Book Volume 3
Foreword
Page: i-ii (2)
Author: Dumitru Baleanu and Jordan Hristov
DOI: 10.2174/9789815051933122010001
Preface
Page: iii-iv (2)
Author: Mehmet Yavuz and Necati Özdemir
DOI: 10.2174/9789815051933122010002
Numerical Procedure and its Applications to the Fractional-Order Chaotic System Represented with the Caputo Derivative
Page: 1-28 (28)
Author: Ndolane Sene*
DOI: 10.2174/9789815051933122010004
PDF Price: $30
Abstract
This chapter focuses on a numerical procedure and its application to a
fractional-order chaotic system. The numerical scheme will discuss the Lyapunov
exponents for the considered model and characterize the chaos’s nature. We will also
use the numerical scheme to depict the phase portraits of the proposed fractional-order
chaotic system and the bifurcation maps. Note that the bifurcation maps are used to
characterize the influence of the different parameters of our considered fractional
model. The impact of the initial conditions and the coexisting attractors will also be
analyzed. With the coexistence, the new types of attractors will be discovered for our
considered model. To confirm the investigations in this chapter, the proposed model
will be applied to the electrical modeling. Therefore, the circuit schematic of the
considered fractional model will be implemented in real-world problems. And we
notice good agreement between the theoretical results and the results obtained after
Multisim simulations. The stability of the equilibrium points of the presented model
will also be focused on details and will permit us to delimit the chaotic region in
general.
A New Method of Multistage Optimal Homotopy Asymptotic Method for Solution of Fractional Optimal Control Problem
Page: 29-60 (32)
Author: Oluwaseun O. Okundalaye*, Necati Özdemir and Wan A. M. Othman
DOI: 10.2174/9789815051933122010005
PDF Price: $30
Abstract
This paper deals with a recent approximate analytical approach of the
multistage optimal homotopy asymptotic method (MOHAM) for fractional optimal
control problems (FOCPs). In this paper, FOCPs are developed in terms of a
conformable derivative operator (CDO) sense. It is validated that the right CDO
appears naturally in the formulation even when the system dynamics are described
with the left CDO only. The CDO is employed to enlarge the stability region of the
dynamical systems of the optimal control problems (OCPs). The necessary and
transversal conditions are achieved using a Hamiltonian technique. The results
demonstrated that as the fractional-order solution derivative tends to integer-order 1,
the formulations lead to integer-order system solutions. Numerical results and a
comparison with the exact solution and other approximate analytical solutions in
fractional order are given to validate the efficiency of the MOHAM. Some numerical
examples are included to demonstrate the effectiveness and applicability of the new
technique.
Complex Chaotic Fractional-order Finance System in Price Exponent with Control and Modeling
Page: 61-84 (24)
Author: Muhammad Farman, Parvaiz Ahmad Naik*, Aqeel Ahmad, Ali Akgul and Muhammad Umer Saleem
DOI: 10.2174/9789815051933122030006
PDF Price: $30
Abstract
The present chapter proposes modeling of complex fractional-order chaotic
ifnancial system with control. Here, we have added critical minimum interest rate ‘d’
as a new parameter to get a novel stable ifnancial model. The fractional derivatives.
are taken in Caputo and Caputo-Fabrizio sense for the proposed ifnance system. Dynamical models in ifnancial system with complicated behavior provide a new.
perspective as result of trends and actual behavior of internal structure of the ifnancial.
system. A theoretical stabilization of the equilibria, as well as the numerical
simulations, are obtained. Furthermore, with sensitivity analysis, a certain threshold
estimation of the basic reproductive number has been made. Also, the stability
analysis of the model, together with uniqueness of the special solutions is provided.
The concept of controllability and observability for the linearized control model is
used for feedback control. The solution of the proposed fractional-order model has
been procured by employing different numerical techniques with comparison among
the solutions. The convergence analysis is carried out for the accuracy of the applied
scheme. Finally, some numerical simulations are given for three fractional-order
chaotic systems to verify the efectiveness for the obtained results. The fractal,
stochastic processes and prediction are used in particular mechanism of its application
to the macro and micro processes.
The Duhamel Method in Transient Heat Conduction: A Rendezvous of Classics and Modern Fractional Calculus
Page: 85-107 (23)
Author: Jordan Hristov*
DOI: 10.2174/9789815051933122030007
PDF Price: $30
Abstract
This chapter presents an attempt to demonstrate that the Duhamel theorem applicable for time-dependent boundary conditions (or time-dependent source terms) of heat conduction in a finite domain and the use of the Fourier method of separation of variable (superposition version) naturally lead to appearance of the Caputo- Fabrizio operators in the solution. The fractional orders of the emerging series of Caputo-Fabrizio operators are directly related to the eigenvalues determined by the Fourier’s method. The general expression of the solution in terms of Caputo-Fabrizio operators has been developed followed by four examples.
