An Entropy Paradox Free Fractional Diffusion Equation
A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/4/236 |
| _version_ | 1827672397416759296 |
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| author | Manuel Duarte Ortigueira |
| author_facet | Manuel Duarte Ortigueira |
| author_sort | Manuel Duarte Ortigueira |
| collection | DOAJ |
| description | A new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formulae used for the stable distributions. The entropy paradox problem was studied and clarified through the Rényi entropy: in the extreme wave regime the entropy is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. In passing, Tsallis and Rényi entropies for stable distributions are introduced and exemplified. |
| first_indexed | 2024-03-10T04:05:51Z |
| format | Article |
| id | doaj.art-fc421f0488ae41fea9df8309b3904a09 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:05:51Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-fc421f0488ae41fea9df8309b3904a092023-11-23T08:24:09ZengMDPI AGFractal and Fractional2504-31102021-11-015423610.3390/fractalfract5040236An Entropy Paradox Free Fractional Diffusion EquationManuel Duarte Ortigueira0CTS-UNINOVA and DEE, NOVA School of Science and Technology, 2829-517 Caparica, PortugalA new look at the fractional diffusion equation was done. Using the unified fractional derivative, a new formulation was proposed, and the equation was solved for three different order cases: neutral, dominant time, and dominant space. The solutions were expressed by generalizations of classic formulae used for the stable distributions. The entropy paradox problem was studied and clarified through the Rényi entropy: in the extreme wave regime the entropy is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. In passing, Tsallis and Rényi entropies for stable distributions are introduced and exemplified.https://www.mdpi.com/2504-3110/5/4/236diffusion equationShannon entropyTsallis entropyRényi entropystable distributionunified fractional derivative |
| spellingShingle | Manuel Duarte Ortigueira An Entropy Paradox Free Fractional Diffusion Equation Fractal and Fractional diffusion equation Shannon entropy Tsallis entropy Rényi entropy stable distribution unified fractional derivative |
| title | An Entropy Paradox Free Fractional Diffusion Equation |
| title_full | An Entropy Paradox Free Fractional Diffusion Equation |
| title_fullStr | An Entropy Paradox Free Fractional Diffusion Equation |
| title_full_unstemmed | An Entropy Paradox Free Fractional Diffusion Equation |
| title_short | An Entropy Paradox Free Fractional Diffusion Equation |
| title_sort | entropy paradox free fractional diffusion equation |
| topic | diffusion equation Shannon entropy Tsallis entropy Rényi entropy stable distribution unified fractional derivative |
| url | https://www.mdpi.com/2504-3110/5/4/236 |
| work_keys_str_mv | AT manuelduarteortigueira anentropyparadoxfreefractionaldiffusionequation AT manuelduarteortigueira entropyparadoxfreefractionaldiffusionequation |