Fractional Calculus and Applied Analysis最新文献

{"title":"Fractal approximation via Phillips-type (p, q)-Bernstein operators","authors":"Ibrahim Islam, Naved Alam, Aaqib Ayoub Bhat, Asif Khan","doi":"10.1007/s13540-026-00536-3","DOIUrl":"https://doi.org/10.1007/s13540-026-00536-3","url":null,"abstract":"This paper presents a new class of <inline-formula><alternatives><mml:math><mml:mi>α</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}$$alpha $$end{document}</tex-math></alternatives></inline-formula>-fractal functions generated using Phillips-type (<italic>p</italic>, <italic>q</italic>)-Bernstein operators as base functions. For a continuous function <italic>f</italic> on a compact interval and a nonzero scaling vector <inline-formula><alternatives><mml:math><mml:mi>α</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}$$alpha $$end{document}</tex-math></alternatives></inline-formula>, a sequence <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">}</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}$${f_n^{(p,q,alpha )}}$$end{document}</tex-math></alternatives></inline-formula> of <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</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}$$(p,q,alpha )$$end{document}</tex-math></alternatives></inline-formula>-fractal approximants is constructed, which converges uniformly to <italic>f</italic> without the conventional requirement of vanishing scaling vectors. The study also explores <inline-formula><alternatives><mml:math><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</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}$$(p,q,alpha )$$end{document}</tex-math></alternatives></inline-formula>-fractal functions within Hölder spaces, confirming the existence, ","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"15 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687769","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 a class of time-fractional diffusion equations with exponential nonlinearity involving the ψdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$psi $$end{document}-Caputo fractional derivative and the ϕdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$phi $$end{document}-fractional Laplacian","authors":"Yacine Arioua, Brahim Benaissi","doi":"10.1007/s13540-026-00550-5","DOIUrl":"https://doi.org/10.1007/s13540-026-00550-5","url":null,"abstract":"This paper investigates the local existence of solutions for a class of <inline-formula><alternatives><mml:math><mml:mi>d</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}$$d$$end{document}</tex-math></alternatives></inline-formula>-dimensional nonlinear time-fractional diffusion equations involving the <inline-formula><alternatives><mml:math><mml:mi>ψ</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}$$psi $$end{document}</tex-math></alternatives></inline-formula>-Caputo fractional derivative and the <inline-formula><alternatives><mml:math><mml:mi>ϕ</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}$$phi $$end{document}</tex-math></alternatives></inline-formula>-fractional Laplacian. By means of the generalized Laplace transform, the Fourier transform with respect to <inline-formula><alternatives><mml:math><mml:mi>ϕ</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}$$phi $$end{document}</tex-math></alternatives></inline-formula>, and Fox <inline-formula><alternatives><mml:math><mml:mi>H</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}$$H$$end{document}</tex-math></alternatives></inline-formula>-functions, we derive the fundamental solution for the corresponding linear problem. Moreover, the local existence and uniqueness of mild solutions are rigorously established for the associated nonlinear problem.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"82 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687959","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":"Separation method of semi-fixed variables for solving a generalized time-fractional reaction-diffusion equation","authors":"Weiguo Rui, Chaoling Li, Weijun He","doi":"10.1007/s13540-026-00549-y","DOIUrl":"https://doi.org/10.1007/s13540-026-00549-y","url":null,"abstract":"It is well known that investigations on exact solutions of nonlinear fractional partial differential equations (PDEs) are very difficult compare with those investigations on integer-order nonlinear PDEs. In this paper, a new method called the separation method of semi-fixed variables together with the dynamical system method is introduced. The connection and difference between the classical separation method of variables and the separation method of semi-fixed variables are compared in details. As example, a generalized nonlinear time-fractional reaction-diffusion equation with higher-order terms is studied under the definition of Riemann-Liouville fractional derivative. In different parametric regions, different kinds of phase portraits of the system derived from the generalized model are presented. Existence and dynamic properties of solutions of the generalized model are investigated. In some special parametric conditions, some exact solutions of the generalized equation are obtained. In the absence of the reaction term, a very strange and interesting phenomenon of the model is found.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"26 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687960","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":"Lie symmetry analysis of two–dimensional fractional integro–differential equations","authors":"Soheila Pashayi, Sedaghat Shahmorad","doi":"10.1007/s13540-026-00545-2","DOIUrl":"https://doi.org/10.1007/s13540-026-00545-2","url":null,"abstract":"A symmetry group analysis is studied for solving 2D-fractional integro-differential equations (2D-FIDEs) based on solving equations with nonlocal structures. A new prolongation formula is presented for 2D-FIDFs. The resulting determining equations which contain fractional derivatives and double integrals are solved for the first time in this paper. Some illustrative examples are given to clarify the method.