Oscillatory Heat Transfer Due to the Cattaneo-Hristov Model on the Real Line
Page: 108-123 (16)
Author: Derya Avci* and Beyza Billur İskender Eroğlu
DOI: 10.2174/9789815051933122030008
PDF Price: $30
Abstract
This chapter aims to discuss the analytical solutions for heat waves
observed in Cattaneo-Hristov heat conduction modelled with Caputo-Fabrizio
fractional derivative. This operator includes a non-singular exponential kernel and
also requires physically interpretable initial conditions for its Laplace transform
property. These provide significant advantages to obtain analytical solutions. Two
different types of harmonic heat sources are assumed to elicit heat waves. The
analytical solutions are obtained by applying Laplace transform with respect to the
time variable and the exponential Fourier transform with respect to spatial coordinate.
The temperature curves for varying values of the fractional parameter, angular
frequency, and the velocity of the moving heat source are drawn using MATLAB.
Optimal Homotopy Analysis of a Nonlinear Fractional-order Model for HTLV-1 Infection of CD4+ T-Cells
Page: 124-161 (38)
Author: Mohammad Ghoreishi, Parvaiz Ahmad Naik* and Mehmet Yavuz
DOI: 10.2174/9789815051933122030009
PDF Price: $30
Abstract
In this chapter, a series solution of a nonlinear fractional-order mathematical model of human T-cells lymphotropic virus-1 (HTLV-1) infection of CD4+ T-cells is obtained by using a strong and capable technique so-called Homotopy Analysis Method (HAM). The proposed model is a system of nonlinear ordinary differential equations that divides CD4+ T-cells into four components: uninfected cells, latently infected cells, actively infected cells and leukemia cells. The fractional model is more general than the classical one, as in the fractional model, the next state depends not only upon its current state but also upon all of its historical states. The homotopy analysis method (HAM) is applied for a strongly nonlinear fractional-order system as it utilizes a simple method to adjust and control the convergence region of the infinite series solution by using an auxiliary parameter and allows to obtain a oneparametric family of explicit series solutions. By using the homotopy series solutions, firstly, several β-curves are plotted to demonstrate the regions of convergence, then the square residual errors are obtained for different values of these regions. Secondly, the numerical solutions are presented to show the accuracy of the applied homotopy analysis method. In this chapter, a detailed proof of the convergence of this method for nonlinear fractional-order model of HTLV-1 infection of CD4+ T-cells is also given. The results indicate that the HAM is accurate and capable to obtain an accurate approximate analytical solution for HTLV-1 infection of CD4+ T-cells.
Behavior Analysis and Asymptotic Stability of the Traveling Wave Solution of the Kaup-Kupershmidt Equation for Conformable Derivative
Page: 162-185 (24)
Author: Hülya Durur, Asıf Yokuş* and Mehmet Yavuz
DOI: 10.2174/9789815051933122030010
PDF Price: $30
Abstract
This article suggests solving the traveling wave solutions of the timefractional Kaup-Kupershmidt (KK) equation via 1/G' -expansion and sub-equation methods. Non-local fractional derivatives have some advantages over local fractional derivatives. The most important of these advantages are the chain rule and the Leibniz rule. The conformable derivative, which has a local fractional derivative feature, is taken into account in this study. Different types of traveling wave solutions of the time-fractional KK equation have been produced by using the important benefits of the time-dependent conformable derivative operator. These wave types are dark, singular, rational, trigonometric and hyperbolic type solitons. 2D, 3D and contour graphics are presented by giving arbitrary values to the constants in the solutions produced by analytical methods. These presented graphs represent the shape of the standing wave at any given moment. Besides, the advantages and disadvantages of the two analytical methods are discussed and presented in the result and discussion section. In addition, wave behavior analysis for different velocity values of the dark soliton produced by the analytical method is analyzed by simulation. The conditional convergence and asymptotic stability of the dark soliton discussed are analyzed. Computer software is also used in operations such as drawing graphs, complex operations, and solving algebraic equation systems.
