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Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method
In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer International Publishing
2016
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4993744/ https://www.ncbi.nlm.nih.gov/pubmed/27610294 http://dx.doi.org/10.1186/s40064-016-2853-6 |
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author | Bahşı, Ayşe Kurt Yalçınbaş, Salih |
author_facet | Bahşı, Ayşe Kurt Yalçınbaş, Salih |
author_sort | Bahşı, Ayşe Kurt |
collection | PubMed |
description | In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method. |
format | Online Article Text |
id | pubmed-4993744 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2016 |
publisher | Springer International Publishing |
record_format | MEDLINE/PubMed |
spelling | pubmed-49937442016-09-08 Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method Bahşı, Ayşe Kurt Yalçınbaş, Salih Springerplus Research In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method. Springer International Publishing 2016-08-22 /pmc/articles/PMC4993744/ /pubmed/27610294 http://dx.doi.org/10.1186/s40064-016-2853-6 Text en © The Author(s) 2016 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Research Bahşı, Ayşe Kurt Yalçınbaş, Salih Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title | Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title_full | Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title_fullStr | Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title_full_unstemmed | Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title_short | Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method |
title_sort | numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via fibonacci collocation method |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4993744/ https://www.ncbi.nlm.nih.gov/pubmed/27610294 http://dx.doi.org/10.1186/s40064-016-2853-6 |
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