Pub Date : 2023-12-15DOI: 10.3390/fractalfract7120885
Victor Manuel Garcia-de-los-Rios, Jose Alberto Arano-Martínez, M. Trejo-Valdez, Martha Leticia Hernández-Pichardo, M. A. Vidales-Hurtado, C. Torres‐Torres
A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize ZnO micro-sized crystals, while a spray pyrolysis technique was employed to prepare ZnO nanostructures. A contrast in the fractional electrical behavior and photoconductivity was identified for the samples studied. A positive nonlinear refractive index was measured on the nanoscale sample using the z-scan technique, which endows it with a dominant real part for the third-order optical nonlinearity. The absence of nonlinear optical absorption, along with a strong optical Kerr effect in the ZnO nanostructures, shows favorable perspectives for their potential use in the development of all-optical switching devices. Fractional models for predicting electronic and nonlinear interactions in nanosystems could pave the way for the development of optoelectronic circuits and ultrafast functions controlled by ZnO photo technology.
本文研究了氧化锌纳米材料中产生导电和多光子效应的光学诱导机制。分析了纯微型和纳米结构氧化锌样品表现出的光电导、电和非线性光学现象。研究采用水热法合成氧化锌微小晶体，同时采用喷雾热解技术制备氧化锌纳米结构。所研究的样品在分数电行为和光电导性方面形成了鲜明对比。使用 Z 扫描技术测量了纳米级样品的正非线性折射率，这使其具有三阶光学非线性的主要实部。氧化锌纳米结构中不存在非线性光吸收以及强烈的光学克尔效应，这为其在全光开关器件开发中的潜在应用提供了有利的前景。用于预测纳米系统中电子和非线性相互作用的分数模型可为开发由氧化锌光电技术控制的光电电路和超快功能铺平道路。
{"title":"Fractional Photoconduction and Nonlinear Optical Behavior in ZnO Micro and Nanostructures","authors":"Victor Manuel Garcia-de-los-Rios, Jose Alberto Arano-Martínez, M. Trejo-Valdez, Martha Leticia Hernández-Pichardo, M. A. Vidales-Hurtado, C. Torres‐Torres","doi":"10.3390/fractalfract7120885","DOIUrl":"https://doi.org/10.3390/fractalfract7120885","url":null,"abstract":"A fractional description for the optically induced mechanisms responsible for conductivity and multiphotonic effects in ZnO nanomaterials is studied here. Photoconductive, electrical, and nonlinear optical phenomena exhibited by pure micro and nanostructured ZnO samples were analyzed. A hydrothermal approach was used to synthetize ZnO micro-sized crystals, while a spray pyrolysis technique was employed to prepare ZnO nanostructures. A contrast in the fractional electrical behavior and photoconductivity was identified for the samples studied. A positive nonlinear refractive index was measured on the nanoscale sample using the z-scan technique, which endows it with a dominant real part for the third-order optical nonlinearity. The absence of nonlinear optical absorption, along with a strong optical Kerr effect in the ZnO nanostructures, shows favorable perspectives for their potential use in the development of all-optical switching devices. Fractional models for predicting electronic and nonlinear interactions in nanosystems could pave the way for the development of optoelectronic circuits and ultrafast functions controlled by ZnO photo technology.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"86 5","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138999545","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}
Pub Date : 2023-12-14DOI: 10.3390/fractalfract7120883
F. Tawfiq, F. Tchier, Luminița-Ioana Cotîrlă
In this study, we begin by examining the τ-fractional differintegral operator and proceed to establish a novel subclass in the open unit disk E. The determination of the nth coefficient bound for functions within this recently established class is accomplished by the use of the Faber polynomial expansion approach. Additionally, we examine the behavior of the initial coefficients of bi-close-to-convex functions defined by the τ-fractional differintegral operator, which may exhibit unexpected reactions. We established connections between our current research and prior studies in order to validate our significant findings.
