Pub Date : 2019-02-18DOI: 10.1515/9783110571622-003
R. Hilfer
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.
{"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":"70 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 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":"29 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 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":"91 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 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":"27 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 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":"1 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 10.1515/9783110571622-016
M. Meerschaert, H. Scheffler
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.
{"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":"31 1","pages":"0"},"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}
Pub Date : 2019-02-18DOI: 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":"46 1","pages":"0"},"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}
Pub Date : 2018-07-04DOI: 10.1515/9783110571622-013
M. Ammi, Delfim F. M. Torres
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.
{"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":"98 1","pages":"0"},"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}
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