On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives
In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operat...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/3/117 |
| _version_ | 1797519236294246400 |
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| author | Briceyda B. Delgado Jorge E. Macías-Díaz |
| author_facet | Briceyda B. Delgado Jorge E. Macías-Díaz |
| author_sort | Briceyda B. Delgado |
| collection | DOAJ |
| description | In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. |
| first_indexed | 2024-03-10T07:40:02Z |
| format | Article |
| id | doaj.art-d4ca14bcf14445c4b609d51d1015f1b2 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T07:40:02Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-d4ca14bcf14445c4b609d51d1015f1b22023-11-22T13:09:51ZengMDPI AGFractal and Fractional2504-31102021-09-015311710.3390/fractalfract5030117On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional DerivativesBriceyda B. Delgado0Jorge E. Macías-Díaz1Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, MexicoDepartamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, MexicoIn this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.https://www.mdpi.com/2504-3110/5/3/117fractional div-curl systemsHelmholtz decomposition theoremRiemann–Liouville derivativeCaputo derivativefractional vector operators |
| spellingShingle | Briceyda B. Delgado Jorge E. Macías-Díaz On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives Fractal and Fractional fractional div-curl systems Helmholtz decomposition theorem Riemann–Liouville derivative Caputo derivative fractional vector operators |
| title | On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives |
| title_full | On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives |
| title_fullStr | On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives |
| title_full_unstemmed | On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives |
| title_short | On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives |
| title_sort | on the general solutions of some non homogeneous div curl systems with riemann liouville and caputo fractional derivatives |
| topic | fractional div-curl systems Helmholtz decomposition theorem Riemann–Liouville derivative Caputo derivative fractional vector operators |
| url | https://www.mdpi.com/2504-3110/5/3/117 |
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