Fractional Calculus and Applied Analysis最新文献

This article proposes a predictor-corrector scheme for solving the fractional differential equations ({}_0^C D_t^alpha y(t) = f(t,y(t)), alpha >0) with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though ({}_0^C D_t^alpha y(t)) is not smooth at (t=0). Numerical examples are carried out to verify the theoretical analysis.

This study delves into the origin, evolution, and practical applications of fractional difference inequalities based on recent literature. The review provides an overview of existing inequalities proposed under various definitions. Furthermore, to enhance this potent mathematical tool, a series of new inequalities have been introduced. Additionally, leveraging renowned Lyapunov functions in continuous-time domain, their discrete-time counterparts have been formulated. Moreover, several new potential Lyapunov functions have been identified. This review aims to aid readers in selecting suitable inequalities and Lyapunov functions to analyze the stability of nabla fractional order systems.

In this paper, the problem of identifying Multiple-Input-Single-Output (MISO) systems with fractional models from noisy input-output available data is studied. The proposed idea is to use Higher-Order-Statistics (HOS), like fourth-order cumulants (foc), instead of noisy measurements. Thus, a fractional fourth-order cumulants based-simplified and refined instrumental variable algorithm (frac-foc-sriv) is first developed. Assuming that all differentiation orders are known a priori, it consists in estimating the linear coefficients of all Single-Input-Single-Output (SISO) sub-models composing the MISO model. Then, the frac-foc-sriv algorithm is combined with a nonlinear optimization technique to estimate all the parameters: coefficients and orders. The performances of the developed algorithms are analyzed using numerical examples. Thanks to fourth-order cumulants, which are insensitive to Gaussian noise, and the iterative strategy of the instrumental variable algorithm, the parameters estimation is consistent.

Unstable first order time delay systems are frequently encountered in industrial applications, such as chemical plants, hydraulic processes, in satellite communications or economic systems, to name just a few. The control of such processes is a challenging issue. In this paper a filtered Smith Predictor control structure is used to compensate for the process time delays and to ensure the stability of the overall closed loop system. A simplified type of a fractional order PI controller is then designed to meet zero steady state error and an overshoot requirement. The tuning is based on the root locus analysis of the closed loop fractional order system. Simulation examples are provided to validate the proposed method and to demonstrate the efficiency of the proposed control method. Comparisons with two existing methods are included to highlight the possibility of using the proposed method as an alternative solution for controlling these types of processes.

Let (({{mathcal {X}}},d,mu )) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral (I_beta ) associated with admissible functions and its commutators. Similarly to (I_beta ), corresponding results for Calderón–Zygmund operators T associated with admissible functions are also included in this article.

A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability.

Various constitutive models have been proposed to quantify a wide range of non-Newtonian fluids, but there is lack of a systematic classification and evaluation of these competing models, such as the quantitative comparison between the classical integer-order constitutive models and the newly proposed fractional derivative equations for non-Newtonian fluids. This study reviews constitutive equation models for non-Newtonian fluids, including time-independent fluids, viscoelastic fluids, and time-dependent fluids. A comparison between fractional derivative non-Newtonian fluid constitutive equations and traditional constitutive equations is also provided. Results show that the space fractional derivative model is equivalent to some classical constitutive models under reasonable assumptions. Further discussions are made from the perspective of the industrial and biomedical applications of non-Newtonian fluids. Advantages and limitations of the constitutive models are also explored to help users to select proper models for real-world applications.

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain (varOmega ) such that V has no critical point in (partial varOmega ). To overcome the “lack of compactness" of the problem, we combine Del Pino-Felmer’s penalization technique with Moser’s iteration method and some ideas from Alves [1].

In order to accurately capture non-local properties and long-term memory effects, this study combines the tempered fractional-order operator with delayed neural networks to investigate its stability, leveraging the introduced decay term of the tempered fractional-order operator. Firstly, the discrete-time tempered fractional-order neural networks model (DTFNNs) is presented. Furthermore, in an effort to better understand the dynamic behavior of complex systems, solutions to discrete-time tempered fractional non-homogeneous equations are obtained. The stability conditions for systems are subsequently established, contributing novel insights to the field. To validate the robustness of these conditions, numerical experiments are conducted, underscoring the practical relevance of the proposed model.

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker (L^{p})-Carathéodory condition. Existence of solutions on an interval [0, T] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities.