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A time-space fractional parabolic type problem: weak, strong and classical solutions
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-12 DOI: 10.1007/s13540-024-00363-4
Dariusz Idczak
{"title":"A time-space fractional parabolic type problem: weak, strong and classical solutions","authors":"Dariusz Idczak","doi":"10.1007/s13540-024-00363-4","DOIUrl":"https://doi.org/10.1007/s13540-024-00363-4","url":null,"abstract":"<p>We use a generalized Riemann-Liouville type derivative of an abstract function of one variable and existence of a weak solution to an abstract fractional parabolic problem on [0, <i>T</i>] containing Riemann-Liouville derivative of a function of one variable and spectral fractional powers of a weak Dirichlet-Laplace operator to study existence of a strong solution to this problem. Our goal in this regard is to provide conditions that allow the transition from a weak to a strong solution. Next, we passage from the abstract problem to a classical one on <span>([0,T]times varOmega )</span>, containing partial (with respect to time <span>(tin [0,T],)</span>) Riemann-Liouville derivative of the unknown real-valued function of two variables and fractional powers of a weak Dirichlet-Laplacian of this function (with respect to spatial variable <span>(xin varOmega )</span>). The most important in this regard is a theorem on the relation of the fractional derivatives of an abstract function of one variable and real-valued one of two variables.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"62 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142816153","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 a mixed partial Caputo derivative and its applications to a hyperbolic partial fractional differential equation
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-10 DOI: 10.1007/s13540-024-00358-1
Rafał Kamocki, Cezary Obczyński
{"title":"On a mixed partial Caputo derivative and its applications to a hyperbolic partial fractional differential equation","authors":"Rafał Kamocki, Cezary Obczyński","doi":"10.1007/s13540-024-00358-1","DOIUrl":"https://doi.org/10.1007/s13540-024-00358-1","url":null,"abstract":"<p>We propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle <span>(P=[0,a]times [0,b])</span> (<span>(a&gt;0, b&gt;0)</span>). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, <i>On the single partial Caputo derivatives for functions of two variables</i>, Periodica Mathematica Hungarica 87(2), (2023), 324–339].</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"12 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142804883","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
Strong stationarity for non-smooth control problems with fractional semi-linear elliptic equations in dimension $$Nle 3$$
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-04 DOI: 10.1007/s13540-024-00359-0
Cyrille Kenne, Gisèle Mophou, Mahamadi Warma
{"title":"Strong stationarity for non-smooth control problems with fractional semi-linear elliptic equations in dimension $$Nle 3$$","authors":"Cyrille Kenne, Gisèle Mophou, Mahamadi Warma","doi":"10.1007/s13540-024-00359-0","DOIUrl":"https://doi.org/10.1007/s13540-024-00359-0","url":null,"abstract":"<p>In this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order <i>s</i> with <span>(sin (0,1))</span>. The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains non-smooth terms. As the mappings involved may not be Gâteaux differentiable, we use a regularization technique to regularize these nonlinear terms, aiming to obtain Gâteaux differentiable mappings. By employing this regularization technique, we are able to derive the first-order optimality condition for the regularized control problem by using the associated adjoint system. Furthermore, we conduct a limit analysis on the regularized term resulting in an optimality system for the non-smooth problem of C-stationary type. Subsequently, we establish a primal optimality condition, specifically B-stationarity. Under the assumption of “constraint qualification”, we derive the strong stationarity conditions for the non-smooth optimization problem with control constraints and establish the equivalence between B-stationarity and strong stationarity conditions.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"10 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142777021","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
Study on the diffusion fractional m-Laplacian with singular potential term
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-04 DOI: 10.1007/s13540-024-00360-7
Wen-Shuo Yuan, Bin Ge, Yu-Hang Han, Qing-Hai Cao
