Fractional Calculus and Applied Analysis最新文献

{"title":"Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order","authors":"Edoardo Proietti Lippi, Caterina Sportelli","doi":"10.1007/s13540-026-00512-x","DOIUrl":"https://doi.org/10.1007/s13540-026-00512-x","url":null,"abstract":"We establish the existence of a ground state solution for the fractional Choquard equation governed by the superposition operator <disp-formula><alternatives><mml:math display=\"block\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>s</mml:mi></mml:msup><mml:mi>u</mml:mi><mml:mspace width=\"0.166667em\"></mml:mspace><mml:mi>d</mml:mi><mml:mi>μ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ int _{[0, 1]} (-varDelta )^s u, dmu (s), $$end{document}</tex-math></alternatives></disp-formula>and in presence of a confining potential. Here, <inline-formula><alternatives><mml:math><mml:mi>μ</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mu $$end{document}</tex-math></alternatives></inline-formula> denotes a signed measure on the interval of fractional exponents [0, 1]. The presence of the superposition operator asks for the problem to be addressed with a special care. However, the wide generality of this setting allows to provide entirely new existence results in several special cases of interest, including, e.g., the mixed operator one <inline-formula><alternatives><mml:math><mml:mrow><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$-varDelta + (-varDelta )^s$$end{document}</tex-math></alternatives></inline-formula>. We point out that the possibility of considering operators “with the wrong sign\" is also a complete novelty.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"59 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642118","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}

{"title":"Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators","authors":"Joel E. Restrepo","doi":"10.1007/s13540-026-00513-w","DOIUrl":"https://doi.org/10.1007/s13540-026-00513-w","url":null,"abstract":"We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642117","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}

{"title":"Fractional scalar products as probes of chaos and fractal dimension","authors":"Octavian Postavaru, Savin Treanţă, Antonela Toma, Bogdan Sebacher","doi":"10.1007/s13540-026-00510-z","DOIUrl":"https://doi.org/10.1007/s13540-026-00510-z","url":null,"abstract":"We develop a fractional-weighted functional-analytic framework for the analysis of chaotic dynamics in which the governing equations remain classical while the geometry of the underlying Hilbert space is modified. Specifically, we introduce a family of fractional scalar products with singular weights derived from the Riemann–Liouville kernel, generating weighted Hilbert spaces that emphasize late-time dynamics and long-term correlations. Within this framework, the fractional parameter &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; plays a dual role by controlling temporal localization in the scalar product and acting as an effective probe of dynamical complexity. By embedding trajectories of the classical Lorenz system into these spaces, we show that the value &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;mml:mo movablelimits=\"true\"&gt;min&lt;/mml:mo&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha _{min }$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; minimizing the normalized fractional norm exhibits a clear nonlinear correlation with the Kaplan–Yorke dimension &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;D&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"italic\"&gt;KY&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$D_{KY}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of the attractor, thereby establishing &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;mml:mo movablelimits=\"true\"&gt;min&lt;/mml:mo&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha _{min }$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; as a functional proxy for fractal complexity without modifying the underlying dynamics. To support analysis and computation, we construct orthogonal and complete basis systems adapted to the fractional geometry, including weighted Gram–Schmidt bases and Jacobi polynomial expansions, which enable efficient spectral approximation of chaotic signals and reveal intrinsic temporal asymmetries not captured by standard &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:m","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"115 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642119","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}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Second-order asymptotics of fractional Gagliardo seminorms as <ns0:math><ns0:mrow><ns0:mi>s</ns0:mi> <ns0:mo>→</ns0:mo> <ns0:msup><ns0:mn>1</ns0:mn> <ns0:mo>-</ns0:mo></ns0:msup> </ns0:mrow> </ns0:math> and convergence of the associated gradient flows.","authors":"Andrea Kubin, Valerio Pagliari, Antonio Tribuzio","doi":"10.1007/s13540-025-00472-8","DOIUrl":"https://doi.org/10.1007/s13540-025-00472-8","url":null,"abstract":"<p><p>We study the second-order asymptotic expansion of the <i>s</i>-fractional Gagliardo seminorm as <math><mrow><mi>s</mi> <mo>→</mo> <msup><mn>1</mn> <mo>-</mo></msup> </mrow> </math> in terms of a higher-order nonlocal functional. We prove a Mosco-convergence result for the energy functionals and characterize the domain of the limit, which can be interpreted as the space of functions of differentiability order \"logarithmically larger\" than one. This is also motivated by the fact that the first variation of the limit is the formal derivative of the (scaled) <i>s</i>-fractional Laplacian as <math><mrow><mi>s</mi> <mo>→</mo> <msup><mn>1</mn> <mo>-</mo></msup> </mrow> </math> . In the second part of the paper, we show that the <math><msup><mi>L</mi> <mn>2</mn></msup> </math> -gradient flows of the energies converge to the <math><msup><mi>L</mi> <mn>2</mn></msup> </math> -gradient flows of the Mosco-limit.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"29 1","pages":"101-130"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12890977/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146182832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Renormalized solutions for a non-local evolution equation with variable exponent","authors":"Le Xuan Truong, Nguyen Thanh Long, Nguyen Ngoc Trong, Tan Duc Do","doi":"10.1007/s13540-025-00425-1","DOIUrl":"https://doi.org/10.1007/s13540-025-00425-1","url":null,"abstract":"<p>We establish the existence and uniqueness of a renormalized solution to an evolution equation featuring the non-local fractional <i>p</i>(<i>x</i>, <i>y</i>)-Laplacian and nonnegative <span>(L^1)</span>-data. The definition of renormalized solutions is adapted to the non-local nature to bypass the use of chain rules which is unavailable. The fractional <i>p</i>(<i>x</i>, <i>y</i>)-Laplacian well encapsulates the fractional <i>p</i>-Laplacian with a constant exponent <i>p</i>. Hence our result extends [25] to the setting of variable exponents.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"33 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144123065","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}

