scholarly journals ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

2017 ◽  
Vol 23 (2) ◽  
pp. 32-43
Author(s):  
M. V. Kukushkin
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Fractional Differentiation ◽  
Qualitative Properties ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Integral Equality ◽  
Definition Of

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

scholarly journals Formal Chaos Existing Conditions on a Transmission Line Circuit with a Piecewise Linear Resistor

2021 ◽  
Vol 11 (20) ◽  
pp. 9672
Author(s):  
Kazuya Ozawa ◽  
Kaito Isogai ◽  
Hideo Nakano ◽  
Hideaki Okazaki
Keyword(s):  
Transmission Line ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Voltage Source ◽  
One Dimensional ◽  
In Series ◽  
Load Resistor ◽  
Dc Bias ◽  
Definition Of ◽  
Bifurcation Behavior

By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal have been actively studied. Bifurcation results or chaotic states in the circuits have been reported. On the other hand, many weak or strong definitions such that a 1-D map is mathematically chaotic are still being studied. In such definitions, the definition of formal chaos is well known as being the most traditional and most definite. However, formal chaos existences have not been rigorously proven in such circuits. In this paper, a general lossless transmission circuit is considered first with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal. Secondly, the method for deriving a 1-D map describing the behavior of the circuit is summarized. Thirdly, to provide a basis of chaotic application for the 1-D map, the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos are discussed. Furthermore, by using Maple, formal chaos existences and bifurcation behavior of 1-D maps are presented. By using the Lyapunov exponent, the observability of formal chaos in such bifurcation processes is outlined. Finally, the principal results and the future works are summarized.

scholarly journals The boundary value problemfor one-dimensional fractional differential equationof advection-diffusion

Vestnik MGSU ◽  
2015 ◽  
pp. 16-22
Author(s):  
Leyla Magametovna Isaeva
Keyword(s):  
Fluid Flow ◽  
Fractional Calculus ◽  
Boundary Value Problems ◽  
Fractional Integral ◽  
Boundary Value ◽  
Long Term Memory ◽  
Term Memory ◽  
One Dimensional ◽  
Fractional Differential

The use of fractional derivatives for describing and studying the physical processes of stochastic transport has become one of the most popular fields of physics in the recent years, many of the problems of fluid flow in highly-porous (fractal) environments also lead to the need to study boundary value problems for the equations of fractional order.The paper considers one of the boundary value problems for one-dimensional differential equation of fractional order. Using the Fourier method, the solution to this problem was explicitly written. The author also studied the qualitative properties of the solutions of the boundary value problem. It was proved that, in the case of going to infinity, the limit of the decisions recorded in the form of the function and the limit of the derivative of this solution tend to zero.The results can find application in the theory of fluid flow in a fractal environment and in order to simulate changes in temperature.Fractional integrals and derivatives of fractional integral-differential equations find wide application in contemporary studies of theoretical physics, mechanics and applied mathematics. Fractional calculus is a very powerful tool for describing the physical systems, which have memory and are non-local. Many processes in complex systems are non-locality and have long-term memory. The fractional integral operators and the fractional differential operators allow describing some of these properties. The use of fractional calculus will be useful for obtaining the dynamic models, in which integraldifferential operators describe the power of long-term memory and time coordinate and three-dimensional nonlocality for medium and complex processes.

Hadamard’s inequality and its extensions for conformable fractional integrals of any order α>0

2018 ◽  
Vol 27 (2) ◽  
pp. 197-206
Author(s):  
ERHAN SET ◽  
◽  
AHMET OCAK AKDEMIR ◽  
I. MUMCU ◽  
◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Hadamard’S Inequality ◽  
Conformable Fractional Integral ◽  
Definition Of ◽  
Appl Math

Recently the authors Abdeljawad [Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66] and Khalil et al. [Khalil, R., Horani, M. Al., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70] introduced a new and simple well-behaved concept of fractional integral called conformable fractional integral. In this article, we establish Hermite-Hadamard’s inequalities for conformable fractional integral. We also give extensions of Hermite-Hadamard type inequalities for conformable fractional integrals.

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.

