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Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces 贝索夫空间中分数卷积椭圆和抛物线算子的定性特性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-21 DOI: 10.1007/s13540-024-00302-3
Veli Shakhmurov, Rishad Shahmurov
{"title":"Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces","authors":"Veli Shakhmurov, Rishad Shahmurov","doi":"10.1007/s13540-024-00302-3","DOIUrl":"https://doi.org/10.1007/s13540-024-00302-3","url":null,"abstract":"<p>The maximal <span>(B_{p,q}^{s})</span>-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in <span>( B_{p,q}^{s})</span> and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the <span>(B_{p,q}^{s})</span>-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"36 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141439862","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 approximation theoretic revamping of fractal interpolation surfaces on triangular domains 三角形域上分形插值面的近似理论翻新
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-18 DOI: 10.1007/s13540-024-00305-0
P. Viswanathan
{"title":"An approximation theoretic revamping of fractal interpolation surfaces on triangular domains","authors":"P. Viswanathan","doi":"10.1007/s13540-024-00305-0","DOIUrl":"https://doi.org/10.1007/s13540-024-00305-0","url":null,"abstract":"<p>The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called <span>(alpha )</span>-fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the <span>(alpha )</span>-fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on <span>({mathcal {C}}(Delta , {mathbb {R}}))</span>, the space of all real-valued continuous functions defined on a triangular domain <span>(Delta )</span>, is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of <span>(alpha )</span>-fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for <span>({mathcal {C}}(Delta , {mathbb {R}}))</span>.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"44 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141425454","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
Identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like hyper-Bessel operator 用卡普托类超贝塞尔算子同时识别时间分形扩散方程的源项和初值
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-17 DOI: 10.1007/s13540-024-00304-1
Fan Yang, Ying Cao, XiaoXiao Li
{"title":"Identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like hyper-Bessel operator","authors":"Fan Yang, Ying Cao, XiaoXiao Li","doi":"10.1007/s13540-024-00304-1","DOIUrl":"https://doi.org/10.1007/s13540-024-00304-1","url":null,"abstract":"<p>In this paper, we consider the inverse problem for identifying the source term and the initial value of time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator. Firstly, we prove that the problem is ill-posed and give the conditional stability result. Then, we choose the Tikhonov regularization method to solve this ill-posed problem, and give the error estimates under a priori and a posteriori regularization parameter selection rules. Finally, we give numerical examples to illustrate the effectiveness of this method.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"47 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141334395","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 high order predictor-corrector method with non-uniform meshes for fractional differential equations 分数微分方程的非均匀网格高阶预测器-校正器方法
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-12 DOI: 10.1007/s13540-024-00303-2
Farzaneh Mokhtarnezhadazar
{"title":"A high order predictor-corrector method with non-uniform meshes for fractional differential equations","authors":"Farzaneh Mokhtarnezhadazar","doi":"10.1007/s13540-024-00303-2","DOIUrl":"https://doi.org/10.1007/s13540-024-00303-2","url":null,"abstract":"<p>This article proposes a predictor-corrector scheme for solving the fractional differential equations <span>({}_0^C D_t^alpha y(t) = f(t,y(t)), alpha &gt;0)</span> with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though <span>({}_0^C D_t^alpha y(t))</span> is not smooth at <span>(t=0)</span>. Numerical examples are carried out to verify the theoretical analysis.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315724","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 difference inequalities for possible Lyapunov functions: a review 可能的 Lyapunov 函数的分数差分不等式：综述
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-12 DOI: 10.1007/s13540-024-00298-w
Yiheng Wei, Linlin Zhao, Xuan Zhao, Jinde Cao
{"title":"Fractional difference inequalities for possible Lyapunov functions: a review","authors":"Yiheng Wei, Linlin Zhao, Xuan Zhao, Jinde Cao","doi":"10.1007/s13540-024-00298-w","DOIUrl":"https://doi.org/10.1007/s13540-024-00298-w","url":null,"abstract":"<p>This study delves into the origin, evolution, and practical applications of fractional difference inequalities based on recent literature. The review provides an overview of existing inequalities proposed under various definitions. Furthermore, to enhance this potent mathematical tool, a series of new inequalities have been introduced. Additionally, leveraging renowned Lyapunov functions in continuous-time domain, their discrete-time counterparts have been formulated. Moreover, several new potential Lyapunov functions have been identified. This review aims to aid readers in selecting suitable inequalities and Lyapunov functions to analyze the stability of nabla fractional order systems.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"24 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315605","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
Continuous-time MISO fractional system identification using higher-order-statistics 利用高阶统计量识别连续时间 MISO 分数系统
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-11 DOI: 10.1007/s13540-024-00297-x
Manel Chetoui, Mohamed Aoun, Rachid Malti
