Asymptotical stability analysis of conformable fractional systems

Qi, Yongfang; Wang, Xuhuan

doi:10.1080/16583655.2019.1701390

Cited by 21 publications

(2 citation statements)

References 36 publications

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“…It can be found many authors focus on using conformable derivative operators to solve a real-life problem [31][32][33][34]. The CDO conserves many features of classical-order derivatives [35][36][37]. We include here more reasons for using fractional derivative and conformable derivative operator.…”

Section: Conformable Derivative Operatormentioning

confidence: 99%

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

2022

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In providing an accurate approximate analytical solution to the non-linear system of fractional-order susceptible-infected-recovered epidemic model (FOSIREM) of childhood disease has been a challenge because no norm guarantees the convergence of the infinite series solution. We compute an accurate approximate analytical solution using the optimal homotopy asymptotic method (OHAM). The fractional differential equations operator (FDEO) is given as conformable derivative operator (CDO). We show the basic idea of the proposed method, the CDO sense, equilibrium points, local asymptotic stability, reproduction number, and the convergence analysis of the proposed method. Numerical results and comparisons with other approximate analytical solutions are given to validate the efficiency of the method. The proposed method speedily converges to the exact solution as the fractional-order derivative approaches 1 proved an excellent tool for solving and predicting the model.

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Section: Conformable Derivative Operatormentioning

confidence: 99%

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

2022

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“…Zhao et al established the multivariate theory of GCFD and illustrated the conformable Maxwell equations, and theorems for Conformable Gauss's, Green's, and Stokes's Theorem, see Zhao et al 22 Though, this new local fractional derivative fails some properties as pointed out in previous studies, [23][24][25][26] it seems to account for many deficiencies of some of the earlier proposed definitions which are of great importance in applied sciences and therefore suitable for more applications. Some authors followed this work and explored potential applications in various fields such as the control theory of dynamical systems, [27][28][29][30] mathematical biology and epidemiology, 19,[31][32][33][34] mechanics, 16,35,36 systems of linear and nonlinear conformable fractional differential equations (CFDEs), [37][38][39] quantum mechanics, 40,41 variational calculus, 42,43 arbitrary time scale problems, [44][45][46] modelling of diffusion, 47,48 stochastic process, 49,50 and optics. 51 Some analytical and numerical methods have attracted great interest and became an important tool for differential equations with CFDs, (see previous studies ).…”

Section: Introductionmentioning

confidence: 99%

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Mortazaasl

Akbarfam

2020

Math Methods in App Sciences

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In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of r-CFSLPs, we discuss two types of CFSLPs which include left-and right-sided CFDs, each of order ∈ (n, n + 1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order ∈ (0, 1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and-II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. KEYWORDS conformable fractional derivatives and integrals, eigenvalues and eigenfunctions, fixed-point theorem, fractional diffusion, fractional Sturm-Liouville operators, transform methods JEL CLASSIFICATION

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Controllability, observability, and stability of φ‐conformable fractional linear dynamical systems

Sadek

2024

Asian Journal of Control

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This paper focuses on two main aspects. Firstly, we introduce a generalized form of conformable fractional derivatives called ‐conformable fractional derivatives along with their corresponding integrals . We also present several theorems related to these derivatives. Secondly, we investigate the observability, controllability, and stability of a fractional dynamical system known as the ‐conformable fractional dynamical system ( ‐CF‐DS). We establish the connection between controllability, the controllability matrix, and the fractional ‐conformable fractional differential Lyapunov equation ( ‐CF‐DLE). We prove that the ‐CF‐DS is stable if and only if all eigenvalues of the matrix have negative real parts. Moreover, we provide theorems regarding the observability of an ‐CF‐DS. To demonstrate the efficacy of our results, we include illustrative examples.

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Asymptotical stability analysis of conformable fractional systems

Cited by 21 publications

References 36 publications

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

Efficient Solution of Fractional-Order SIR Epidemic Model of Childhood Diseases With Optimal Homotopy Asymptotic Method

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Controllability, observability, and stability of φ‐conformable fractional linear dynamical systems

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