{"title":"Approximate solutions for fractional stochastic integro-differential equation with short memory driven by non-instantaneous impulses","authors":"Surendra Kumar, Paras Sharma","doi":"10.1007/s13540-025-00415-3","DOIUrl":"https://doi.org/10.1007/s13540-025-00415-3","url":null,"abstract":"<p>The current study discusses the approximate solutions for a class of fractional stochastic integro-differential equation with short memory driven by non-instantaneous impulses (NIIs) defined on a separable Hilbert space. The approximation to the nonlinear functions is obtained using orthogonal projection operator. The existence and convergence of the sequence of approximate solutions is proved using a fixed point theorem and analytic semigroup theory. Moreover, we show that finite-dimensional approximations converge, guaranteeing both computational feasibility and theoretical soundness. This study emphasises on short-memory systems, which are very significant for modelling fading memory effects. We demonstrate the practical importance and versatility of our method by applying it on fractional stochastic Burgers’ and subdiffusion equations.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"129 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144104730","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}

{"title":"On the three dimensional generalized Navier-Stokes equations with damping","authors":"Nguyen Thi Le, Le Tran Tinh","doi":"10.1007/s13540-025-00421-5","DOIUrl":"https://doi.org/10.1007/s13540-025-00421-5","url":null,"abstract":"<p>In this paper, we consider the long time behavior of solutions of the three dimensional (3D) generalized Navier-Stokes equations with damping. This family of 3D generalized Navier-Stokes equations with damping can be viewed as an interpolation model between subcritical (if <span>(alpha &gt;frac{5}{4})</span>), critical (if <span>(alpha =frac{5}{4})</span>), and supercritical dissipations (if <span>(alpha &lt;frac{5}{4})</span>) and it may reduce to many models by varying the parameters. First, in a periodic bounded domain, we study the existence and uniqueness of weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since our system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu called the evolutionary system to obtain various attractors and their properties. Moreover, the determining wavenumbers are also investigated here and this is the first result for a fractional equation.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"40 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144097505","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}

