer Order ◽  
Convolution Operation ◽  
Order Case ◽  
Insight Into

The primary work of this paper is to investigate some potential properties of Grünwald–Letnikov discrete fractional calculus. By employing a concise and convenient description, this paper not only establishes excellent relationships between fractional difference/sum and the integer order case but also generalizes the Z-transform and convolution operation.

scholarly journals Synthesis and Evaluation of Cyclic Acetals of Serine Hydroxylamine for Amide-Forming KAHA Ligations

Synthesis ◽  
2019 ◽  
Vol 51 (05) ◽  
pp. 1273-1283 ◽  
Author(s):  
Simon Baldauf ◽  
Jeffrey Bode
Keyword(s):  
Amino Acid ◽  
Amino Acid Residue ◽  
Amino Acid Residues ◽  
High Demand ◽  
Cyclic Acetals ◽  
Ligation Product ◽  
Canonical Amino Acid ◽  
The Stability ◽  
Peptide Segments ◽  
Derivatives Of

The α-ketoacid–hydroxylamine (KAHA) ligation allows the coupling of unprotected peptide segments. The most widely used variant employs a 5-membered cyclic hydroxylamine that forms a homoserine ester as the primary ligation product. While very effective, monomers that give canonical amino acid residues are in high demand. In order to preserve the stability and reactivity of cyclic hydroxylamines, but form a canonical amino acid residue upon ligation, we sought to prepare cyclic derivatives of serine hydroxylamine. An evaluation of several cyclization strategies led to cyclobutanone ketals as the leading structures. The preparation, stability, and amide-forming ligation of these serine-derived ketals are described.

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