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Keywords = Trotter-Kato theorem

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13 pages, 309 KiB  
Article
Stability of Two Kinds of Discretization Schemes for Nonhomogeneous Fractional Cauchy Problem
by Xiaoping Xu and Lei Xu
Symmetry 2023, 15(7), 1355; https://doi.org/10.3390/sym15071355 - 3 Jul 2023
Viewed by 946
Abstract
The full discrete approximation of solutions of nonhomogeneous fractional equations is considered in this paper. The methods of iteration, finite differences and projection are applied to obtain desired formulas of explicit- and implicit-difference schemes for discretization schemes. The stability of two difference schemes [...] Read more.
The full discrete approximation of solutions of nonhomogeneous fractional equations is considered in this paper. The methods of iteration, finite differences and projection are applied to obtain desired formulas of explicit- and implicit-difference schemes for discretization schemes. The stability of two difference schemes is also discussed using the Trotter–Kato theorem. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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21 pages, 1098 KiB  
Article
Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method
by Zhuoqian Chen, Houbao Xu and Huixia Huo
Mathematics 2022, 10(21), 4117; https://doi.org/10.3390/math10214117 - 4 Nov 2022
Cited by 1 | Viewed by 1631
Abstract
First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited [...] Read more.
First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited by the order of the differences in L1 space. Therefore, how to design a computational scheme for the first-order linear IDEs with computational efficiency becomes an urgent problem to be solved. To this end, a polynomial approximation scheme based on the shifted Legendre spectral collocation method is proposed in this paper. First, we transform the first-order linear IDEs into an Cauchy problem for consideration. Second, by decomposing the system operator, we rewrite the Cauchy problem into a more general form for approximating. Then, by using the shifted Legendre spectral collocation method, we construct a computational scheme and write it into an abstract version. The convergence of the scheme is proven in the sense of L1-norm by employing Trotter-Kato theorem. At the end of this paper, we summarize the usage of the scheme into an algorithm and present some numerical examples to show the applications of the algorithm. Full article
(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)
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13 pages, 1009 KiB  
Article
Dynamic Analysis of Software Systems with Aperiodic Impulse Rejuvenation
by Huixia Huo, Houbao Xu and Zhuoqian Chen
Mathematics 2022, 10(2), 197; https://doi.org/10.3390/math10020197 - 9 Jan 2022
Viewed by 1602
Abstract
This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt [...] Read more.
This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt a difference scheme and establish the convergence of this scheme by applying the Trotter–Kato theorem to obtain the system’s dynamical solution. Moreover, the instantaneous availability as an important evaluation index for software systems is derived, and its range is also estimated. At last, numerical examples are shown to illustrate the validity of theoretical results. Full article
(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)
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