Stability and Logarithmic Decay of the Solution to Hadamard-Type Fractional Differential Equation

Li, Changpin; Li, Zhiqiang

doi:10.1007/s00332-021-09691-8

Cited by 53 publications

(23 citation statements)

References 30 publications

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“…x i (t)| t�a � x ai , i � 1, 2, 3. Same as [16,17], we can directly investigate the matrix A in the following three cases: (14) or (25), the solution of system (39) is…”

Section: -D Casementioning

confidence: 99%

“…In this subsection, we will study the stability of system (12) with n-dimensional case. Similar to [16,17], we directly investigate the following two cases:…”

Section: -D Casementioning

confidence: 99%

“…In the last five decades, fractional differential equations (FDEs) have attracted great attention from researchers since FDEs may acquire more accurate results than the integerorder differential equations, especially for the description of materials and processes charactering memorial and hereditary properties [1][2][3][4]. Some authors studied works on existence, uniqueness, and data dependence of solution to FDEs [5][6][7][8][9][10][11], while another scholars focused on stability results of FDEs (including linear and nonlinear cases) with different types of fractional derivatives [12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…After that, the stability analysis for the case of Riemann-Liouville derivative was presented by Qian et al [20] based on the eigenvalue of system matrix. Recently, Li and Li investigated the stability of linear FDEs involving the Hadamard and Caputo-Hadamard derivatives [16], as well as the ψ-Riemann-Liouville and ψ-Caputo derivatives [17]. e stability mentioned above for linear FDEs is a Lyapunov stability, however, we mainly consider the structural stability for nonlinear FDEs.…”

Section: Introductionmentioning

confidence: 99%

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Studying the Stability of the ψ‐Hilfer Fractional Differential System

Yang

2022

Complexity

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This paper devotes to the study on the stability and decay of solution to fractional differential system involving the ψ -Hilfer fractional derivative of order α ∈ 0,1 and type β ∈ 0,1 . We first derive the solution of linear system by using the generalized Laplace transform, which can be represented by the form of Mittag-Leffler function. Then, in terms of the asymptotic expansion of the Mittag-Leffler function, stability properties of linear system are analyzed in more detail. Finally, we construct a linearization theorem and determine the stability near the equilibrium for the autonomous nonlinear differential system with the ψ -Hilfer derivative.

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“…x i (t)| t�a � x ai , i � 1, 2, 3. Same as [16,17], we can directly investigate the matrix A in the following three cases: (14) or (25), the solution of system (39) is…”

Section: -D Casementioning

confidence: 99%

“…In this subsection, we will study the stability of system (12) with n-dimensional case. Similar to [16,17], we directly investigate the following two cases:…”

Section: -D Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Studying the Stability of the ψ‐Hilfer Fractional Differential System

Yang

2022

Complexity

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“…In [16], the authors estimate the upper bound of the modified Lyapunov exponents for Caputo-Hadamard FDEs by virtue of the Gronwall's inequality and the expansion of the Mittag-Leffler function. In [15], the authors study the asymptotic stability and logarithmic decay of the solutions to HFDEs by applying the modified Laplace transform. Besides, by constructing suitable Banach space along with lower and upper solutions method, the existence of blow-up phenomena in HFDEs has been proved in [21].…”

Section: Introductionmentioning

confidence: 99%

Finite-time stability of Hadamard fractional differential equations in weighted Banach spaces

Ма

2022

Nonlinear Dyn

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The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definition of finite-time stability of HFDEs in compatible Banach space are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria for finite-time stability of linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results.

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Blowup for semilinear fractional diffusion system with Caputo–Hadamard derivative

Yang

2022

Math Methods in App Sciences

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The main aim of this paper is to study the blowing-up behavior of the solution for semilinear fractional diffusion system with the Caputo-Hadamard derivative and the fractional Laplacian. We construct a mild solution of the semilinear system by using the fundamental solutions and then prove the local existence and uniqueness of the mild solution by virtue of the fixed point argument. Next, the definition of a weak solution is introduced by the test function, and the mild solution can be verified to be a weak solution. Finally, we show the finite time blowup and global solution to the considered system in terms of the contraction mapping principle.

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Stability and Logarithmic Decay of the Solution to Hadamard-Type Fractional Differential Equation

Cited by 53 publications

References 30 publications

Studying the Stability of the ψ‐Hilfer Fractional Differential System

Studying the Stability of the ψ‐Hilfer Fractional Differential System

Finite-time stability of Hadamard fractional differential equations in weighted Banach spaces

Blowup for semilinear fractional diffusion system with Caputo–Hadamard derivative

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