Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators
Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solut...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-08-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/6/8/436 |
| _version_ | 1827627974194626560 |
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| author | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs |
| author_facet | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs |
| author_sort | Christopher N. Angstmann |
| collection | DOAJ |
| description | Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications. |
| first_indexed | 2024-03-09T13:24:19Z |
| format | Article |
| id | doaj.art-6334ea6358c548b687287bf44ebc819e |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T13:24:19Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-6334ea6358c548b687287bf44ebc819e2023-11-30T21:26:02ZengMDPI AGFractal and Fractional2504-31102022-08-016843610.3390/fractalfract6080436Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential OperatorsChristopher N. Angstmann0Stuart-James M. Burney1Bruce I. Henry2Byron A. Jacobs3School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaSchool of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaSchool of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaDepartment of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg 2092, South AfricaMotivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications.https://www.mdpi.com/2504-3110/6/8/436fractional calculusCaputo derivativeRiemann-Liouville derivativeintegro-differential equations |
| spellingShingle | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators Fractal and Fractional fractional calculus Caputo derivative Riemann-Liouville derivative integro-differential equations |
| title | Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators |
| title_full | Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators |
| title_fullStr | Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators |
| title_full_unstemmed | Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators |
| title_short | Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators |
| title_sort | solutions of initial value problems with non singular caputo type and riemann liouville type integro differential operators |
| topic | fractional calculus Caputo derivative Riemann-Liouville derivative integro-differential equations |
| url | https://www.mdpi.com/2504-3110/6/8/436 |
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