Unified predictor–corrector method for fractional differential equations with general kernel functions

Wu, Guo–Cheng; Kong, Hua; Luo, Maokang; Fu, Hui; Huang, Lanlan

doi:10.1007/s13540-022-00029-z

Cited by 28 publications

(11 citation statements)

References 18 publications

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“…It is also worth noting that the results in this paper can be extended to semilinear cases, and many fractional neural network equations hold such a structure. Recently, Wu et al [20][21][22] studied the conclusions of general fractional calculus, and put forward a numerical method to solve fractional differential equations and fractional delay differential equations with general kernels, respectively. Moreover, Wu et al [23] gave a uniform delayed Mittag-Leffler function without segmented definition which is helpful to analyze the basic properties of the solution, such as monotonicity, differentiability and so on.…”

Section: Discussionmentioning

confidence: 99%

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Analyses of solutions of Riemann‐Liouville fractional oscillatory differential equations with pure delay

Pan

Fan

2023

Math Methods in App Sciences

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This paper first finishes off the exact solutions of Riemann‐Liouville fractional differential time‐delay oscillatory system of order ρ∈false(1,2false)$$ \rho \in \left(1,2\right) $$ by using two newly defined delayed perturbations of Mittag‐Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved.

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Section: Discussionmentioning

confidence: 99%

“…The study of such problems is very meaningful. Furthermore, we may consider some new problems by combining the ideas of references [20][21][22][23].…”

Section: Discussionmentioning

confidence: 99%

Analyses of solutions of Riemann‐Liouville fractional oscillatory differential equations with pure delay

Pan

Fan

2023

Math Methods in App Sciences

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“…which says that the controllability Grammian [ ] G 1, 2 is +ve definite and, by theorem 3.2, the linear part of (15)…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Aydi et al examined the existence and uniqueness of positive solutions for a fractional thermostat model for both cases of concave and convex source terms by utilizing ψ-Caputo fractional derivative in [9]. For reference, [10][11][12][13][14][15][16] these are certain articles on differential equations that heavily rely on the ψ-Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Selvam¹,

Vellappandi²,

Govindaraj³

2023

Phys. Scr.

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The idea behind this study is to investigate the controllability of dynamical systems in terms of the ψ-Caputo fractional derivative. The Grammian matrix is used to get at necessary and sufficient controllability requirements for linear systems, which are characterized by the Mittag-Leffler functions, while the fixed point approach is used to arrive at adequate controllability criteria for nonlinear systems. The novelty of this research is to inquire into the controllability concepts by utilizing the ψ-Caputo fractional derivative. Since ψ-Caputo fractional derivatives have the advantage of capturing memory effects as well as increasing the accuracy of anticipating real-world scenarios. A few numerical examples are offered to help better understand the theoretical results.

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“…etc. Recently, some scholars have paid much attention to theories and applications such as Laplace transform [8] and predictor-corrector method [27]. Moreover, the physical meaning of the kernel function g(t) are explained by use of continuous time random walk theory [4], and the boundedness condition is provided to guarantee that the general fractional integral is well defined [3].…”

Section: Introductionmentioning

confidence: 99%

Fractional uncertain differential equations with general memory effects: Existences and alpha-path solutions

Luo

Wu²,

Huang³

2022

NAMC

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General fractional calculus is popular recently. Fractional uncertain differential equations (FUDEs) with general memory effects are proposed in this paper. Firstly, existence and uniqueness theorems of solution for general fractional uncertain differential equations (GFUDEs) is presented, and the analytic solution of a linear one is given. Then the concept of α-path is introduced, and relationship between solution of GFUDE and corresponding α-path is also discussed. In addition, a theorem is proved to obtain the expected value of a monotonic function related to solutions of GFUDEs. Finally, a numerical example is given to better understand the significance of general memory effects. This paper provides more types of FUDEs to better describe some phenomena in uncertain environments.

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Unified predictor–corrector method for fractional differential equations with general kernel functions

Cited by 28 publications

References 18 publications

Analyses of solutions of Riemann‐Liouville fractional oscillatory differential equations with pure delay

Analyses of solutions of Riemann‐Liouville fractional oscillatory differential equations with pure delay

Controllability of fractional dynamical systems with ψ-Caputo fractional derivative

Fractional uncertain differential equations with general memory effects: Existences and alpha-path solutions

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