在本研究中，我们首先研究了τ-分数差分算子，并进而在开放单位盘 E 中建立了一个新的子类。此外，我们还研究了由τ-分数差分算子定义的双接近凸函数的初始系数行为，它们可能会表现出意想不到的反应。我们建立了当前研究与先前研究之间的联系，以验证我们的重要发现。
{"title":"Faber Polynomial Coefficient Inequalities for a Subclass of Bi-Close-To-Convex Functions Associated with Fractional Differential Operator","authors":"F. Tawfiq, F. Tchier, Luminița-Ioana Cotîrlă","doi":"10.3390/fractalfract7120883","DOIUrl":"https://doi.org/10.3390/fractalfract7120883","url":null,"abstract":"In this study, we begin by examining the τ-fractional differintegral operator and proceed to establish a novel subclass in the open unit disk E. The determination of the nth coefficient bound for functions within this recently established class is accomplished by the use of the Faber polynomial expansion approach. Additionally, we examine the behavior of the initial coefficients of bi-close-to-convex functions defined by the τ-fractional differintegral operator, which may exhibit unexpected reactions. We established connections between our current research and prior studies in order to validate our significant findings.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"44 11","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138974626","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}
Pub Date : 2023-12-14DOI: 10.3390/fractalfract7120884
Y. Alruwaily, Kuppusamy Venkatachalam, El-sayed El-hady
The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements.
{"title":"Some Results on Fractional Boundary Value Problem for Caputo-Hadamard Fractional Impulsive Integro Differential Equations","authors":"Y. Alruwaily, Kuppusamy Venkatachalam, El-sayed El-hady","doi":"10.3390/fractalfract7120884","DOIUrl":"https://doi.org/10.3390/fractalfract7120884","url":null,"abstract":"The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"180 3","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139002339","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}
Pub Date : 2023-12-13DOI: 10.3390/fractalfract7120882
Mehboob Alam, S. Haq, Ihteram Ali, M. Ebadi, S. Salahshour
In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.
{"title":"Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation","authors":"Mehboob Alam, S. Haq, Ihteram Ali, M. Ebadi, S. Salahshour","doi":"10.3390/fractalfract7120882","DOIUrl":"https://doi.org/10.3390/fractalfract7120882","url":null,"abstract":"In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"82 9","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139005718","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}
Pub Date : 2023-12-13DOI: 10.3390/fractalfract7120881
Ivo Petráš
This paper presents an innovative Mittag–Leffler two-dimensional filter and its application in image processing. The proposed filter leverages the utilization of a Mittag–Leffler function within the probability density function. It introduces three adjustable filter parameters that enable the manipulation of the curve shape and the filter’s forgetting factor. Moreover, a two-dimensional Mittag–Leffler distribution was defined and used for the first time in an image filter. By conducting a comparative analysis against conventional filtering techniques, the paper showcases the distinct advantages of the proposed filter through illustrative examples. Additionally, the paper provides detailed implementation explanations and presents the Matlab function corresponding to the proposed two-dimensional filter.
{"title":"Novel Low-Pass Two-Dimensional Mittag–Leffler Filter and Its Application in Image Processing","authors":"Ivo Petráš","doi":"10.3390/fractalfract7120881","DOIUrl":"https://doi.org/10.3390/fractalfract7120881","url":null,"abstract":"This paper presents an innovative Mittag–Leffler two-dimensional filter and its application in image processing. The proposed filter leverages the utilization of a Mittag–Leffler function within the probability density function. It introduces three adjustable filter parameters that enable the manipulation of the curve shape and the filter’s forgetting factor. Moreover, a two-dimensional Mittag–Leffler distribution was defined and used for the first time in an image filter. By conducting a comparative analysis against conventional filtering techniques, the paper showcases the distinct advantages of the proposed filter through illustrative examples. Additionally, the paper provides detailed implementation explanations and presents the Matlab function corresponding to the proposed two-dimensional filter.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"55 5","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139004200","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}
