{"title":"卡普托方程作为非经典扩散过程中粒子扩散的模型:一个特例","authors":"","doi":"10.35741/issn.0258-2724.58.4.91","DOIUrl":null,"url":null,"abstract":"The main objective of this article is to present the application of the Mittag-Leffler function in anomalous diffusion modeling using the Caputo equation. Anomalous diffusion refers to non-classical diffusion processes in which traditional models fail to accurately capture the dynamics. The Caputo equation, which involves a fractional derivative of order in the Caputo sense, provides a powerful tool for describing these types of phenomena. The model we consider in this research presents particle diffusion , which represents the concentration or density of particles at position and time . The Laplacian operator captures the spatial diffusion process, and the diffusion coefficient governs the diffusion rate. An important aspect of this study is the integration of the Mittag-Leffler function, which arises in the solution of the Caputo equation. By solving the Caputo equation with the appropriate initial and boundary conditions, the concentration profile is obtained. The Mittag-Leffler function plays a key role in this solution because it accurately captures the memory-dependent behavior and the nonlocal nature of anomalous diffusion. A distinctive feature of this model is the presence of the fractional derivative of order in the Caputo sense, which captures the memory-dependent behavior and non-local nature of the diffusion process, allowing the representation of anomalous diffusion phenomena. In this paper, an important contribution is evidenced in the use of the Mittag-Leffler function to explore the behavior of anomalous diffusion processes and obtain information about the complex dynamics of particle propagation in various physical systems. Keywords: The Mittag-Leffler Function, Fractional Derivative, Anomalous Diffusion Processes, The Caputo Equation DOI: https://doi.org/10.35741/issn.0258-2724.58.4.91","PeriodicalId":35772,"journal":{"name":"Xinan Jiaotong Daxue Xuebao/Journal of Southwest Jiaotong University","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"THE CAPUTO EQUATION AS A MODEL OF PARTICLE SPREADING IN A NON-CLASSICAL DIFFUSION PROCESS: A SPECIAL CASE\",\"authors\":\"\",\"doi\":\"10.35741/issn.0258-2724.58.4.91\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main objective of this article is to present the application of the Mittag-Leffler function in anomalous diffusion modeling using the Caputo equation. Anomalous diffusion refers to non-classical diffusion processes in which traditional models fail to accurately capture the dynamics. The Caputo equation, which involves a fractional derivative of order in the Caputo sense, provides a powerful tool for describing these types of phenomena. The model we consider in this research presents particle diffusion , which represents the concentration or density of particles at position and time . The Laplacian operator captures the spatial diffusion process, and the diffusion coefficient governs the diffusion rate. An important aspect of this study is the integration of the Mittag-Leffler function, which arises in the solution of the Caputo equation. By solving the Caputo equation with the appropriate initial and boundary conditions, the concentration profile is obtained. The Mittag-Leffler function plays a key role in this solution because it accurately captures the memory-dependent behavior and the nonlocal nature of anomalous diffusion. A distinctive feature of this model is the presence of the fractional derivative of order in the Caputo sense, which captures the memory-dependent behavior and non-local nature of the diffusion process, allowing the representation of anomalous diffusion phenomena. In this paper, an important contribution is evidenced in the use of the Mittag-Leffler function to explore the behavior of anomalous diffusion processes and obtain information about the complex dynamics of particle propagation in various physical systems. Keywords: The Mittag-Leffler Function, Fractional Derivative, Anomalous Diffusion Processes, The Caputo Equation DOI: https://doi.org/10.35741/issn.0258-2724.58.4.91\",\"PeriodicalId\":35772,\"journal\":{\"name\":\"Xinan Jiaotong Daxue Xuebao/Journal of Southwest Jiaotong University\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Xinan Jiaotong Daxue Xuebao/Journal of Southwest Jiaotong University\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.35741/issn.0258-2724.58.4.91\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Multidisciplinary\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Xinan Jiaotong Daxue Xuebao/Journal of Southwest Jiaotong University","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.35741/issn.0258-2724.58.4.91","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Multidisciplinary","Score":null,"Total":0}
THE CAPUTO EQUATION AS A MODEL OF PARTICLE SPREADING IN A NON-CLASSICAL DIFFUSION PROCESS: A SPECIAL CASE
The main objective of this article is to present the application of the Mittag-Leffler function in anomalous diffusion modeling using the Caputo equation. Anomalous diffusion refers to non-classical diffusion processes in which traditional models fail to accurately capture the dynamics. The Caputo equation, which involves a fractional derivative of order in the Caputo sense, provides a powerful tool for describing these types of phenomena. The model we consider in this research presents particle diffusion , which represents the concentration or density of particles at position and time . The Laplacian operator captures the spatial diffusion process, and the diffusion coefficient governs the diffusion rate. An important aspect of this study is the integration of the Mittag-Leffler function, which arises in the solution of the Caputo equation. By solving the Caputo equation with the appropriate initial and boundary conditions, the concentration profile is obtained. The Mittag-Leffler function plays a key role in this solution because it accurately captures the memory-dependent behavior and the nonlocal nature of anomalous diffusion. A distinctive feature of this model is the presence of the fractional derivative of order in the Caputo sense, which captures the memory-dependent behavior and non-local nature of the diffusion process, allowing the representation of anomalous diffusion phenomena. In this paper, an important contribution is evidenced in the use of the Mittag-Leffler function to explore the behavior of anomalous diffusion processes and obtain information about the complex dynamics of particle propagation in various physical systems. Keywords: The Mittag-Leffler Function, Fractional Derivative, Anomalous Diffusion Processes, The Caputo Equation DOI: https://doi.org/10.35741/issn.0258-2724.58.4.91