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Fractional Calculus and Applied Analysis最新文献

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Overview of fractional calculus and its computer implementation in Wolfram Mathematica 分数微积分及其在 Wolfram Mathematica 中的计算机实现概述
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-09-11 DOI: 10.1007/s13540-024-00332-x
Oleg Marichev, Elina Shishkina
{"title":"Overview of fractional calculus and its computer implementation in Wolfram Mathematica","authors":"Oleg Marichev, Elina Shishkina","doi":"10.1007/s13540-024-00332-x","DOIUrl":"https://doi.org/10.1007/s13540-024-00332-x","url":null,"abstract":"<p>This survey aims to present various approaches to non-integer integrals and derivatives and their practical implementation within Wolfram Mathematica. It begins by short discussion of historical moments and applications related to fractional calculus. Different methods for handling non-integer powers of differentiation operators are presented, along with generalizations of fractional integrals and derivatives. The survey also delves into the diverse applications of fractional calculus in physics, engineering, medicine, and numerical calculations. Essential details of fractional integro-differentiation implemented in Wolfram Mathematica are highlighted. The Hadamard regularization of Riemann-Liouville operator is utilized as the foundation for creating the arbitrary order of integro-differential operator in Mathematica. The survey describes the application of fractional integro-differentiation to Taylor series expansions near zero using Hadamard regularization and the use of the Meijer <i>G</i>-function for evaluating derivatives of complex orders. We conclude with a discussion on applying fractional integro-differentiation to “differential constants” and provide generic formulas for fractional differentiation. The extensive list of references underscores the vast body of works on fractional calculus.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"52 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142170844","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
Non-confluence for SDEs driven by fractional Brownian motion with Markovian switching 具有马尔可夫切换的分数布朗运动驱动的 SDE 的非汇合问题
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-09-09 DOI: 10.1007/s13540-024-00334-9
Zhi Li, Benchen Huang, Liping Xu
{"title":"Non-confluence for SDEs driven by fractional Brownian motion with Markovian switching","authors":"Zhi Li, Benchen Huang, Liping Xu","doi":"10.1007/s13540-024-00334-9","DOIUrl":"https://doi.org/10.1007/s13540-024-00334-9","url":null,"abstract":"<p>In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter <span>(Hin (1/2,1))</span>. By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present two important corollaries on the non-confluence property by the Poisson equation and <i>M</i>-matrix, respectively, which can verify the non-confluence property more effectively than the general condition. Finally, we provide an example to illustrate the practical usefulness of our theoretical results.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"48 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160425","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
Stepanov-like weighted pseudo S-asymptotically Bloch type periodicity and applications to stochastic evolution equations with fractional Brownian motions 斯捷潘诺夫类加权伪 S-渐近布洛赫型周期性及其在分数布朗运动随机演化方程中的应用
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-09-06 DOI: 10.1007/s13540-024-00333-w
Amadou Diop, Mamadou Moustapha Mbaye, Yong-Kui Chang, Gaston Mandata N’Guérékata
{"title":"Stepanov-like weighted pseudo S-asymptotically Bloch type periodicity and applications to stochastic evolution equations with fractional Brownian motions","authors":"Amadou Diop, Mamadou Moustapha Mbaye, Yong-Kui Chang, Gaston Mandata N’Guérékata","doi":"10.1007/s13540-024-00333-w","DOIUrl":"https://doi.org/10.1007/s13540-024-00333-w","url":null,"abstract":"<p>In this paper, we introduce the concept of Stepanov-like (weighted) pseudo <i>S</i>-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo <i>S</i>-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo <i>S</i>-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"46 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142144251","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
An efficient numerical method to the stochastic fractional heat equation with random coefficients and fractionally integrated multiplicative noise 具有随机系数和分数积分乘法噪声的随机分数热方程的高效数值方法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-09-06 DOI: 10.1007/s13540-024-00335-8
Xiao Qi, Chuanju Xu
{"title":"An efficient numerical method to the stochastic fractional heat equation with random coefficients and fractionally integrated multiplicative noise","authors":"Xiao Qi, Chuanju Xu","doi":"10.1007/s13540-024-00335-8","DOIUrl":"https://doi.org/10.1007/s13540-024-00335-8","url":null,"abstract":"<p>This paper studies the stochastic time-fractional heat diffusion equation involving a Caputo derivative in time of order <span>(alpha in (frac{1}{2},1])</span>, driven simultaneously by a random diffusion coefficient field and fractionally integrated multiplicative noise. First, the well-posedness of the underlying problem is established by proving the existence, uniqueness, and stability of the mild solution. Then a spatio-temporal discretization method based on a Milstein exponential integrator scheme and finite element method is constructed and analyzed. The strong convergence rate of the fully discrete solution is derived. Numerical experiments are finally reported to confirm the theoretical result.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"23 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142144300","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
Dirichlet problems with fractional competing operators and fractional convection 带分数竞争算子和分数对流的 Dirichlet 问题
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-09-04 DOI: 10.1007/s13540-024-00331-y
Laura Gambera, Salvatore Angelo Marano, Dumitru Motreanu