Pub Date : 2023-12-12DOI: 10.3390/fractalfract7120880
Yinfeng Wang, Jilong Lu, Ying Shi, Ning Wang, Liguo Han
The propagation of Rayleigh waves is usually accompanied by dispersion, which becomes more complex with inherent attenuation. The accurate simulation of Rayleigh waves in attenuation media is crucial for understanding wave mechanisms, layer thickness identification, and parameter inversion. Although the vacuum formalism or stress image method (SIM) combined with the generalized standard linear solid (GSLS) is widely used to implement the numerical simulation of Rayleigh waves in attenuation media, this type of method still has its limitations. First, the GSLS model cannot split the velocity dispersion and amplitude attenuation term, thus limiting its application in the Q-compensated reverse time migration/full waveform inversion. In addition, GSLS-model-based wave equation is usually numerically solved using staggered-grid finite-difference (SGFD) method, which may result in the numerical dispersion due to the harsh stability condition and poses complexity and computational burden. To overcome these issues, we propose a high-accuracy Rayleigh-waves simulation scheme that involves the integration of the fractional viscoelastic wave equation and vacuum formalism. The proposed scheme not only decouples the amplitude attenuation and velocity dispersion but also significantly suppresses the numerical dispersion of Rayleigh waves under the same grid sizes. We first use a homogeneous elastic model to demonstrate the accuracy in comparison with the analytical solutions, and the correctness for a viscoelastic half-space model is verified by comparing the phase velocities with the dispersive images generated by the phase shift transformation. We then simulate several two-dimensional synthetic models to analyze the effectiveness and applicability of the proposed method. The results show that the proposed method uses twice as many spatial step sizes and takes 0.6 times that of the GSLS method (solved by the SGFD method) when achieved at 95% accuracy.
{"title":"High-Accuracy Simulation of Rayleigh Waves Using Fractional Viscoelastic Wave Equation","authors":"Yinfeng Wang, Jilong Lu, Ying Shi, Ning Wang, Liguo Han","doi":"10.3390/fractalfract7120880","DOIUrl":"https://doi.org/10.3390/fractalfract7120880","url":null,"abstract":"The propagation of Rayleigh waves is usually accompanied by dispersion, which becomes more complex with inherent attenuation. The accurate simulation of Rayleigh waves in attenuation media is crucial for understanding wave mechanisms, layer thickness identification, and parameter inversion. Although the vacuum formalism or stress image method (SIM) combined with the generalized standard linear solid (GSLS) is widely used to implement the numerical simulation of Rayleigh waves in attenuation media, this type of method still has its limitations. First, the GSLS model cannot split the velocity dispersion and amplitude attenuation term, thus limiting its application in the Q-compensated reverse time migration/full waveform inversion. In addition, GSLS-model-based wave equation is usually numerically solved using staggered-grid finite-difference (SGFD) method, which may result in the numerical dispersion due to the harsh stability condition and poses complexity and computational burden. To overcome these issues, we propose a high-accuracy Rayleigh-waves simulation scheme that involves the integration of the fractional viscoelastic wave equation and vacuum formalism. The proposed scheme not only decouples the amplitude attenuation and velocity dispersion but also significantly suppresses the numerical dispersion of Rayleigh waves under the same grid sizes. We first use a homogeneous elastic model to demonstrate the accuracy in comparison with the analytical solutions, and the correctness for a viscoelastic half-space model is verified by comparing the phase velocities with the dispersive images generated by the phase shift transformation. We then simulate several two-dimensional synthetic models to analyze the effectiveness and applicability of the proposed method. The results show that the proposed method uses twice as many spatial step sizes and takes 0.6 times that of the GSLS method (solved by the SGFD method) when achieved at 95% accuracy.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"26 7","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139008824","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}
Pub Date : 2023-12-12DOI: 10.3390/fractalfract7120879
M. Merad, B. Meftah, Abdelkader Moumen, Mohamed Bouye
This paper’s major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided.
