Complex systems have characteristics that give rise to the emergence of rare and extreme events. This paper addresses an example of such type of crisis, namely the spread of the new Coronavirus ...disease 2019 (COVID-19). The study deals with the statistical comparison and visualization of country-based real-data for the period December 31, 2019, up to April 12, 2020, and does not intend to address the medical treatment of the disease. Two distinct approaches are considered, the description of the number of infected people across time by means of heuristic models fitting the real-world data, and the comparison of countries based on hierarchical clustering and multidimensional scaling. The computational and mathematical modeling lead to the emergence of patterns, highlighting similarities and differences between the countries, pointing toward the main characteristics of the complex dynamics.
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory ...and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation.As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.
•The fractional Cattaneo model is formulated to describe the heat flow in a porous medium.•The hybrid scheme based on the RBF-PU method is proposed to approximate the model.•Unconditional stability ...and convergence of the time discretization formulation are proved using energy method.•Numerical results are conducted to validate the theoretical findings.
The generalized Cattaneo model describes the heat conduction system in the perspective of time-nonlocality. This paper proposes an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model applied to the heat flow in a porous medium. The fractional derivative is formulated in the Caputo sense with order 1<α<2. First, a finite difference technique of convergence order O(δt3−α) is adopted to achieve the temporal discretization. The unconditional stability of the method and its convergence are analysed using the discrete energy technique. Then, a local meshless method based on the radial basis function partition of unity collocation is employed to obtain a full discrete algorithm. The matrices produced using this localized scheme are sparse and, therefore, they are not subject to ill-conditioning and do not pose a large computational burden. Two examples illustrate in computational terms of the accuracy and effectiveness of the proposed method.
Reactive power dispatch is a vital problem in the operation, planning and control of power system for obtaining a fixed economic load expedition. An optimal dispatch reduces the grid congestion ...through the minimization of the active power loss. This strategy involves adjusting the transformer tap settings, generator voltages and reactive power sources, such as flexible alternating current transmission systems (FACTS). The optimal dispatch improves the system security, voltage profile, power transfer capability and overall network efficiency. In the present work, a fractional evolutionary approach achieves the desired objectives of reactive power planning by incorporating FACTS devices. Two compensation arrangements are possible: the shunt type compensation, through Static Var compensator (SVC) and the series compensation through the Thyristor controlled series compensator (TCSC). The fractional order Darwinian Particle Swarm Optimization (FO-DPSO) is implemented on the standard IEEE 30, IEEE 57 and IEEE 118 bus test systems. The power flow analysis is used for determining the location of TCSC, while the voltage collapse proximity indication (VCPI) method identifies the location of the SVC. The superiority of the FO-DPSO is demonstrated by comparing the results with those obtained by other techniques in terms of measure of central tendency, variation indices and time complexity.
•A fractal model of the LC-electric circuit is derived from local fractional calculus.•The relaxation oscillator in the fractal LC-electric circuit is considered.•The exact solution for the model is ...analyzed by the local fractional Laplace transform.•Comparative results among the different operators are presented.
A non-differentiable model of the LC-electric circuit described by a local fractional differential equation of fractal dimensional order is addressed in this article. From the fractal electrodynamics point of view, the relaxation oscillator, defined on Cantor sets in LC-electric circuit, and its exact solution using the local fractional Laplace transform are obtained. Comparative results among local fractional derivative, Riemann–Liouville fractional derivative and conventional derivative are discussed. Local fractional calculus is proposed as a new tool suitable for the study of a large class of electric circuits.
•This study presents a novel fractional order adaptive algorithm, called MIFLMS.•The MIFLMS extends the scalar innovation into a vector innovation.•It reveals a faster convergence speed than the FLMS ...with no noticeable increase in the computational burden.•The proposed algorithm effectively estimates the parameters of nonlinear systems.•The MIFLMS is more robust than the FLMS and provide consistent accurate and convergent performance.
The development of procedures based on fractional calculus is an emerging research area. This paper presents a new perspective regarding the fractional least mean square (FLMS) adaptive algorithm, called multi innovation FLMS (MIFLMS). We verify that the iterative parameter adaptation mechanism of the FLMS uses merely the current error value (scalar innovation). The MIFLMS expands the scalar innovation into a vector innovation (error vector) by considering data over a fixed window at each iteration. Therefore, the MIFLMS yields better convergence speed than the standard FLMS by increasing the length of innovation vector. The superior performance of the MIFLMS is verified through parameter identification problem of input nonlinear systems. The statistical performance indices based on multiple independent trials confirm the consistent accuracy and reliability of the proposed scheme.
In this article we propose a new fractional derivative without singular kernel. We consider the potential application for modeling the steady heat-conduction problem. The analytical solution of the ...fractional-order heat flow is also obtained by means of the Laplace transform.
In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre ...polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.
•The time-fractional telegraph model as a useful description of the neutron transport process is generated.•A new hybrid scheme based LRBF-FD method is formulated to approximate the time-fractional ...telegraph model.•The LRBF-FD method useful for complex domains with acceptable accuracy was proposed.•The stability and convergence are examined using the energy method.
This paper focusses on the numerical solution of the nonlinear time-fractional telegraph equation formulated in the Caputo sense. This model is a useful description of the neutron transport process inside the core of a nuclear reactor. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, a semi-discrete algorithm is obtained by means of a difference approach with the accuracy O(τ3−β), where 1<β<2 is the fractional-order derivative. At a second stage, a full-discretization is obtained by an efficient augmented local radial basis function finite difference (LRBF-FD). This method approximates the derivatives of an unknown function at a given point named as center, based on the finite difference at each local-support domain, instead of applying the entire set of points. The technique produces a sparse matrix system, reduces the computational effort and avoids the ill-conditioning derived from the global collocation. The unconditional stability and convergence of the time-discretized formulation are demonstrated and confirmed numerically. The numerical results highlight the accuracy and the validity of the method.
This paper studies the robust stability analysis for a class of memristive-based neural networks (NN). The NN consists of a fractional order neutral type quaternion-valued leaky integrator echo state ...with parameter uncertainties and time-varying delays. First, the quaternion-valued leaky integrator echo state NN with QUAD vector field activation function is transformed into a real-valued system using a linear mapping function. Then, the Lyapunov–Krasovskii functional is adopted to derive the sufficient conditions on the existence and uniqueness of Filippov solution of the NN equilibrium point. The delay-dependent robust stability analysis of such NN is provided with the help of linear matrix inequality technique. Finally, the theoretical results are validated by means of a numerical example.
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