Design and Application of Digital Filter Based on Calculus Computing Concept

2014 ◽  
Vol 513-517 ◽  
pp. 3151-3155
Author(s):  
Yan Chun Zhao
Keyword(s):  
Fractional Calculus ◽  
Digital Filter ◽  
Digital Filters ◽  
Engineering Analysis ◽  
Analysis Software ◽  
One Dimensional ◽  
The One ◽  
Definition Of ◽  
Object Of Study ◽  
The Mathematical Model

Calculus has been widely applied in engineering fields. The development of Integer order calculus theory is more mature in the project which can obtain fractional calculus theory through the promotion of integration order. It extends the flexibility of calculation and achieves the engineering analysis of multi-degree of freedom. According to fractional calculus features and the characteristics of fractional calculus, this paper treats the frequency domain as the object of study and gives the fractional calculus definition of the frequency characteristics. It also designs the mathematical model of fractional calculus digital filters using Fourier transform and Laplace transform. At last, this paper stimulates and analyzes numerical filtering of fractional calculus digital filter circuit using matlab general numerical analysis software and FDATool filter toolbox provided by matlab. It obtains the one-dimensional and two-dimensional filter curves of fractional calculus method which achieves the fractional Calculus filter of complex digital filter.

PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950084 ◽  
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
One Dimensional ◽  
Zero Measure ◽  
Liouville Fractional Derivative ◽  
Fractal Functions ◽  
Dimension One

In this paper, fractal dimensions of fractional calculus of continuous functions defined on [Formula: see text] have been explored. Continuous functions with Box dimension one have been divided into five categories. They are continuous functions with bounded variation, continuous functions with at most finite unbounded variation points, one-dimensional continuous functions with infinite but countable unbounded variation points, one-dimensional continuous functions with uncountable but zero measure unbounded variation points and one-dimensional continuous functions with uncountable and non-zero measure unbounded variation points. Box dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions has been proved to be with Box dimension one. Continuous functions on [Formula: see text] are divided as local fractal functions and fractal functions. According to local structure and fractal dimensions, fractal functions are composed of regular fractal functions, irregular fractal functions and singular fractal functions. Based on previous work, upper Box dimension of any continuous functions has been proved to be no less than upper Box dimension of their Riemann–Liouville fractional integral. Fractal dimensions of Riemann–Liouville fractional derivative of certain continuous functions have been investigated elementary.

A Philosophical Analysis To Uncover The Meaning And Terminology Of Person In Indonesian Criminal Law Context

2017 ◽  
Vol 1 (1) ◽  
pp. 56
Author(s):  
Nani Mulyati ◽  
Topo Santoso ◽  
Elwi Danil
Keyword(s):  
Criminal Law ◽  
Sufficient Conditions ◽  
Criminal Code ◽  
Legal Person ◽  
Philosophical Approach ◽  
Legal Personality ◽  
Legal Subject ◽  
Long Time ◽  
Definition Of ◽  
Broad Meaning

The definition of person and non-person always change through legal history. Long time ago, law did not recognize the personality of slaves. Recently, it accepted non-human legal subject as legitimate person before the law. This article examines sufficient conditions for being person in the eye of law according to its particular purposes, and then, analyses the meaning of legal person in criminal law. In order to do that, scientific methodology that is adopted in this research is doctrinal legal research combined with philosophical approach. Some theories regarding person and legal person were analysed, and then the concept of person was associated with the accepted definition of legal person that is adopted in the latest Indonesian drafted criminal code. From the study that has been done, can be construed that person in criminal law concerned with norm adressat of the rule, as the author of the acts or omissions, and not merely the holder of rights. It has to be someone or something with the ability to think rationally and the ability to be responsible for the choices he/she made. Drafted penal code embraces human and corporation as its norm adressat. Corporation defined with broad meaning of collectives. Consequently, it will include not only entities with legal personality, but also associations without legal personality. Furthermore, it may also hold all kind of collective namely states, states bodies, political parties, state’s corporation, be criminally liable.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

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 Quantitative Experimental and Theoretical Investigations on the Interaction of Shock Waves with Magnetic Fields

1969 ◽  
Vol 24 (10) ◽  
pp. 1449-1457
Author(s):  
H. Klingenberg ◽  
F. Sardei ◽  
W. Zimmermann
Keyword(s):  
Magnetic Fields ◽  
Shock Waves ◽  
Physical Model ◽  
Spectroscopic Method ◽  
Experimental Results ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Theoretical Investigations ◽  
Theoretical Predictions ◽  
Interaction Of Shock Waves

Abstract In continuation of the work on interaction between shock waves and magnetic fields 1,2 the experiments reported here measured the atomic and electron densities in the interaction region by means of an interferometric and a spectroscopic method. The transient atomic density was also calculated using a one-dimensional theory based on the work of Johnson3 , but modified to give an improved physical model. The experimental results were compared with the theoretical predictions.

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