A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel

This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam Rassias, and generalized Hyers-Ulam-Rassias stability. We provide some test problems for illustrating our analysis.

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“…Using property (P 2 ) of the Green's function W(t, s) given in Lemma 2 and inequality (16) in inequality (15), we obtain…”

Section: Lemmamentioning

confidence: 99%

“…On the other hand, the area devoted to establishing a procedure for numerical solutions has been investigated very well. See [15][16][17][18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Study of implicit delay fractional differential equations under anti-periodic boundary conditions

2020

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“…Other approaches to solving fractional differential equations are the numerical method based on the Adams-Bashforth-Moulton scheme, the Lagrange polynomial interpolation or the homotopy perturbation transform [65,66,42,30,41]. In these papers, the authors show how to solve variable-order fractional systems, so that variable-order Mittag-Leffler observers can be proposed intuitively.…”

Section: Review Of Numerical Aspectsmentioning

confidence: 99%

Stability of Fractional Nonlinear Systems with Mittag-Leffler Kernel and Design of State Observers

Martínez-Fuentes,

Delfín-Prieto

2020

Preprint

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Atangana and Baleanu proposed a new fractional derivative with non-local and nosingular MittagLeffler kernel to solve some problems proposed by researchers in the field of fractional calculus. This new derivative is better to describe essential aspects of non-local dynamical systems. We present some results regarding Lyapunov stability theory, particularly the Lyapunov Direct Method for fractional-order systems modeled with Atangana-Baleanu derivatives and some significant inequalities that help to develop the theoretical analysis. As applications in control theory, some algorithms of state estimation are proposed for linear and nonlinear fractional-order systems.

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“…In [14,15], authors derived numerical approaches for the numerical integration of FDEs, which are a generalization of many known methods in the literature, such as the Adams-Bashforth approach. Adams-Bashforth methods were implemented to solve nonlinear FDDEs [16,17]. In addition, Daftardar-Gejji et al recently introduced the predictor-corrector method for solving FDEs [18].…”

Section: Introductionmentioning

confidence: 99%

A Reliable Approach for Solving Delay Fractional Differential Equations

et al. 2022

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In this paper, we study a class of second-order delay fractional differential equations with a variable-order Caputo derivative. This type of equation is an extension to ordinary delay equations which are used in the modeling of several biological systems such as population dynamics, epidemiology, and immunology. Usually, fractional differential equations are difficult to solve analytically, and with fractional derivatives of variable-order, they become more challenging. Therefore, the need for reliable numerical techniques is worth investigating. To solve this type of equation, we derive a new approach based on the operational matrix. We use the shifted Chebyshev polynomials of the second kind as the basis for the approximate solutions. A convergence analysis is discussed and the uniform convergence of the approximate solutions is proven. Several examples are discussed to illustrate the efficiency of the presented approach. The computed errors, figures, and tables show that the approximate solutions converge to the exact ones by considering only a few terms in the expansion, and illustrate the novelty of the presented approach.

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A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel

Cited by 9 publications

References 26 publications

Study of implicit delay fractional differential equations under anti-periodic boundary conditions

Study of implicit delay fractional differential equations under anti-periodic boundary conditions

Stability of Fractional Nonlinear Systems with Mittag-Leffler Kernel and Design of State Observers

A Reliable Approach for Solving Delay Fractional Differential Equations

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