Fractional Calculus and Applied Analysis最新文献

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Spatial $$beta $$ -fractional output stabilization of bilinear systems with a time $$alpha $$ -fractional-order 时间分阶双线性系统的空间分阶输出稳定问题
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-11-15 DOI: 10.1007/s13540-024-00354-5
Mustapha Benoudi, Rachid Larhrissi
{"title":"Spatial $$beta $$ -fractional output stabilization of bilinear systems with a time $$alpha $$ -fractional-order","authors":"Mustapha Benoudi, Rachid Larhrissi","doi":"10.1007/s13540-024-00354-5","DOIUrl":"https://doi.org/10.1007/s13540-024-00354-5","url":null,"abstract":"<p>This research aims to investigate the stabilization problem of the Riemann-Liouville spatial <span>(beta )</span>-fractional output with order <span>(beta in (0, 1))</span> for a class of bilinear dynamical systems with a time Caputo <span>(alpha )</span>-fractional derivative. Initially, we provide definitions and establish the well-posedness of the problem addressed. Furthermore, we introduce a feedback control strategy that ensures both weak and strong stabilization of the <span>(beta )</span>-fractional output, under a broad set of sufficient conditions. Additionally, we present numerical computations to elucidate the effectiveness of the obtained results.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"43 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A second-order fitted scheme for time fractional telegraph equations involving weak singularity 涉及弱奇异性的时间分数电报方程的二阶拟合方案
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-11-14 DOI: 10.1007/s13540-024-00355-4
Caixia Ou, Dakang Cen, Zhibo Wang, Seakweng Vong
{"title":"A second-order fitted scheme for time fractional telegraph equations involving weak singularity","authors":"Caixia Ou, Dakang Cen, Zhibo Wang, Seakweng Vong","doi":"10.1007/s13540-024-00355-4","DOIUrl":"https://doi.org/10.1007/s13540-024-00355-4","url":null,"abstract":"<p>In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition, the improved regularity of solution is derived. Furthermore, to overcome the weak singularity and enhance convergence precision, a second-order fitted scheme based on <i>L</i>2-<span>(1_sigma )</span> approximation and order reduction method is applied to such problems, which is an improvement for the work [6]. Ultimately, examples are presented to verify the effectiveness of our theoretical results.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"6 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142637871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unification of popular artificial neural network activation functions 统一流行的人工神经网络激活函数
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-30 DOI: 10.1007/s13540-024-00347-4
Mohammad Mostafanejad
{"title":"Unification of popular artificial neural network activation functions","authors":"Mohammad Mostafanejad","doi":"10.1007/s13540-024-00347-4","DOIUrl":"https://doi.org/10.1007/s13540-024-00347-4","url":null,"abstract":"<p>We present a unified representation of the most popular neural network activation functions. Adopting Mittag-Leffler functions of fractional calculus, we propose a flexible and compact functional form that is able to interpolate between various activation functions and mitigate common problems in training deep neural networks such as vanishing and exploding gradients. The presented gated representation extends the scope of fixed-shape activation functions to their adaptive counterparts whose shape can be learnt from the training data. The derivatives of the proposed functional form can also be expressed in terms of Mittag-Leffler functions making it suitable for backpropagation algorithms. By training an array of neural network architectures of different complexities on various benchmark datasets, we demonstrate that adopting a unified gated representation of activation functions offers a promising and affordable alternative to individual built-in implementations of activation functions in conventional machine learning frameworks.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"108 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142556280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete-time general fractional calculus 离散时间一般分数微积分
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-21 DOI: 10.1007/s13540-024-00350-9
Alexandra V. Antoniouk, Anatoly N. Kochubei
{"title":"Discrete-time general fractional calculus","authors":"Alexandra V. Antoniouk, Anatoly N. Kochubei","doi":"10.1007/s13540-024-00350-9","DOIUrl":"https://doi.org/10.1007/s13540-024-00350-9","url":null,"abstract":"<p>In general fractional calculus (GFC), the counterpart of the fractional time derivative is a differential-convolution operator whose integral kernel satisfies some additional conditions, under which the Cauchy problem for the corresponding time-fractional equation is not only well-posed, but has properties similar to those of classical evolution equations of mathematical physics. In this work, we develop the GFC approach for the discrete-time fractional calculus. In particular, we define within GFC the appropriate resolvent families and use them to solve the discrete-time Cauchy problem with an appropriate analog of the Caputo fractional derivative.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"19 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The well-posedness analysis in Besov-type spaces for multi-term time-fractional wave equations 多期时间分式波方程在贝索夫类型空间中的拟合分析
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-21 DOI: 10.1007/s13540-024-00348-3
Yubin Liu, Li Peng
{"title":"The well-posedness analysis in Besov-type spaces for multi-term time-fractional wave equations","authors":"Yubin Liu, Li Peng","doi":"10.1007/s13540-024-00348-3","DOIUrl":"https://doi.org/10.1007/s13540-024-00348-3","url":null,"abstract":"<p>In this paper, we consider the initial value problems for multi-term time-fractional wave equations in the framework of Besov spaces, which can be described the Couette flow of viscoelastic fluid. Considering the initial data in Besov spaces, we obtain some results about the local well-posedness and the blow-up of mild solutions for the proposed problem. Further, we extend these results to Besov–Morrey spaces.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"21 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the computation of the Mittag-Leffler function of fractional powers of accretive operators 关于分数幂增量算子的米塔格-勒弗勒函数的计算
