Search Results (31)

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the $p$-proportional $\omega$-weighted $\kappa$-Hilfer derivative includes an exponential function, $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, and then we consider a non-instantaneous impulse differential inclusion containing $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$ with order $\sigma \in (1,2)$ and of kind $\rho \in [0,1]$ in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the $(p,\omega ,\kappa )$-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the $(p,\omega ,\kappa )$-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

In this paper, a new class of coupled hybrid systems of proportional sequential $\psi$-Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel’ski$\breve{\mathrm{i}}$’s fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential $\psi$-Hilfer fractional differential equations. Full article

The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. We investigate a special class of exact solutions and accurate analytical approximations for the SIRVD and SIRD compartment models. While the SIRVD and SIRD equations pose complex integro-differential equations for the rate of new infections and the fractions as a function of time, a simpler approach considers determining equations for the sum of ratios for given variations. This approach enables us to derive fully exact analytical solutions for the SIRVD and SIRD models. For nonlinear models with a high-dimensional parameter space, such as the SIRVD and SIRD models, analytical solutions, exact or accurately approximative, are of high importance and interest, not only as suitable benchmarks for numerical codes, but especially as they allow us to understand the critical behavior of epidemic outbursts as well as the decisive role of certain parameters. In the second part of our study, we apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using an a posteriori approach. We demonstrate that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates. The case of stationary ratios allows one to construct a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave. Full article

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues. Full article

The Ebola virus continues to be the world’s biggest cause of mortality, especially in developing countries, despite the availability of safe and effective immunization. In this paper, we construct a fractional-order Ebola virus model to check the dynamical transmission of the disease as it is impacted by immunization, learning, prompt identification, sanitation regulations, isolation, and mobility limitations with a constant proportional Caputo (CPC) operator. The existence and uniqueness of the proposed model’s solutions are discussed using the results of fixed-point theory. The stability results for the fractional model are presented using Ulam–Hyers stability principles. This paper assesses the hybrid fractional operator by applying methods to invert proportional Caputo operators, calculate CPC eigenfunctions, and simulate fractional differential equations computationally. The Laplace–Adomian decomposition method is used to simulate a set of fractional differential equations. A sustainable and unique approach is applied to build numerical and analytic solutions to the model that closely satisfy the theoretical approach to the problem. The tools in this model appear to be fairly powerful, capable of generating the theoretical conditions predicted by the Ebola virus model. The analysis-based research given here will aid future analysis and the development of a control strategy to counteract the impact of the Ebola virus in a community. Full article

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties. Full article

The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this study is that, to the best of the authors’ knowledge, this work presents the first state feedback control design for GPFO systems. The suggested state feedback controller is intended to ensure the system’s generalized Mittag Leffler (GML) stability and to deliver optimal performance. The findings of this paper show that the proposed strategy is effective in stabilizing Generalized Proportional Fractional Order Nonlinear Systems. A numerical example is presented to demonstrate the usefulness of the stated theoretical conclusions. Full article

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations. Full article

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results. Full article

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations. Full article

In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results. Full article

A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary parameter are obtained. In the study of Ulam-type stability, this parameter was chosen to depend on the solution of the corresponding fractional differential inequality. We provide sufficient conditions for Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the given problem on a finite interval. As a partial case, sufficient conditions for Ulam-type stability for a couple of multi-term delay, Caputo fractional differential equations are obtained. An example is illustrating the results. Full article

In this article, starting from a benchmark represented by a Direct Current-to-Direct Current (DC-DC) three-phase power electronic converter used as an interface and interconnection between the grid and a DC microgrid, we compare the performances of a series of control structures—starting with the classical proportional integrator (PI) type and continuing with more advanced ones, such as sliding mode control (SMC), integer-order synergetic, and fractional-order (FO) controllers—in terms of maintaining the constant DC voltage of the DC microgrid. We present the topology and the mathematical modeling using differential equations and transfer functions of the DC-DC three-phase power electronic converter that provides the interface between the grid and a DC microgrid. The main task of the presented control systems is to maintain the DC voltage supplied to the microgrid at an imposed constant value, regardless of the total value of the current absorbed by the consumers connected to the DC microgrid. We present the elements of fractional calculus that were used to synthesize a first set of FO PI, FO tilt-integral-derivative (TID), and FO lead-lag controllers with Matlab R2021b and the Fractional-order Modeling and Control (FOMCON) toolbox, and these controllers significantly improved the control system performance of the DC-DC three-phase power electronic converter compared to classical PI controllers. The next set of proposed and synthesized controllers were based on SMC, together with its more general and flexible synergetic control variant, and both integer-order and FO controllers were developed. The proposed control structures are cascade control structures combining the SMC properties of robustness and control over nonlinear systems for the outer voltage control loop with the use of properly tuned synergetic controllers to obtain faster response time for the inner current control loop. To achieve superior performance, this type of cascade control also used a properly trained reinforcement learning-twin delayed deep deterministic policy gradient (RL-TD3) agent, which provides correction signals overlapping with the command signals of the current and voltage controllers. We present the Matlab/Simulink R2021b implementations of the synthesized controllers and the RL-TD3 agent, along with the results of numerical simulations performed for the comparison of the performance of the control structures. Full article

A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with different orders is obtained. A mild solution of the boundary value problem for the multi-term nonlinear couple of the given fractional equations is defined. The connection between the mild solution and the solution of the studied problem is discussed. As a partial case, several results for the nonlocal boundary value problem for the linear and non-linear multi-term Caputo fractional differential equations are provided. The results generalize several known results in the literature. Full article

In this paper, nonlinear differential equations with a generalized proportional Caputo fractional derivative and finite delay are studied in this paper. The eventual presence of impulses in the equations is considered, and the statement of initial value problems in three cases is defined: namely non-instantaneous impulses, instantaneous impulses and no impulses. The relations between these three cases are discussed. Additionally, some stability properties are investigated. We apply the Mittag–Leffler function which plays a vital role and which gives well-known bounds on the norm of the solutions. The symmetry of this function about a line and the bounds is a property that plays an important role in stability. Several sufficient conditions are presented via appropriate new comparison results and the modified Razumikhin method. The results generalize several known results in the literature. Full article