Fractional Calculus: New Applications in Understanding Nonlinear Phenomena

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.

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. 

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.

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.

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. 

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.

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.

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.

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. 

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.