Finite difference and finite element methods for partial differential equations on fractals
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measu...
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| Format: | Article |
| Language: | Spanish |
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Universidad Industrial de Santander
2022-09-01
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| Series: | Revista Integración |
| Subjects: | |
| Online Access: | https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/13850 |
| _version_ | 1797987878264897536 |
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| author | Luis F. Contreras H. Juan Galvis |
| author_facet | Luis F. Contreras H. Juan Galvis |
| author_sort | Luis F. Contreras H. |
| collection | DOAJ |
| description |
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree.
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| first_indexed | 2024-04-11T07:54:08Z |
| format | Article |
| id | doaj.art-cd796cab743e47499d43b281b55976b5 |
| institution | Directory Open Access Journal |
| issn | 0120-419X 2145-8472 |
| language | Spanish |
| last_indexed | 2024-04-11T07:54:08Z |
| publishDate | 2022-09-01 |
| publisher | Universidad Industrial de Santander |
| record_format | Article |
| series | Revista Integración |
| spelling | doaj.art-cd796cab743e47499d43b281b55976b52022-12-22T04:35:59ZspaUniversidad Industrial de SantanderRevista Integración0120-419X2145-84722022-09-01402Finite difference and finite element methods for partial differential equations on fractalsLuis F. Contreras H.0Juan Galvis1Universidad Nacional de ColombiaUniversidad Nacional de Colombia In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree. https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/13850Fractal diffusionLaplacian on a fractalRenormalization constant |
| spellingShingle | Luis F. Contreras H. Juan Galvis Finite difference and finite element methods for partial differential equations on fractals Revista Integración Fractal diffusion Laplacian on a fractal Renormalization constant |
| title | Finite difference and finite element methods for partial differential equations on fractals |
| title_full | Finite difference and finite element methods for partial differential equations on fractals |
| title_fullStr | Finite difference and finite element methods for partial differential equations on fractals |
| title_full_unstemmed | Finite difference and finite element methods for partial differential equations on fractals |
| title_short | Finite difference and finite element methods for partial differential equations on fractals |
| title_sort | finite difference and finite element methods for partial differential equations on fractals |
| topic | Fractal diffusion Laplacian on a fractal Renormalization constant |
| url | https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/13850 |
| work_keys_str_mv | AT luisfcontrerash finitedifferenceandfiniteelementmethodsforpartialdifferentialequationsonfractals AT juangalvis finitedifferenceandfiniteelementmethodsforpartialdifferentialequationsonfractals |