On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel
Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a univers...
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MDPI AG
2024-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/11/653 |
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| author | Zelin Liu Xiaobin Yu Yajun Yin |
| author_facet | Zelin Liu Xiaobin Yu Yajun Yin |
| author_sort | Zelin Liu |
| collection | DOAJ |
| description | Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the <i>α</i>-th power of a differential operator corresponds precisely to the <i>α</i>-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures. |
| format | Article |
| id | doaj-art-5fa3a81244664258b0e3456e8aeb5ee0 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-5fa3a81244664258b0e3456e8aeb5ee02025-08-20T01:53:41ZengMDPI AGFractal and Fractional2504-31102024-11-0181165310.3390/fractalfract8110653On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine KernelZelin Liu0Xiaobin Yu1Yajun Yin2Department of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinaDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinaDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinaTraditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the <i>α</i>-th power of a differential operator corresponds precisely to the <i>α</i>-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.https://www.mdpi.com/2504-3110/8/11/653fractional calculusphysical fractal spaceSonine kerneloperational calculuscommutativity |
| spellingShingle | Zelin Liu Xiaobin Yu Yajun Yin On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel Fractal and Fractional fractional calculus physical fractal space Sonine kernel operational calculus commutativity |
| title | On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel |
| title_full | On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel |
| title_fullStr | On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel |
| title_full_unstemmed | On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel |
| title_short | On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel |
| title_sort | on the rigorous correspondence between operator fractional powers and fractional derivatives via the sonine kernel |
| topic | fractional calculus physical fractal space Sonine kernel operational calculus commutativity |
| url | https://www.mdpi.com/2504-3110/8/11/653 |
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