{"title":"Exponential sampling type neural network Kantorovich operators based on Hadamard fractional integral","authors":"Purshottam N. Agrawal, Behar Baxhaku","doi":"10.1007/s13540-025-00418-0","DOIUrl":"https://doi.org/10.1007/s13540-025-00418-0","url":null,"abstract":"<p>This study introduces a novel family of exponential sampling type neural network Kantorovich operators, leveraging Hadamard fractional integrals to significantly enhance function approximation capabilities. By incorporating a flexible parameter <span>(alpha )</span>, derived from fractional Hadamard integrals, and utilizing exponential sampling, introduced to tackle exponentially sampled data, our operators address critical limitations of existing methods, providing substantial improvements in approximation accuracy. We establish fundamental convergence theorems for continuous functions and demonstrate effectiveness in <i>p</i>th Lebesgue integrable spaces. Approximation degrees are quantified using logarithmic moduli of continuity, asymptotic expansions, and Peetre’s <i>K</i>-functional for <i>r</i>-times continuously differentiable functions. A Voronovskaja-type theorem confirms higher-order convergence via linear combinations. Extensions to multivariate cases are proven for convergence in <span>({L}_{{p}})</span>-spaces <span>((1le {p}&lt;infty ).)</span> MATLAB algorithms and illustrative examples validate theoretical findings, confirming convergence, computational efficiency, and operator consistency. We analyze the impact of various sigmoidal activation functions on approximation errors, presented via tables and graphs for one and two-dimensional cases. To demonstrate practical utility, we apply these operators to image scaling, focusing on the “Butterfly” dataset. With fractional parameter <span>(alpha =2)</span>, our operators, activated by a parametric sigmoid function, consistently outperform standard interpolation methods. Significant improvements in Structural Similarity Index Measure (SSIM) and Peak Signal-to-Noise Ratio (PSNR) are observed at <span>({m}=128)</span>, highlighting the operators’ efficacy in preserving image quality during upscaling. These results, combining theoretical rigor, computational validation, and practical application to image scaling, showcase the performance advantage of our proposed operators. By integrating fractional calculus and neural network theory, this work advances constructive approximation and image processing.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"3 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143940082","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}

{"title":"Stability analysis of Hilfer fractional stochastic switched dynamical systems with non-Gaussian process and impulsive effects","authors":"Rajesh Dhayal, Quanxin Zhu","doi":"10.1007/s13540-025-00420-6","DOIUrl":"https://doi.org/10.1007/s13540-025-00420-6","url":null,"abstract":"<p>This paper is devoted to exploring a new class of Hilfer fractional stochastic switched dynamical systems with the Rosenblatt process and abrupt changes, where the abrupt changes occur suddenly at specific points and extend over finite time intervals. Initially, we established solvability outcomes for the proposed switched dynamical systems by employing the fractional calculus, fixed point method, and Mittag-Leffler function. Moreover, we derived the Ulam-Hyers stability criteria for considered switched dynamical systems. Finally, we provide an example to illustrate the obtained results.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"118 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143927345","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}

{"title":"On the multivariate generalized counting process and its time-changed variants","authors":"Kuldeep Kumar Kataria, Manisha Dhillon","doi":"10.1007/s13540-025-00419-z","DOIUrl":"https://doi.org/10.1007/s13540-025-00419-z","url":null,"abstract":"<p>In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator <i>etc.</i> Several distributional properties that include the probability generating function, probability mass function and their governing differential equations are obtained for these variants. It is shown that some of these time-changed processes are Lévy and for such processes we derived the associated Lévy measure. The explicit expressions for the covariance and codifference of the component processes for some of these time-changed variants are obtained. An application of the multivariate generalized space fractional counting process to a shock model is discussed.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"35 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920901","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}