{"title":"Study on the diffusion fractional m-Laplacian with singular potential term","authors":"Wen-Shuo Yuan, Bin Ge, Yu-Hang Han, Qing-Hai Cao","doi":"10.1007/s13540-024-00360-7","DOIUrl":"https://doi.org/10.1007/s13540-024-00360-7","url":null,"abstract":"<p>This paper addresses the questions of well-posedness to fractional <i>m</i>-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity: </p><span>$$begin{aligned} left| xright| ^{-2s}partial _t u+(-varDelta )_{m}^{s} u+ (-varDelta )^{s} partial _t u!=!u|u|^{-2} R(u), end{aligned}$$</span><p>where <span>(R(u)=left| uright| ^{r}ln (|u|))</span>. Guided by the made assumptions, we arrive at the conclusions of the local and global solvability of solutions within the framework of Galerkin approximation. In addition, this study considers weak solutions’ asymptotic stability and explosion in finite time. Significantly, we not only figure out the relationship between the non-local fractional operator and singular potential term, but generalize and improve earlier results in the literature.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"32 6 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776409","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
Hardy–Hénon fractional equation with nonlinearities involving exponential critical growth
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-04 DOI: 10.1007/s13540-024-00361-6
Eudes M. Barboza, Olímpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana
{"title":"Hardy–Hénon fractional equation with nonlinearities involving exponential critical growth","authors":"Eudes M. Barboza, Olímpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana","doi":"10.1007/s13540-024-00361-6","DOIUrl":"https://doi.org/10.1007/s13540-024-00361-6","url":null,"abstract":"<p>In this paper, our goal is to study the following class of Hardy–Hénon type problems </p><span>$$begin{aligned} left{ begin{array}{rclcl}displaystyle (-Delta )^{1/2} u&amp; =&amp; lambda |x|^{mu } u+|x|^{alpha }f(u)&amp; text{ in }&amp; (-1,1), u&amp; =&amp; 0&amp; text{ on }&amp; mathbb {R}setminus (-1,1), end{array}right. end{aligned}$$</span><p>when <span>(mu ge alpha {&gt;-1})</span>, and the nonlinearity <i>f</i> has exponential critical growth in the sense of the Trudinger-Moser inequality. This way, with an appropriate change of variable, due to the behavior of the weight <span>(|x|^{alpha })</span>, one can obtain a version of this inequality in the radial context that allows us to treat the problem with a nonlocal framework and a critical exponent depending on <span>(alpha )</span>. When <span>(alpha &gt;0)</span>, we have a Hénon problem and this exponent becomes larger than usual. This fact is a counterpart for Ni [37] result for the local case and <span>(mathbb {R}^N)</span> (<span>(Nge 3))</span>. If <span>(-1&lt;alpha &lt;0)</span>, we have a Hardy equation. In this case, the exponent is smaller than usual, but we treat a problem with a singularity at 0. The main difficulty is to overcome the lack of compactness inherent to problems involving nonlinearities with critical growth. For this, we apply the variational methods to control the minimax level using Moser functions (see [48]). Then, we guarantee the existence of at least one radial solution under suitable hypotheses for the constants <span>(lambda , mu ,alpha )</span>, as well as, on <i>f</i>, combined with the interaction of the spectrum of the fractional Laplacian operator via the Mountain Pass Theorem or the Linking Theorem (see [41, 42]). Thus, we study a version of problems treated in [3] for nonlinearities with exponential critical growth and in [25] in a nonlocal context.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"25 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776408","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 definition of fractional k-dimensional measure: bridging the gap between fractional length and fractional area
IF 3 2区 数学
Fractional Calculus and Applied Analysis Pub Date : 2024-12-02 DOI: 10.1007/s13540-024-00351-8
Cornelia Mihaila, Brian Seguin
{"title":"A definition of fractional k-dimensional measure: bridging the gap between fractional length and fractional area","authors":"Cornelia Mihaila, Brian Seguin","doi":"10.1007/s13540-024-00351-8","DOIUrl":"https://doi.org/10.1007/s13540-024-00351-8","url":null,"abstract":"<p>Here we introduce a notion of fractional <i>k</i>-dimensional measure, <span>(0le k&lt;n)</span>, that depends on a parameter <span>(sigma )</span> that lies between 0 and 1. When <span>(k=n-1)</span> this coincides with the notions of fractional area and perimeter, and when <span>(k=1)</span> this coincides with the notion of fractional length. It is shown that, when multiplied by the factor <span>(1-sigma )</span>, this <span>(sigma )</span>-measure converges to the <i>k</i>-dimensional Hausdorff measure up to a multiplicative constant that is computed exactly. We also mention several future directions of research that could be pursued using the fractional measure introduced.</p>","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"46 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142760543","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 collection of correct fractional calculus for discontinuous functions
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
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