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-21 DOI: 10.1007/s13540-024-00349-2
Eleonora Denich, Paolo Novati
{"title":"On the computation of the Mittag-Leffler function of fractional powers of accretive operators","authors":"Eleonora Denich, Paolo Novati","doi":"10.1007/s13540-024-00349-2","DOIUrl":"https://doi.org/10.1007/s13540-024-00349-2","url":null,"abstract":"<p>This paper deals with the computation of the two parameter Mittag-Leffler function of operators by exploiting its Stieltjes integral representation and then by using a single exponential transform together with the sinc rule. Whenever the parameters of the function do not allow this representation, we resort to the Dunford-Taylor one. The error analysis is kept in the framework of unbounded accretive operators in order to make it a useful tool for the solution of fractional differential equations. The theory is also used to design a rational Krylov method.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"44 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids 一般非均匀网格上双面空间分数扩散方程的快速分数块中心有限差分法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-18 DOI: 10.1007/s13540-024-00346-5
Meijie Kong, Hongfei Fu
{"title":"A fast fractional block-centered finite difference method for two-sided space-fractional diffusion equations on general nonuniform grids","authors":"Meijie Kong, Hongfei Fu","doi":"10.1007/s13540-024-00346-5","DOIUrl":"https://doi.org/10.1007/s13540-024-00346-5","url":null,"abstract":"<p>In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, this discretization still results in an unstructured dense coefficient matrix with huge memory requirement and computational complexity. To address this issue, a fast version fractional BCFD algorithm by employing the well-known sum-of-exponentials (SOE) approximation technique is also proposed. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only <span>({mathcal {O}}(MN_{exp}))</span> operations per iteration without losing any accuracy compared to the direct solvers, where <span>(N_{exp}ll M)</span> is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in <span>({mathcal {O}}(MN_{exp}))</span> memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"2 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fractional Wiener chaos: Part 1 分数维纳混沌第 1 部分
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-08 DOI: 10.1007/s13540-024-00343-8
Elena Boguslavskaya, Elina Shishkina
{"title":"Fractional Wiener chaos: Part 1","authors":"Elena Boguslavskaya, Elina Shishkina","doi":"10.1007/s13540-024-00343-8","DOIUrl":"https://doi.org/10.1007/s13540-024-00343-8","url":null,"abstract":"<p>In this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as <span>({mathcal {H}}_alpha (x,y))</span>, referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, <span>({mathcal {H}}_alpha (W_t,t))</span>, demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"53 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142385848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Variable-order fractional 1-Laplacian diffusion equations for multiplicative noise removal 用于消除乘法噪声的变阶分数 1-Laplacian 扩散方程
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-08 DOI: 10.1007/s13540-024-00345-6
Yuhang Li, Zhichang Guo, Jingfeng Shao, Yao Li, Boying Wu
{"title":"Variable-order fractional 1-Laplacian diffusion equations for multiplicative noise removal","authors":"Yuhang Li, Zhichang Guo, Jingfeng Shao, Yao Li, Boying Wu","doi":"10.1007/s13540-024-00345-6","DOIUrl":"https://doi.org/10.1007/s13540-024-00345-6","url":null,"abstract":"<p>This paper deals with a class of fractional 1-Laplacian diffusion equations with variable orders, proposed as a model for removing multiplicative noise in images. The well-posedness of weak solutions to the proposed model is proved. To overcome the essential difficulties encountered in the approximation process, we place particular emphasis on studying the density properties of the variable-order fractional Sobolev spaces. Numerical experiments demonstrate that our model exhibits favorable performance across the entire image.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"23 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142385746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems 用于计算能源系统中功率流问题解决方案的分数阶派生牛顿-拉斐森方法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-10-08 DOI: 10.1007/s13540-024-00342-9
Francisco Damasceno Freitas, Laice Neves de Oliveira
{"title":"A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems","authors":"Francisco Damasceno Freitas, Laice Neves de Oliveira","doi":"10.1007/s13540-024-00342-9","DOIUrl":"https://doi.org/10.1007/s13540-024-00342-9","url":null,"abstract":"<p>Some nonlinear real-valued equations have no solution in the real number field, and only roots of this nature are of practical interest. However, complex roots associated with the solution may introduce an interpretation of the physical problem analysis. This paper investigates the solution of nonlinear equations exploiting fractional order derivative (FOD) calculus resources. The theory addresses a solver that considers the FOD and Newton-Raphson method. The problem is extended to a multivariate fractional order derivative (MFOD) method so that a set of nonlinear equations can be resolved iteratively. Applications to the computation of the power flow problem in energy systems are used to illustrate some types of equations and solutions. The MFOD uses a numerical limits technique to determine a Jacobian required for the fractional application. This work demonstrates how real-valued nonlinear equations with complex roots can be solved by the MFOD Newton-Raphson approach. The results indicate the potentiality of the method reaches complex-valued solutions despite the iterative process starting with a real-valued guess. The complex-valued results are interpreted considering the connection between the imaginary part of a root and the divergence of the classical Newton-Raphson (CNR) method.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"56 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142385763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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