scholarly journals The Four-Parameters Wright Function of the Second kind and its Applications in FC

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 970 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
Wide Spectrum ◽  
Operational Calculus ◽  
Fractional Diffusion ◽  
Wright Function ◽  
Scale Invariant ◽  
First Case

In this survey paper, we present both some basic properties of the four-parameters Wright function and its applications in Fractional Calculus. For applications in Fractional Calculus, the four-parameters Wright function of the second kind is especially important. In the paper, three case studies illustrating a wide spectrum of its applications are presented. The first case study deals with the scale-invariant solutions to a one-dimensional time-fractional diffusion-wave equation that can be represented in terms of the Wright function of the second kind and the four-parameters Wright function of the second kind. In the second case study, we consider a subordination formula for the solutions to a multi-dimensional space-time-fractional diffusion equation with different orders of the fractional derivatives. The kernel of the subordination integral is a special case of the four-parameters Wright function of the second kind. Finally, in the third case study, we shortly present an application of an operational calculus for a composed Erdélyi-Kober fractional operator for solving some initial-value problems for the fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. In particular, we present an example with an explicit solution in terms of the four-parameters Wright function of the second kind.

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.

scholarly journals Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications

2021 ◽  
Vol 24 (2) ◽  
pp. 518-540
Author(s):  
Hafiz Muhammad Fahad ◽  
Arran Fernandez
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
General Setting ◽  
Fractional Differentiation ◽  
Fractional Calculus Operators ◽  
Liouville Fractional Derivative ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

Some Pioneers of the Application of Fractional Calculus

2013 ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
Alexandra M. S. F. Galhano
Keyword(s):  
Twentieth Century ◽  
Fractional Calculus ◽  
Research And Development ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Time Line ◽  
Seminal Work ◽  
Electrical Relaxation ◽  
Intense Research

In the last decades fractional calculus (FC) became an area of intense research and development. This paper goes back and draws the time line of FC. We recall also important pioneers that started to apply FC during the beginning of the twentieth century. First we remember Oliver Heaviside, a pioneer in several areas of mathematics, physics and engineering. His operational calculus revealed expressions involving fractional derivatives, a premonition of the progresses that emerged later in the last decades of the twentieth century. Second it is addressed the Cole-Cole electrical relaxation. This model was a seminal work in the application of FC and plays nowadays an important role in many areas of physics and engineering.

Scales and Units

2018 ◽  
Author(s):  
Kathryn M. de Luna
Keyword(s):  
Language Change ◽  
Historical Linguistics ◽  
Bantu Languages ◽  
The North ◽  
First Case ◽  
Linguistic Evidence ◽  
River Region ◽  
History Of ◽  
Alternative Approaches

This chapter uses two case studies to explore how historians study language movement and change through comparative historical linguistics. The first case study stands as a short chapter in the larger history of the expansion of Bantu languages across eastern, central, and southern Africa. It focuses on the expansion of proto-Kafue, ca. 950–1250, from a linguistic homeland in the middle Kafue River region to lands beyond the Lukanga swamps to the north and the Zambezi River to the south. This expansion was made possible by a dramatic reconfiguration of ties of kinship. The second case study explores linguistic evidence for ridicule along the Lozi-Botatwe frontier in the mid- to late 19th century. Significantly, the units and scales of language movement and change in precolonial periods rendered visible through comparative historical linguistics bring to our attention alternative approaches to language change and movement in contemporary Africa.

Applications of random graphs

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Social Networks ◽  
Random Graphs ◽  
Interaction Network ◽  
Power Grids ◽  
Protein Protein Interaction ◽  
Topological Features ◽  
First Case ◽  
The Stability ◽  
Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

Investigating the potential of redundant manipulators in narrow channels

2020 ◽  
pp. 095440622096451
Author(s):  
Ashish Singla ◽  
Jyotindra Narayan ◽  
Himanshu Arora
Keyword(s):  
Case Studies ◽  
Fuzzy Inference ◽  
Redundant Manipulators ◽  
Redundancy Resolution ◽  
Inference System ◽  
Narrow Channels ◽  
First Case ◽  
Task Priority ◽  
Patient Side

In this paper, an attempt has been made to investigate the potential of redundant manipulators, while tracking trajectories in narrow channels. The behavior of redundant manipulators is important in many challenging applications like under-water welding in narrow tanks, checking the blockage in sewerage pipes, performing a laparoscopy operation etc. To demonstrate this snake-like behavior, redundancy resolution scheme is utilized using two different approaches. The first approach is based on the concept of task priority, where a given task is split and prioritize into several subtasks like singularity avoidance, obstacle avoidance, torque minimization, and position preference over orientation etc. The second approach is based on Adaptive Neuro Fuzzy Inference System (ANFIS), where the training is provided through given datasets and the results are back-propagated using augmentation of neural networks with fuzzy logics. Three case studies are considered in this work to demonstrate the redundancy resolution of serial manipulators. The first case study of 3-link manipulator is attempted with both the approaches, where the objective is to track the desired trajectory while avoiding multiple obstacles. The second case study of 7-link manipulator, tracking trajectory in a narrow channel, is investigated using the concept of task priority. The realistic application of minimum-invasive surgery (MIS) based trajectory tracking is considered as the third case study, which is attempted using ANFIS approach. The 5-link spatial redundant manipulator, also known as a patient-side manipulator being developed at CSIR-CSIO, Chandigarh is used to track the desired surgical cuts. Through the three case studies, it is well demonstrated that both the approaches are giving satisfactory results.