{"title":"Dirichlet problems with fractional competing operators and fractional convection","authors":"Laura Gambera, Salvatore Angelo Marano, Dumitru Motreanu","doi":"10.1007/s13540-024-00331-y","DOIUrl":"https://doi.org/10.1007/s13540-024-00331-y","url":null,"abstract":"<p>In this paper, the existence of weak solutions to some Dirichlet problems with fractional competing operators and distributional Riesz fractional gradient is investigated. Due to the nature of driving operators, the most known techniques, basically based on ellipticity and monotonicity, are no longer applicable. Generalized solutions (in a suitable sense) are obtained via an approximation procedure and a corollary of the Brouwer fixed point theorem.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"11 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142138156","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 calculus for distributions 分布的分数微积分
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-08-29 DOI: 10.1007/s13540-024-00306-z
R. Hilfer, T. Kleiner
{"title":"Fractional calculus for distributions","authors":"R. Hilfer, T. Kleiner","doi":"10.1007/s13540-024-00306-z","DOIUrl":"https://doi.org/10.1007/s13540-024-00306-z","url":null,"abstract":"<p>Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of <img alt=\"\" src=\"//media.springernature.com/lw20/springer-static/image/art%3A10.1007%2Fs13540-024-00306-z/MediaObjects/13540_2024_306_IEq2_HTML.gif\" style=\"width:20px;max-width:none;\"/>-convolution.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"314 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142101136","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
Generalized separation of variable methods with their comparison: exact solutions of time-fractional nonlinear PDEs in higher dimensions 广义变量分离法及其比较：高维度时间分数非线性 PDE 的精确解法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-08-27 DOI: 10.1007/s13540-024-00330-z
P. Prakash, K. S. Priyendhu, R. Sahadevan
{"title":"Generalized separation of variable methods with their comparison: exact solutions of time-fractional nonlinear PDEs in higher dimensions","authors":"P. Prakash, K. S. Priyendhu, R. Sahadevan","doi":"10.1007/s13540-024-00330-z","DOIUrl":"https://doi.org/10.1007/s13540-024-00330-z","url":null,"abstract":"<p>A systematic investigation of the significance and applicability of two different approaches of generalized separation of variable (GSV) methods for time-fractional nonlinear PDEs in <span>((2+1))</span> and <span>((m+1))</span>-dimensions is presented. Also, this work explicitly shows that while constructing the exact solutions of time-fractional nonlinear PDEs in <span>((2+1))</span> and <span>((m+1))</span>-dimensions without and with delay terms, how to overcome unusual (non-standard) properties of singular kernel fractional derivatives such as chain rule, semigroup property, and the Leibniz rule. Moreover, the importance and effectiveness of the two GSV methods have been discussed through the initial and boundary value problems of the time-fractional nonlinear generalized convection-diffusion equation in <span>((2+1))</span>-dimensions. Additionally, the discussed methods extended to find the exact solutions of time-fractional nonlinear PDEs in <span>((2+1))</span> and <span>((m+1))</span>-dimensions involving multiple linear time-delay terms along with appropriate examples. Also, this work investigates the comparative study of the obtained results and solutions of the underlying equations using the two GSV methods, along with the 2D and 3D graphical representations.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"51 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084947","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
Parameter identification in anomalous diffusion equations with nonlocal conditions and weak-valued nonlinearities 具有非局部条件和弱值非线性的反常扩散方程中的参数识别
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-08-26 DOI: 10.1007/s13540-024-00329-6
Nguyen Thi Van Anh, Bui Thi Hai Yen
{"title":"Parameter identification in anomalous diffusion equations with nonlocal conditions and weak-valued nonlinearities","authors":"Nguyen Thi Van Anh, Bui Thi Hai Yen","doi":"10.1007/s13540-024-00329-6","DOIUrl":"https://doi.org/10.1007/s13540-024-00329-6","url":null,"abstract":"<p>The paper deals with a source identification problem of the anomalous diffusion equations from nonlocal final data observations where the nonlinearity probably takes values in Hilbert scales. The existence and uniqueness results are proved by establishing some estimates for resolvent operators and using the embedding theorems. We also study regularity results for this equation in terms of the Hölder continuity of mild solutions. Finally, the multi-term fractional diffusion equations with polynomial nonlinearities and the ultra-slow diffusions are considered as illustrative applications.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"12 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084950","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
Time-fractional discrete diffusion equation for Schrödinger operator 薛定谔算子的时间分数离散扩散方程
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-08-19 DOI: 10.1007/s13540-024-00323-y
Aparajita Dasgupta, Shyam Swarup Mondal, Michael Ruzhansky, Abhilash Tushir
{"title":"Time-fractional discrete diffusion equation for Schrödinger operator","authors":"Aparajita Dasgupta, Shyam Swarup Mondal, Michael Ruzhansky, Abhilash Tushir","doi":"10.1007/s13540-024-00323-y","DOIUrl":"https://doi.org/10.1007/s13540-024-00323-y","url":null,"abstract":"<p>This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, <span>(mathcal {H}_{hbar ,V}:=-hbar ^{-2}mathcal {L}_{hbar }+V)</span> on the lattice <span>(hbar mathbb {Z}^{n},)</span> where <i>V</i> is a positive multiplication operator and <span>(mathcal {L}_{hbar })</span> is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit <span>(hbar rightarrow 0)</span>.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"89 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142007532","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
Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition 具有非 Lipschitz 条件的随机卡普托分数微分方程的平均原理
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-08-14 DOI: 10.1007/s13540-024-00308-x
Zhongkai Guo, Xiaoying Han, Junhao Hu
{"title":"Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition","authors":"Zhongkai Guo, Xiaoying Han, Junhao Hu","doi":"10.1007/s13540-024-00308-x","DOIUrl":"https://doi.org/10.1007/s13540-024-00308-x","url":null,"abstract":"<p>In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"19 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141986587","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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