Stability of Linear Multiple‎ Different Order Caputo Fractional System ‎‏

In this paper, we introduce a new equivalent system to the higher order Caputo fractional system ‎‎(CFS) . This equivalent system has multiple order Caputo fractional derivatives ‎‎(CFDs). These CFDs are lying between zero and one.‎ As well ‎as, we find the fundamental solution for linear CFS with multiple order CFDs. Also, we introduce new criteria of studying the stability (asymptotic stability) of the ‎linear CFS with multiple order CFDs. These criteria can be applied in three cases: ‎the first, all CFDs is lying between zero and one. The second, all CFDs are lying ‎between one and two. Finally, some of CFDs are lying between zero and one, ‎and the rest of these derivatives are lying between one and two. The criteria are ‎depending on the position of eigenvalues of the matrix system in the complex plane. ‎These criteria are considered as a generalized of the classical criteria which is ‎used to study the stability of linear first ODEs. Also, these criteria are considered ‎as generalized of the criteria which used to study the stability same order CFS in ‎case when all CFDs lying between zero and one, also in case when all CFDs lying ‎between one and two. Several examples are given to show the behavior of the ‎solution near the equilibrium point.‎ Keywords: Caputo fractional derivatives; Linear Caputo fractional system ‎; Fundamental solution Stability analysis

Cauchy Problem for Fractional Ricatti‎ Differential Equations ‎Type with Alpha Order Caputo Fractional Derivatives

In this paper, we investigate solution of the fractional Ricatti differential equations (FRDEs) with alpha order Caputo fractional derivatives. In fact, FRDEs are analogous of the Ricatti‎ ordinary differential equations. The multi power series method is used to obtain a useful formula that is implemented to find an explicit solution of Cauchy problem for FRDEs without solving any integral. This formula is explicit and easy to compute by using Maple software to get explicit solution. Also, it is shown that the proposed formula can be used to solve the Cauchy problem for Ricatti‎ ordinary differential equations

A novel numerical method for solving optimal control problems using fourth-degree hat functions

This paper focuses on solving a class of nonlinear optimal control problems by constructing novel hat functions based on fourth-order polynomials, namely, fourth-degree hat functions (FDHFs). The FDHFs are used in this method to approximate the state equations and cost function. In fact, the FDHFs enable us to turn the optimal control problem under consideration into a nonlinear optimal control problem with unknown coefficients that is easy to solve by any numerical method. The main advantages of the proposed method are its simplicity, ease of application, low computational expense, and avoidance of numerical integration. Several examples have been discussed to demonstrate the efficacy and applicability of the suggested method

Integro-differential equations: Numerical solution by a new operational matrix based on fourth-order hat functions

This article aims to present a new approach based on the operational matrix method for solving integro-differential equations using numerical techniques. In this method, we employ fourth-degree hat functions (FDHFs) to construct operational matrices. The approach involves two main steps. First, we utilize FDHFs to create operational matrices, which allows us to transform the given problem into a system of algebraic equations. The second step involves solving these algebraic equations numerically. Additionally, we provide an analysis of the errors involved and compare the proposed method with existing techniques. The results demonstrate that the proposed method outperforms its counterparts, highlighting its superiority

Solving Volterra integral equations via fourth-degree hat functions

The goal of this paper is to develop a novel operation matrix approach to solving Volterra integral equations by constructing fourth-degree hat functions and investigating their properties. This approach requires turning the problem under dissection into a set of algebraic equations and then solving it by any numerical method. In addition, we demonstrate that the convergence of this approach is of the order of O(h5), and numerical results show that it is practical and useful for dealing with such problems

Stability of nonlinear q-fractional dynamical systems on time scale

This paper adopts a new terminology, “delta q-Mittag-Leffler stability”, for studying the stability of nonlinear q-fractional dynamical systems on the time scale. In fact, the idea of delta q-Mittag-Leffler stability is inspired by the idea of Mittag-Leffler stability, which is designed to investigate the stability of fractional dynamical systems. The sufficient conditions for delta q-Mittag-Leffler stability of considered dynamical systems with Caputo delta q-derivatives have been introduced

Studying of COVID-19 fractional model: Stability analysis

This article focuses on the recent epidemic caused by COVID-19 and takes into account several measures that have been taken by governments, including complete closure, media coverage, and attention to public hygiene. It is well known that mathematical models in epidemiology have helped determine the best strategies for disease control. This motivates us to construct a fractional mathematical model that includes quarantine categories as well as government sanctions. In this article, we prove the existence and uniqueness of positive bounded solutions for the suggested model. Also, we investigate the stability of the disease-free and endemic equilibriums by using the basic reproduction number (BRN). Moreover, we investigate the stability of the considering model in the sense of Ulam–Hyers criteria. To underpin and demonstrate this study, we provide a numerical simulation, whose results are consistent with the analysis presented in this article