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"46 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687963","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":"The impact of Richardson extrapolation on the asymptotic global error estimation of uniform L2-1σdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$text {L2-1}_sigma $$end{document} scheme for subdiffusion equation with multiple delays","authors":"Vladimir G. Pimenov, Ahmed S. Hendy, Mohammad Shahbazi Asl, Anatoly A. Alikhanov","doi":"10.1007/s13540-026-00542-5","DOIUrl":"https://doi.org/10.1007/s13540-026-00542-5","url":null,"abstract":"A numerical algorithm with convergence order &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;O&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;Δ&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;3&lt;/mml:mn&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:msup&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;h&lt;/mml:mi&gt;&lt;mml:mn&gt;4&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&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}$$O(varDelta ^{3-alpha }+h^{4})$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is developed for solving subdiffusion equations with multiple time delays, where &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi&gt;Δ&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}$$varDelta $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;italic&gt;h&lt;/italic&gt; denote the temporal and spatial step sizes, respectively. The proposed approach is primarily designed to investigate and demonstrate the improvement produced by applying Richardson extrapolation to the uniform L2-&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mi&gt;σ&lt;/mml:mi&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}$$1_{sigma }$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; scheme. Starting from the basic L2-&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mi&gt;σ&lt;/mml:mi&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}$$1_{sigma }$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; method, which conventionally provides second-order temporal convergence, we first derive the asymptotic expansion of its global error. Based on this expansion, a Richardson extrapolation technique in the time direction is constructed to enhance the temporal accuracy. Auxiliary lemmas and theorems are established and proved to characterize the asymptotic behavior of the global error of the uniform L2-&lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mi&gt;σ&lt;/mml:mi&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}$$1_{sigma }$$end{doc","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"37 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687995","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":"Asymptotic synchronization for Riemann-Liouville fractional-order delayed BAM neural networks by the exponential-type controllers","authors":"Junli You, Zhengqiu Zhang","doi":"10.1007/s13540-026-00543-4","DOIUrl":"https://doi.org/10.1007/s13540-026-00543-4","url":null,"abstract":"This paper considers the globally asymptotic synchronization (GAS) of drive-response Riemann-Liouville fractional-order BAM neural networks (DRRLFOBAMNNS). By using the integral inequality techniques, L’Hospital rule and integral mean-valued theorem, two new criteria are derived assuring the GAS for discussed DRRLFOBAMNNS under the exponential-type controllers &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;t&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:msup&gt;&lt;/mml:msup&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}$$e^{t^{alpha _1}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;t&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;β&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:msup&gt;&lt;/mml:msup&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}$$e^{t^{beta _1}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. The exponential-type controllers play the key role in studying GAS for the considered systems. The exponential-type controllers &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;t&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;α&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:msup&gt;&lt;/mml:msup&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}$$e^{t^{alpha _1}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;e&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;t&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;β&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:msup&gt;&lt;/mml:msup&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}$$e^{t^{beta _1}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; designed are used to study the synchronization is a novel work. L’Hospital rule together with integral mean-valued theorem is applied to researching the synchronization for DRRLFOBAMNNS is a novel study work. Until now, the parameters in the controllers all are assumed to be negative numbers in the study of synchronization for neural networks (NNS), however, the parameters in our controllers in the paper are assumed to all be positive, which extends the valued-field of the parameters and makes the limits of the error variables be all ze","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"26 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687768","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":"An inverse source problem for nonlocal Fokker-Planck equations involving nonlinear force fields","authors":"Tran Dinh Ke, Dang Thi Phuong Thanh","doi":"10.1007/s13540-026-00528-3","DOIUrl":"https://doi.org/10.1007/s13540-026-00528-3","url":null,"abstract":"We study a source identification problem for a class of nonlocal Fokker-Planck equations, where the force field depends on the state function. Our aim is to find some suitable conditions ensuring the solvability of the problem and the Hölder regularity of solutions. The main ingredient of our analysis is the regularity estimates, established for resolvent operators in Hilbert scales, that allow us to employ the fixed point argument. Some specific cases of kernel function are discussed.