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Mathematical and physical interpretations of fractional derivatives and integrals 分数阶导数和积分的数学和物理解释
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-003
R. Hilfer
{"title":"Mathematical and physical interpretations of fractional derivatives and integrals","authors":"R. Hilfer","doi":"10.1515/9783110571622-003","DOIUrl":"https://doi.org/10.1515/9783110571622-003","url":null,"abstract":"Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point.","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127127130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 24
Fractional Fourier transform 分数傅里叶变换
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-009
Yuri Luchko
{"title":"Fractional Fourier transform","authors":"Yuri Luchko","doi":"10.1515/9783110571622-009","DOIUrl":"https://doi.org/10.1515/9783110571622-009","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128734450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Fractional Laplace operator and its properties 分数阶拉普拉斯算子及其性质
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-007
M. Kwasnicki
{"title":"Fractional Laplace operator and its properties","authors":"M. Kwasnicki","doi":"10.1515/9783110571622-007","DOIUrl":"https://doi.org/10.1515/9783110571622-007","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129601331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 26
Mittag-Leffler function: properties and applications Mittag-Leffler函数:属性和应用
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-011
R. Gorenflo, F. Mainardi, S. Rogosin
{"title":"Mittag-Leffler function: properties and applications","authors":"R. Gorenflo, F. Mainardi, S. Rogosin","doi":"10.1515/9783110571622-011","DOIUrl":"https://doi.org/10.1515/9783110571622-011","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122109502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 31
Inverse subordinators and time fractional equations 逆从属和时间分数方程
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-017
M. Meerschaert, Erkan Nane, P. Vellaisamy
{"title":"Inverse subordinators and time fractional equations","authors":"M. Meerschaert, Erkan Nane, P. Vellaisamy","doi":"10.1515/9783110571622-017","DOIUrl":"https://doi.org/10.1515/9783110571622-017","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133228679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Basic FC operators and their properties 基本FC操作符及其属性
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-002
A. Kochubei, Yuri Luchko
{"title":"Basic FC operators and their properties","authors":"A. Kochubei, Yuri Luchko","doi":"10.1515/9783110571622-002","DOIUrl":"https://doi.org/10.1515/9783110571622-002","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126146710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Continuous time random walks and space-time fractional differential equations 连续时间随机漫步和时空分数阶微分方程
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-016
M. Meerschaert, H. Scheffler
{"title":"Continuous time random walks and space-time fractional differential equations","authors":"M. Meerschaert, H. Scheffler","doi":"10.1515/9783110571622-016","DOIUrl":"https://doi.org/10.1515/9783110571622-016","url":null,"abstract":"The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of thismodel is governedbya fractional diffusion equation. If the step lengthof the randomwalk followsapower law,weget a spacefractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives.","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123178907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Applications of the Mellin integral transform technique in fractional calculus 梅林积分变换技术在分数阶微积分中的应用
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-008
Yuri Luchko, V. Kiryakova
{"title":"Applications of the Mellin integral transform technique in fractional calculus","authors":"Yuri Luchko, V. Kiryakova","doi":"10.1515/9783110571622-008","DOIUrl":"https://doi.org/10.1515/9783110571622-008","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130504463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fractional differentiation in p-adic analysis p进分析中的分数阶微分
Basic Theory Pub Date : 2019-02-18 DOI: 10.1515/9783110571622-019
A. Kochubei
{"title":"Fractional differentiation in p-adic analysis","authors":"A. Kochubei","doi":"10.1515/9783110571622-019","DOIUrl":"https://doi.org/10.1515/9783110571622-019","url":null,"abstract":"","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131834434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Analysis of fractional integro-differential equations of thermistor type 热敏电阻型分数阶积分微分方程分析
Basic Theory Pub Date : 2018-07-04 DOI: 10.1515/9783110571622-013
M. Ammi, Delfim F. M. Torres
{"title":"Analysis of fractional integro-differential equations of thermistor type","authors":"M. Ammi, Delfim F. M. Torres","doi":"10.1515/9783110571622-013","DOIUrl":"https://doi.org/10.1515/9783110571622-013","url":null,"abstract":"We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: with Riemann-Liouville, Caputo, and time-scale fractional operators. Existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.","PeriodicalId":385044,"journal":{"name":"Basic Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127095964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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