{"title":"Fractional Maclaurin-Type Inequalities for Multiplicatively Convex Functions","authors":"M. Merad, B. Meftah, Abdelkader Moumen, Mohamed Bouye","doi":"10.3390/fractalfract7120879","DOIUrl":"https://doi.org/10.3390/fractalfract7120879","url":null,"abstract":"This paper’s major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"43 5","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139007217","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}
This paper studies the asymptotic stability of fractional-order neural networks (FONNs) with time delay utilizing a sampled-data controller. Firstly, a novel class of Lyapunov–Krasovskii functions (LKFs) is established, in which time delay and fractional-order information are fully taken into account. Secondly, by combining with the fractional-order Leibniz–Newton formula, LKFs, and other analysis techniques, some less conservative stability criteria that depend on time delay and fractional-order information are given in terms of linear matrix inequalities (LMIs). In the meantime, the sampled-data controller gain is developed under a larger sampling interval. Last, the proposed criteria are shown to be valid and less conservative than the existing ones using three numerical examples.
{"title":"Improved Results on Delay-Dependent and Order-Dependent Criteria of Fractional-Order Neural Networks with Time Delay Based on Sampled-Data Control","authors":"Junzhou Dai, Lianglin Xiong, Haiyang Zhang, Weiguo Rui","doi":"10.3390/fractalfract7120876","DOIUrl":"https://doi.org/10.3390/fractalfract7120876","url":null,"abstract":"This paper studies the asymptotic stability of fractional-order neural networks (FONNs) with time delay utilizing a sampled-data controller. Firstly, a novel class of Lyapunov–Krasovskii functions (LKFs) is established, in which time delay and fractional-order information are fully taken into account. Secondly, by combining with the fractional-order Leibniz–Newton formula, LKFs, and other analysis techniques, some less conservative stability criteria that depend on time delay and fractional-order information are given in terms of linear matrix inequalities (LMIs). In the meantime, the sampled-data controller gain is developed under a larger sampling interval. Last, the proposed criteria are shown to be valid and less conservative than the existing ones using three numerical examples.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"2 9","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138981120","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}
Pub Date : 2023-12-11DOI: 10.3390/fractalfract7120878
Y. Geum
A family of three-point, sixth-order, multiple-zero solvers is developed, and special cases of weight functions are investigated based on polynomials and low-order rational functions. The chosen cases of the proposed iterative method are compared with existing methods. The experiments show the superiority of the proposed schemes in terms of the number of divergent points and the average number of function evaluations per point. The dynamical characteristics of the developed methods, along with their illustrations, are represented with detailed analyses, comparisons, and comments.
{"title":"On Constructing a Family of Sixth-Order Methods for Multiple Roots","authors":"Y. Geum","doi":"10.3390/fractalfract7120878","DOIUrl":"https://doi.org/10.3390/fractalfract7120878","url":null,"abstract":"A family of three-point, sixth-order, multiple-zero solvers is developed, and special cases of weight functions are investigated based on polynomials and low-order rational functions. The chosen cases of the proposed iterative method are compared with existing methods. The experiments show the superiority of the proposed schemes in terms of the number of divergent points and the average number of function evaluations per point. The dynamical characteristics of the developed methods, along with their illustrations, are represented with detailed analyses, comparisons, and comments.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"62 7","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138979359","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}
Pub Date : 2023-12-11DOI: 10.3390/fractalfract7120877
Wei Zhang, Yong He, Zerong Yang
In this research, we focus on the symmetry of an ancient solution for a fractional parabolic equation involving logarithmic Laplacian in an entire space. In the process of studying the property of a fractional parabolic equation, we obtained some maximum principles, such as the maximum principle of anti-symmetric function, narrow region principle, and so on. We will demonstrate how to apply these tools to obtain radial symmetry of an ancient solution.
{"title":"Symmetry of Ancient Solution for Fractional Parabolic Equation Involving Logarithmic Laplacian","authors":"Wei Zhang, Yong He, Zerong Yang","doi":"10.3390/fractalfract7120877","DOIUrl":"https://doi.org/10.3390/fractalfract7120877","url":null,"abstract":"In this research, we focus on the symmetry of an ancient solution for a fractional parabolic equation involving logarithmic Laplacian in an entire space. In the process of studying the property of a fractional parabolic equation, we obtained some maximum principles, such as the maximum principle of anti-symmetric function, narrow region principle, and so on. We will demonstrate how to apply these tools to obtain radial symmetry of an ancient solution.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"90 2","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138981779","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}
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