Journal•ISSN: 1314-2224
Fractional Calculus and Applied Analysis
Springer Science+Business Media
About: Fractional Calculus and Applied Analysis is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Fractional calculus & Mathematics. It has an ISSN identifier of 1314-2224. Over the lifetime, 996 publications have been published receiving 20842 citations. The journal is also known as: Fractional Calculus & Applied Analysis.
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TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.
372 citations
TL;DR: Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations and can be extended to other kinds of themulti-term fractional time-space models with fractional Laplacian.
Abstract: In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.
278 citations
Journal Article•
TL;DR: In this article, a matrix transfer technique (MTT) was proposed for solving the space fractional diception equation (SFDE) with non-homogeneous boundary conditions on a bounded domain.
Abstract: In this paper, a space fractional di®usion equation (SFDE) with non-
homogeneous boundary conditions on a bounded domain is considered. A
new matrix transfer technique (MTT) for solving the SFDE is proposed.
The method is based on a matrix representation of the fractional-in-space
operator and the novelty of this approach is that a standard discretisation
of the operator leads to a system of linear ODEs with the matrix raised
to the same fractional power. Analytic solutions of the SFDE are derived.
Finally, some numerical results are given to demonstrate that the MTT is
a computationally e±cient and accurate method for solving SFDE.
261 citations
Journal Article•
TL;DR: In this article, the design of fractional-order discrete-time controllers is discussed, which are suited for z-transform analysis and discrete time implementation, and the performance of the proposed algorithms with linear and nonlinear systems is also investigated.
Abstract: The theory of fractional calculus goes back to the beginning of the theory of di erential calculus but its inherent complexity postponed the application of the associated concepts. In the last decade the progress in the areas of chaos and fractals revealed subtle relationships with the fractional calculus leading to an increasing interest in the development of the new paradigm. In the area of automatic control preliminary work has already been carried out but the proposed algorithms are restricted to the frequency domain. The paper discusses the design of fractional-order discrete-time controllers. The algorithms studied adopt the time domain, which makes them suited for z -transform analysis and discrete-time implementation. The performance of discrete-time fractional-order controllers with linear and nonlinear systems is also investigated. Mathematics Subject Classi cation: 26A33, 93C15, 93C55, 93C80
238 citations
TL;DR: In this article, a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications is provided, which can help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.
Abstract: Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.
207 citations