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"34 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355587","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":"Uncertain differential equations driven by fractional Liu process","authors":"Zhi Li, Liping Xu","doi":"10.1007/s13540-026-00533-6","DOIUrl":"https://doi.org/10.1007/s13540-026-00533-6","url":null,"abstract":"In this paper, we focus on a class of uncertain differential equations driven by fractional Liu process, which is the twin of fractional Brownian motion in the structure of uncertain theory. By the Carathéodory approximation, we prove the existence and uniqueness of solutions for the considered equations under some non-Lipschitz conditions. Subsequently, we consider the stability properties with respect to initial data and coefficients by establishing some Bihari type fractional inequalities, generalizing some known results. Finally, an example is provided to illustrate the effectiveness of our main results.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"7 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355725","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":"Existence results for a variable order Hadamard fractional problem under weak topology features","authors":"Amor Fahem, Aref Jeribi, Kadda Maazouz, Moussa Daif allah Zaak","doi":"10.1007/s13540-026-00521-w","DOIUrl":"https://doi.org/10.1007/s13540-026-00521-w","url":null,"abstract":"The objective of this research is to establish specific adequate conditions aimed to guarantee the existence of weak solutions for various types of initial value problems, which involve variable order Hadamard fractional derivative. This is accomplished by employing Krasnoselskii’s type fixed point theorem and De Blasi measure of weak noncompactness. Furthermore, an illustrative example is presented to showcase our results.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"34 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355625","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":"Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity","authors":"Kaushik Bal, Stuti Das","doi":"10.1007/s13540-026-00526-5","DOIUrl":"https://doi.org/10.1007/s13540-026-00526-5","url":null,"abstract":"We prove the multiplicity of solutions for the mixed local-nonlocal elliptic equation of the form &lt;disp-formula&gt;&lt;alternatives&gt;&lt;mml:math display=\"block\"&gt;&lt;mml:mrow&gt;&lt;mml:mtable&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mrow&gt;&lt;mml:mtable&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mrow&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;Δ&lt;/mml:mi&gt;&lt;mml:mi&gt;p&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:msubsup&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mi&gt;Δ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;p&lt;/mml:mi&gt;&lt;mml:mi&gt;s&lt;/mml:mi&gt;&lt;/mml:msubsup&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;mml:mtd columnalign=\"left\"&gt;&lt;mml:mrow&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:mfrac&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;mml:mi&gt;γ&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;/mml:mfrac&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mtext&gt;in&lt;/mml:mtext&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mi&gt;Ω&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;/mml:mtr&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;mml:mtd columnalign=\"left\"&gt;&lt;mml:mrow&gt;&lt;mml:mo&gt;&gt;&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mtext&gt;in&lt;/mml:mtext&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mi&gt;Ω&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;/mml:mtr&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;mml:mtd columnalign=\"left\"&gt;&lt;mml:mrow&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:mtext&gt;in&lt;/mml:mtext&gt;&lt;mml:mspace width=\"0.333333em\"&gt;&lt;/mml:mspace&gt;&lt;mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;n&lt;/mml:mi&gt;&lt;/mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"true\"&gt;&lt;/mml:mo&gt;&lt;mml:mi&gt;Ω&lt;/mml:mi&gt;&lt;mml:mo&gt;;&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;/mml:mtr&gt;&lt;/mml:mtable&gt;&lt;/mml:mrow&gt;&lt;/mml:mtd&gt;&lt;/mml:mtr&gt;&lt;/mml:mtable&gt;&lt;/mml:mrow&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}$$begin{aligned} begin{aligned} -varDelta _pu+(-varDelta )_p^s u&amp;=frac{lambda }{u^{gamma }}+u^r text{ in } varOmega , u&amp;&gt;0 text { in } varOmega , u&amp;=0 text{ in } mathbb {R}^n backslash varOmega ; end{aligned} end{aligned}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/disp-formula&gt;where &lt;disp-formula&gt;&lt;alternatives&gt;&lt;mml:math display=\"block\"&gt;&lt;mml:mrow&gt;&lt;mml:mtable&gt;&lt;mml:mtr&gt;&lt;mml:mtd columnalign=\"right\"&gt;&lt;mml:mstyle displaystyle=\"true\" scriptlevel=\"0\"&gt;&lt;mml:mrow&gt;&lt;mml:msubsup&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mo&gt;-&lt;/mml:mo&gt;&lt;mml:mi&gt;Δ&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;p&lt;/mml:mi&gt;&lt;mml:mi&gt;s&lt;/mml:mi&gt;&lt;/mml:msubsup&gt;&lt;mml:mi&gt;u&lt;/mml:mi&gt;&lt;mml:mrow&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"22 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148355727","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}