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A collection of correct fractional calculus for discontinuous functions 一个关于不连续函数的正确分数微积分的集合
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-02 DOI: 10.1007/s13540-024-00356-3
Tian Feng, YangQuan Chen
{"title":"A collection of correct fractional calculus for discontinuous functions","authors":"Tian Feng, YangQuan Chen","doi":"10.1007/s13540-024-00356-3","DOIUrl":"https://doi.org/10.1007/s13540-024-00356-3","url":null,"abstract":"<p>In this paper, an important property of fractional order operators involving discontinuous functions is discussed, First, a pioneering work of impulsive fractional differential equations is recalled to illuminate the incorrectness of notation <span>({^C_{t_k}D}^{q}_t)</span>. Second, a class of piecewise-defined equations with Caputo fractional derivative is contrastively investigated, and it is revealed that the additivity of integration on intervals for integer-order integral does not hold for fractional integrals, not to mention fractional derivatives. Third, by utilizing the Heaviside step function, an interesting property of fractional integral involving piecewise-defined functions is correspondingly presented. Finally, illustrative examples are given for validation of the derived results, which may lead to a new way to reconsider the dynamic behavior of fractional hybrid systems, as well as discontinuous control design for fractional systems.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142760665","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 semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions 受均质 Dirichlet 边界条件约束的半线性扩散 PDE 与变阶时间分数 Caputo 导数
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-11-18 DOI: 10.1007/s13540-024-00352-7
Marian Slodička
{"title":"A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions","authors":"Marian Slodička","doi":"10.1007/s13540-024-00352-7","DOIUrl":"https://doi.org/10.1007/s13540-024-00352-7","url":null,"abstract":"<p>We investigate a semilinear problem for a fractional diffusion equation with variable order Caputo fractional derivative <span>(left( partial _t^{beta (t)} uright) (t))</span> subject to homogeneous Dirichlet boundary conditions. The right-hand side of the governing PDE is nonlinear (Lipschitz continuous) and it contains a weakly singular Volterra operator. The whole process takes place in a bounded Lipschitz domain in <span>({{mathbb {R}}}^d)</span>. We establish the existence of a unique solution in <span>(Cleft( [0,T],L^{2} (varOmega )right) )</span> if <span>(u_0in L^{2} (varOmega ))</span>. Moreover, if <span>(mathcal {L}^{gamma }u_0in L^{2} (varOmega ))</span> for some <span>(0&lt;gamma &lt;1-frac{delta }{beta (0)})</span> (<span>(delta )</span> depends on the right-hand-side of the PDE) then <span>(mathcal {L}^{gamma }uin Cleft( {[}0,T{]},L^{2} (varOmega )right) )</span>.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"177 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142670562","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
Global existence, uniqueness and $$L^{infty }$$ -bound of weak solutions of fractional time-space Keller-Segel system 分数时空凯勒-西格尔系统弱解的全局存在性、唯一性和 $$L^{infty }$ -bound
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-11-15 DOI: 10.1007/s13540-024-00353-6
Fei Gao, Liujie Guo, Xinyi Xie, Hui Zhan
{"title":"Global existence, uniqueness and $$L^{infty }$$ -bound of weak solutions of fractional time-space Keller-Segel system","authors":"Fei Gao, Liujie Guo, Xinyi Xie, Hui Zhan","doi":"10.1007/s13540-024-00353-6","DOIUrl":"https://doi.org/10.1007/s13540-024-00353-6","url":null,"abstract":"<p>This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in <span>({mathbb {R}}^{n})</span>, <span>(nge 2)</span>. The global existence and <span>(L^{infty })</span>-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) <span>(b&gt;1-frac{alpha }{n})</span>, for any initial value and birth rate; (ii) <span>(0&lt;ble 1-frac{alpha }{n})</span>, for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the <span>(L^{infty })</span>-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"24 1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spatial $$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
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