scholarly journals Sensitivity analysis for interpretation of machine learning based segmentation models in cardiac MRI

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Markus J. Ankenbrand ◽  
Liliia Shainberg ◽  
Michael Hock ◽  
David Lohr ◽  
Laura M. Schreiber
Keyword(s):  
Neural Networks ◽  
Sensitivity Analysis ◽  
Cardiac Magnetic Resonance ◽  
Magnetic Resonance ◽  
Cardiac Mri ◽  
Magnetic Resonance Images ◽  
Model Interpretation ◽  
First Case ◽  
Segmentation Models

Abstract Background Image segmentation is a common task in medical imaging e.g., for volumetry analysis in cardiac MRI. Artificial neural networks are used to automate this task with performance similar to manual operators. However, this performance is only achieved in the narrow tasks networks are trained on. Performance drops dramatically when data characteristics differ from the training set properties. Moreover, neural networks are commonly considered black boxes, because it is hard to understand how they make decisions and why they fail. Therefore, it is also hard to predict whether they will generalize and work well with new data. Here we present a generic method for segmentation model interpretation. Sensitivity analysis is an approach where model input is modified in a controlled manner and the effect of these modifications on the model output is evaluated. This method yields insights into the sensitivity of the model to these alterations and therefore to the importance of certain features on segmentation performance. Results We present an open-source Python library (misas), that facilitates the use of sensitivity analysis with arbitrary data and models. We show that this method is a suitable approach to answer practical questions regarding use and functionality of segmentation models. We demonstrate this in two case studies on cardiac magnetic resonance imaging. The first case study explores the suitability of a published network for use on a public dataset the network has not been trained on. The second case study demonstrates how sensitivity analysis can be used to evaluate the robustness of a newly trained model. Conclusions Sensitivity analysis is a useful tool for deep learning developers as well as users such as clinicians. It extends their toolbox, enabling and improving interpretability of segmentation models. Enhancing our understanding of neural networks through sensitivity analysis also assists in decision making. Although demonstrated only on cardiac magnetic resonance images this approach and software are much more broadly applicable.

scholarly journals Some Applications of the Wright Function in Continuum Physics: A Survey

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 198
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Fractional Calculus ◽  
Exponential Function ◽  
Nonlocal Elasticity ◽  
Bessel Functions ◽  
Wright Function ◽  
Mainardi Function ◽  
Integral Relations

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

scholarly journals Novel filter bank-based cooperative spectrum sensing under practical challenges for beyond 5G cognitive radios

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sener Dikmese ◽  
Kishor Lamichhane ◽  
Markku Renfors
Keyword(s):  
Spectrum Sensing ◽  
Filter Bank ◽  
Cooperative Spectrum Sensing ◽  
Minimum Energy ◽  
Cognitive Radios ◽  
Spectrum Management ◽  
Wireless Channel ◽  
First Case ◽  
Channel Effects

AbstractCognitive radio (CR) technology with dynamic spectrum management capabilities is widely advocated for utilizing effectively the unused spectrum resources. The main idea behind CR technology is to trigger secondary communications to utilize the unused spectral resources. However, CR technology heavily relies on spectrum sensing techniques which are applied to estimate the presence of primary user (PU) signals. This paper firstly focuses on novel analysis filter bank (AFB) and FFT-based cooperative spectrum sensing (CSS) techniques as conceptually and computationally simplified CSS methods based on subband energies to detect the spectral holes in the interesting part of the radio spectrum. To counteract the practical wireless channel effects, collaborative subband-based approaches of PU signal sensing are studied. CSS has the capability to relax the problems of both hidden nodes and fading multipath channels. FFT- and AFB-based receiver side sensing methods are applied for OFDM waveform and filter bank-based multicarrier (FBMC) waveform, respectively, the latter one as a candidate beyond-OFDM/beyond-5G scheme. Subband energies are then applied for enhanced energy detection (ED)-based CSS methods that are proposed in the context of wideband, multimode sensing. Our first case study focuses on sensing potential spectral gaps close to relatively strong primary users, considering also the effects of spectral regrowth due to power amplifier nonlinearities. The study shows that AFB-based CSS with FBMC waveform is able to improve the performance significantly. Our second case study considers a novel maximum–minimum energy detector (Max–Min ED)-based CSS. The proposed method is expected to effectively overcome the issue of noise uncertainty (NU) with remarkably lower implementation complexity compared to the existing methods. The developed algorithm with reduced complexity, enhanced detection performance, and improved reliability is presented as an attractive solution to counteract the practical wireless channel effects under low SNR. Closed-form analytic expressions are derived for the threshold and false alarm and detection probabilities considering frequency selective scenarios under NU. The validity of the novel expressions is justified through comparisons with respective results from computer simulations.

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