scholarly journals Numerical Approaches to Fractional Integrals and Derivatives: A Review

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 43 ◽  
Author(s):  
Min Cai ◽  
Changpin Li
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Mathematical Concept ◽  
Fractional Integrals ◽  
Numerical Approximations ◽  
The Real ◽  
Potential Applications ◽  
Fractional Integrals And Derivatives ◽  
Almost All ◽  
Numerical Approaches

Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two main characteristics—singularity and nonlocality—has attracted increasing interest due to its potential applications in the real world. This mathematical concept reveals underlying principles that govern the behavior of nature. The present paper focuses on numerical approximations to fractional integrals and derivatives. Almost all the results in this respect are included. Existing results, along with some remarks are summarized for the applied scientists and engineering community of fractional calculus.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

scholarly journals The effects of repetition frequency on the illusory truth effect

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Aumyo Hassan ◽  
Sarah J. Barber
Keyword(s):  
Real World ◽  
Repetition Frequency ◽  
Processing Fluency ◽  
Fake News ◽  
Theoretical Understanding ◽  
The Real ◽  
Truth Effect ◽  
New Information ◽  
Almost All ◽  
Illusory Truth Effect

AbstractRepeated information is often perceived as more truthful than new information. This finding is known as the illusory truth effect, and it is typically thought to occur because repetition increases processing fluency. Because fluency and truth are frequently correlated in the real world, people learn to use processing fluency as a marker for truthfulness. Although the illusory truth effect is a robust phenomenon, almost all studies examining it have used three or fewer repetitions. To address this limitation, we conducted two experiments using a larger number of repetitions. In Experiment 1, we showed participants trivia statements up to 9 times and in Experiment 2 statements were shown up to 27 times. Later, participants rated the truthfulness of the previously seen statements and of new statements. In both experiments, we found that perceived truthfulness increased as the number of repetitions increased. However, these truth rating increases were logarithmic in shape. The largest increase in perceived truth came from encountering a statement for the second time, and beyond this were incrementally smaller increases in perceived truth for each additional repetition. These findings add to our theoretical understanding of the illusory truth effect and have applications for advertising, politics, and the propagation of “fake news.”

A distributional theory of fractional calculus

1985 ◽  
Vol 99 (3-4) ◽  
pp. 347-357 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Fractional Calculus ◽  
Spectral Properties ◽  
Fractional Integration ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Fractional Integrals And Derivatives

SynopsisIn this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

Some generalized fractional calculus operators and their applications in integral equations

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Om Agrawal
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Systematic Approach ◽  
Fractional Integrals ◽  
Generalized Fractional Calculus ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Generalized Fractional Calculus Operators

AbstractIn this paper, we survey some generalizations of fractional integrals and derivatives and present some of their properties. Using these properties, we show that many integral equations can be solved in a much elegant way. We believe that this will blur the distinction between the integral and differential equations, and provide a systematic approach for the two of these classes.

scholarly journals Katugampola Fractional Calculus With Generalized k−Wright Function

2021 ◽  
Vol 1 (1) ◽  
pp. 34-44
Author(s):  
Ahmad Y. A. Salamooni ◽  
D. D. Pawar
Keyword(s):  
Fractional Calculus ◽  
Fractional Integrals ◽  
Wright Function ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

In this article, we present some properties of the Katugampola fractional integrals and derivatives. Also, we study the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized k−Wright function nΦkm(z).

NaviCam:A Magnifying Glass Approach to Augmented Reality

1997 ◽  
Vol 6 (4) ◽  
pp. 399-412 ◽  
Author(s):  
Jun Rekimoto
Keyword(s):  
Augmented Reality ◽  
Real World ◽  
Video Camera ◽  
Display Device ◽  
Daily Lives ◽  
The Real ◽  
Video Images ◽  
Potential Applications ◽  
Subjective Score ◽  
Tracking Devices

Current augmented reality (AR) systems are not designed to be used in our daily lives. Head-mounted see-through displays are too cumbersome and look too unusual for everyday life. The limited scalability of position-tracking devices limits the use of AR to very restricted environments. This paper proposes a different way to realize AR that can be used in an open environment by introducing the concept of ID awareness and a hand-held video see-through display. Unlike other AR systems that use head-mounted or head-up displays, our approach employs the combination of a palmtop-sized display and a small video camera. A user sees the real world through the display device, with added computer-augmented information. We call this configuration the magnifying glass approach. It has several advantages over traditional head-up or head-mounted configurations. The main advantage is that the user is not required to wear any cumbersome headgear. The user can easily move the display device around like a magnifying glass and compare real and augmented images. The video camera also obtains information related to real-world situations. The system recognizes real-world objects using the video images by reading identification (ID) tags. Based on the recognized ID tag, the system retrieves and displays information about the real-world object to the user. The prototype hand-held device based on our proposed concept is called NaviCam. We describe several potential applications. Our experiments with NaviCam show the great potential of our video see-through palmtop display. It was significantly faster than a head-up configuration, and its subjective score from testers was also higher.

scholarly journals A systematic evaluation of assumptions in centrality measures by empirical flow data

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mareike Bockholt ◽  
Katharina A. Zweig
Keyword(s):  
Real World ◽  
Process Model ◽  
Systematic Evaluation ◽  
Centrality Measures ◽  
The Real ◽  
Link Type ◽  
Network Process ◽  
The One ◽  
Almost All ◽  
The Impact

AbstractWhen considering complex systems, identifying themost importantactors is often of relevance. When the system is modeled as a network, centrality measures are used which assign each node a value due to its position in the network. It is often disregarded that they implicitly assume a network process flowing through a network, and also make assumptions ofhowthe network process flows through the network. A node is then central with respect to this network process (Borgatti in Soc Netw 27(1):55–71, 2005,10.1016/j.socnet.2004.11.008). It has been shown that real-world processes often do not fulfill these assumptions (Bockholt and Zweig, in Complex networks and their applications VIII, Springer, Cham, 2019,10.1007/978-3-030-36683-4_7). In this work, we systematically investigate the impact of the measures’ assumptions by using four datasets of real-world processes. In order to do so, we introduce several variants of the betweenness and closeness centrality which, for each assumption, use either the assumed process model or the behavior of the real-world process. The results are twofold: on the one hand, for all measure variants and almost all datasets, we find that, in general, the standard centrality measures are quite robust against deviations in their process model. On the other hand, we observe a large variation of ranking positions of single nodes, even among the nodes ranked high by the standard measures. This has implications for the interpretability of results of those centrality measures. Since a mismatch of the behaviour of the real network process and the assumed process model does even affect the highly-ranked nodes, resulting rankings need to be interpreted with care.

scholarly journals Fractional derivatives and the fundamental theorem of fractional calculus

2020 ◽  
Vol 23 (4) ◽  
pp. 939-966 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
Finite Interval ◽  
Fractional Integrals ◽  
Important Class ◽  
Left Inverse ◽  
Caputo Fractional Derivatives ◽  
Fractional Integrals And Derivatives ◽  
The One

AbstractIn this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

scholarly journals Fractional Sums and Differences with Binomial Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Fahd Jarad ◽  
Ravi P. Agarwal
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Binomial Coefficients ◽  
Discrete Version ◽  
Binomial Theorem ◽  
Left And Right ◽  
Fractional Integrals And Derivatives ◽  
Dual Identities

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

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