Mathematical Analysis of a Rumor Spreading Model within the Frame of Fractional Derivative
Page: 186-209 (24)
Author: Chandrali Baishya, Sindhu J. Achar and P. Veeresha*
DOI: 10.2174/9789815051933122030011
PDF Price: $30
Abstract
Rumor spreading is a trivial social practice, which has a long history of
affecting society both in a positive and negative way, and modelling of transmission
of rumors has been an attractive area for social and, of late, for physical scientists. In
this chapter, we have modified the rumor-spreading model by incorporating fractional
derivatives in the Caputo sense. To analyze the spread of rumors in social as well as
virtual networks, we have considered four populations, namely, ignorant, spreader,
recaller, and stifler. The existence and uniqueness, and boundedness of the solutions
of the present model have been exhibited theoretically. Numerically, we have
experimented with the effect of fractional derivatives and the density of one
population on the other population by demonstrating the impact of rumor spread with
the change of various parameters.
A Unified Approach for the Fractional System of Equations Arising in the Biochemical Reaction without Singular Kernel
Page: 210-231 (22)
Author: P. Veeresha*, M.S. Kiran, L. Akinyemi and Mehmet Yavuz
DOI: 10.2174/9789815051933122030012
PDF Price: $30
Abstract
The pivotal aim of the present work is to find the solution for the fractional
system of equations arising in the biochemical reaction using q-homotopy analysis
transform method (q-HATM). The hired scheme technique unification of Laplace
transform with q-homotopy analysis method, and fractional derivative defined with
Caputo-Fabrizio (CF) operator. To validate and illustrate the competence of the future
method, we examined the model in terms of fractional order. The fixed-point theorem
hired to demonstrates the existence and uniqueness. Moreover, the physical nature of
achieved solutions has been captured in terms of plots for different order. The
obtained results elucidate that the considered algorithm is easy to implement, highly
methodical, and very effective as well as accurate to analyse the nature of nonlinear
differential equations of fractional order arising in the connected areas of science and
engineering.
Floating Object Induced Hydro-morphological Effects in Approach Channel
Page: 232-250 (19)
Author: Onur Bora*, M. Sedat Kabdaşlı, Nuray Gedik and Emel İrtem
DOI: 10.2174/9789815051933122030013
PDF Price: $30
Abstract
Transversal and diverging waves, return flows, propeller induced jet flows,
and other hydrodynamic effects induced by a floating object may cause significant
movement and/or suspension of bottom and bank sediments in the marine
environment, especially in approach channels. Using the CFD (Computational Fluid
Dynamics) process, the hydro-morphodynamic effects induced by a non-powered
floating object navigating in an approach channel are investigated in this study. The
approach channel dimensions depth, width, and channel slope are determined
according to PIANC (2014) [1]. The floating object locations and velocities are used
in nine different scenarios. In these cases, the floating object is 0.90, 1.10, and 1.30
meters from the bottom of the approach channel, respectively. According to the
findings, when the floating object is located nearest to the bottom and its speed is
fastest, there is a significant amount of sediment suspension and sediment movement
in the channel slope, which is mostly attributed to super-critical return flows. When
the floating object is farthest from the channel bottom and the floating object speed is
lowest, however, there is a noticeable reduction in the acceleration and suspension of
the sediment. As a result, the velocity and location of the floating object, channel
slope, the kinematics of ship-generated waves, and particularly the return flows are
found to have a significant impact on sediment movement and suspension.
Subject Index
Page: 251-259 (9)
Author: Mehmet Yavuz and Necati Özdemir
DOI: 10.2174/9789815051933122030014
Introduction
In the last two decades, many new fractional operators have appeared, often defined using integrals with special functions in the kernel as well as their extended or multivariable forms. Modern operators in fractional calculus have different properties which are comparable to those of classical operators. These have been intensively studied for modelling and analysing real-world phenomena. There is now a growing body of research on new methods to understand natural occurrences and tackle different problems. This book presents ten reviews of recent fractional operators split over three sections: - Chaotic Systems and Control (covers the Caputo fractional derivative, and a chaotic fractional-order financial system) - Heat Conduction (covers the Duhamel theorem for time-dependent source terms, and the Cattaneo-Hristov model for oscillatory heat transfer) - Computational Methods and Their Illustrative Applications (covers mathematical analysis for understanding 5 real-word phenomena: HTLV-1 infection of CD4+ T-cells, traveling waves, rumor-spreading, biochemical reactions, and the computational fluid dynamics of a non-powered floating object navigating in an approach channel) This volume is a resource for researchers in physics, biology, behavioral sciences, and mathematics who are interested in new applications of fractional calculus in the study of nonlinear phenomena.