{"title":"Continuous-time MISO fractional system identification using higher-order-statistics","authors":"Manel Chetoui, Mohamed Aoun, Rachid Malti","doi":"10.1007/s13540-024-00297-x","DOIUrl":"https://doi.org/10.1007/s13540-024-00297-x","url":null,"abstract":"<p>In this paper, the problem of identifying Multiple-Input-Single-Output (MISO) systems with fractional models from noisy input-output available data is studied. The proposed idea is to use Higher-Order-Statistics (HOS), like fourth-order cumulants (<i>foc</i>), instead of noisy measurements. Thus, a fractional fourth-order cumulants based-simplified and refined instrumental variable algorithm (<i>frac-foc-sriv</i>) is first developed. Assuming that all differentiation orders are known a priori, it consists in estimating the linear coefficients of all Single-Input-Single-Output (SISO) sub-models composing the MISO model. Then, the <i>frac-foc-sriv</i> algorithm is combined with a nonlinear optimization technique to estimate all the parameters: coefficients and orders. The performances of the developed algorithms are analyzed using numerical examples. Thanks to fourth-order cumulants, which are insensitive to Gaussian noise, and the iterative strategy of the instrumental variable algorithm, the parameters estimation is consistent.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141309188","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 order control for unstable first order processes with time delays 有时间延迟的不稳定一阶过程的分数阶控制
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-10 DOI: 10.1007/s13540-024-00301-4
Cristina I. Muresan, Isabela Birs
{"title":"Fractional order control for unstable first order processes with time delays","authors":"Cristina I. Muresan, Isabela Birs","doi":"10.1007/s13540-024-00301-4","DOIUrl":"https://doi.org/10.1007/s13540-024-00301-4","url":null,"abstract":"<p>Unstable first order time delay systems are frequently encountered in industrial applications, such as chemical plants, hydraulic processes, in satellite communications or economic systems, to name just a few. The control of such processes is a challenging issue. In this paper a filtered Smith Predictor control structure is used to compensate for the process time delays and to ensure the stability of the overall closed loop system. A simplified type of a fractional order PI controller is then designed to meet zero steady state error and an overshoot requirement. The tuning is based on the root locus analysis of the closed loop fractional order system. Simulation examples are provided to validate the proposed method and to demonstrate the efficiency of the proposed control method. Comparisons with two existing methods are included to highlight the possibility of using the proposed method as an alternative solution for controlling these types of processes.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"70 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304492","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
Weighted boundedness of fractional integrals associated with admissible functions on spaces of homogeneous type 同质类型空间上与可接受函数相关的分数积分的加权有界性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-10 DOI: 10.1007/s13540-024-00300-5
Gaigai Qin, Xing Fu
{"title":"Weighted boundedness of fractional integrals associated with admissible functions on spaces of homogeneous type","authors":"Gaigai Qin, Xing Fu","doi":"10.1007/s13540-024-00300-5","DOIUrl":"https://doi.org/10.1007/s13540-024-00300-5","url":null,"abstract":"<p>Let <span>(({{mathcal {X}}},d,mu ))</span> be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral <span>(I_beta )</span> associated with admissible functions and its commutators. Similarly to <span>(I_beta )</span>, corresponding results for Calderón–Zygmund operators <i>T</i> associated with admissible functions are also included in this article.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"26 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304547","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
Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media 多孔介质问题的 Mittag-Leffler 稳定性和 Lyapunov 稳定性
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-05 DOI: 10.1007/s13540-024-00299-9
Jamilu Hashim Hassan, Nasser-eddine Tatar, Banan Al-Homidan
{"title":"Mittag-Leffler stability and Lyapunov stability for a problem arising in porous media","authors":"Jamilu Hashim Hassan, Nasser-eddine Tatar, Banan Al-Homidan","doi":"10.1007/s13540-024-00299-9","DOIUrl":"https://doi.org/10.1007/s13540-024-00299-9","url":null,"abstract":"<p>A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"78 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264811","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 review of constitutive models for non-Newtonian fluids 非牛顿流体构成模型综述
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-06-04 DOI: 10.1007/s13540-024-00294-0
HongGuang Sun, Yuehua Jiang, Yong Zhang, Lijuan Jiang
{"title":"A review of constitutive models for non-Newtonian fluids","authors":"HongGuang Sun, Yuehua Jiang, Yong Zhang, Lijuan Jiang","doi":"10.1007/s13540-024-00294-0","DOIUrl":"https://doi.org/10.1007/s13540-024-00294-0","url":null,"abstract":"<p>Various constitutive models have been proposed to quantify a wide range of non-Newtonian fluids, but there is lack of a systematic classification and evaluation of these competing models, such as the quantitative comparison between the classical integer-order constitutive models and the newly proposed fractional derivative equations for non-Newtonian fluids. This study reviews constitutive equation models for non-Newtonian fluids, including time-independent fluids, viscoelastic fluids, and time-dependent fluids. A comparison between fractional derivative non-Newtonian fluid constitutive equations and traditional constitutive equations is also provided. Results show that the space fractional derivative model is equivalent to some classical constitutive models under reasonable assumptions. Further discussions are made from the perspective of the industrial and biomedical applications of non-Newtonian fluids. Advantages and limitations of the constitutive models are also explored to help users to select proper models for real-world applications.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"51 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251596","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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