Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations
First-order ordinary differential equations (ODEs) are widely used in various fields of science and engineering to model natural phenomena. This study proposes an extension of these equations using fractional derivatives, specifically the Grünwald-Letnikov definition, to explore their impact on the...
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Universidade Estadual de Londrina
2024-12-01
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| Series: | Semina: Ciências Exatas e Tecnológicas |
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| Online Access: | https://ojs.uel.br/revistas/uel/index.php/semexatas/article/view/51533 |
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| author | Jaque Willian Scotton |
| author_facet | Jaque Willian Scotton |
| author_sort | Jaque Willian Scotton |
| collection | DOAJ |
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First-order ordinary differential equations (ODEs) are widely used in various fields of science and engineering to model natural phenomena. This study proposes an extension of these equations using fractional derivatives, specifically the Grünwald-Letnikov definition, to explore their impact on the behavior of solution curves. The fractional ODE considered is discretized using the finite difference method and solved numerically for different values of the derivative order (α). Tests were conducted to verify mesh independence and the quality of the computational implementation of the method, through which the accuracy and the absence of implementation errors were confirmed. The behavior of the solution curves for different values of α was analyzed, revealing a sharp decrease near the initial point (t = 0) and an almost linear growth at higher values of t, within the considered domain. Additionally, in solving a specific initial value problem with a known analytical solution, it was discovered that the accuracy of the numerical solutions for higher values of α was more dependent on the mesh refinement than the solutions for lower values.
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| format | Article |
| id | doaj-art-e2e429140a9a44fe92e9d24ba9a20297 |
| institution | Kabale University |
| issn | 1676-5451 1679-0375 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Universidade Estadual de Londrina |
| record_format | Article |
| series | Semina: Ciências Exatas e Tecnológicas |
| spelling | doaj-art-e2e429140a9a44fe92e9d24ba9a202972024-12-19T06:51:19ZengUniversidade Estadual de LondrinaSemina: Ciências Exatas e Tecnológicas1676-54511679-03752024-12-014510.5433/1679-0375.2024.v45.51533Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential EquationsJaque Willian Scotton0https://orcid.org/0000-0002-8112-8086Santa Catarina State University First-order ordinary differential equations (ODEs) are widely used in various fields of science and engineering to model natural phenomena. This study proposes an extension of these equations using fractional derivatives, specifically the Grünwald-Letnikov definition, to explore their impact on the behavior of solution curves. The fractional ODE considered is discretized using the finite difference method and solved numerically for different values of the derivative order (α). Tests were conducted to verify mesh independence and the quality of the computational implementation of the method, through which the accuracy and the absence of implementation errors were confirmed. The behavior of the solution curves for different values of α was analyzed, revealing a sharp decrease near the initial point (t = 0) and an almost linear growth at higher values of t, within the considered domain. Additionally, in solving a specific initial value problem with a known analytical solution, it was discovered that the accuracy of the numerical solutions for higher values of α was more dependent on the mesh refinement than the solutions for lower values. https://ojs.uel.br/revistas/uel/index.php/semexatas/article/view/51533differential equationsfractional calculusGrünwald-Letnikov |
| spellingShingle | Jaque Willian Scotton Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations Semina: Ciências Exatas e Tecnológicas differential equations fractional calculus Grünwald-Letnikov |
| title | Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations |
| title_full | Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations |
| title_fullStr | Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations |
| title_full_unstemmed | Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations |
| title_short | Grünwald-Letnikov Fractional Derivative Applied to First-Order Ordinary Differential Equations |
| title_sort | grunwald letnikov fractional derivative applied to first order ordinary differential equations |
| topic | differential equations fractional calculus Grünwald-Letnikov |
| url | https://ojs.uel.br/revistas/uel/index.php/semexatas/article/view/51533 |
| work_keys_str_mv | AT jaquewillianscotton grunwaldletnikovfractionalderivativeappliedtofirstorderordinarydifferentialequations |