A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model
This note tackles the equivalence problem between the fractional and integer order diffusion models. Unlike existing approaches, the existence of a unique integral transformation mapping the solution of the integer order model to a solution of the fractional order model of <inline-formula><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/5/753 |
| _version_ | 1797474439040860160 |
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| author | Claudia A. Pérez-Pinacho Cristina Verde |
| author_facet | Claudia A. Pérez-Pinacho Cristina Verde |
| author_sort | Claudia A. Pérez-Pinacho |
| collection | DOAJ |
| description | This note tackles the equivalence problem between the fractional and integer order diffusion models. Unlike existing approaches, the existence of a unique integral transformation mapping the solution of the integer order model to a solution of the fractional order model of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> is proven. Moreover, the corresponding inverse integral transformation is formally established to guarantee the equivalence and well-posedness of the solutions of these models. Finally, as an example, the solution of a fractional order diffusion model <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, obtained through the solution of its integer order counterpart and the proposed transformation, is compared with the solution derived by using the Fourier transform. |
| first_indexed | 2024-03-09T20:31:05Z |
| format | Article |
| id | doaj.art-647b06328af8443a852daf46b10d589d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T20:31:05Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-647b06328af8443a852daf46b10d589d2023-11-23T23:23:07ZengMDPI AGMathematics2227-73902022-02-0110575310.3390/math10050753A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion ModelClaudia A. Pérez-Pinacho0Cristina Verde1Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City 04510, MexicoInstituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City 04510, MexicoThis note tackles the equivalence problem between the fractional and integer order diffusion models. Unlike existing approaches, the existence of a unique integral transformation mapping the solution of the integer order model to a solution of the fractional order model of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> is proven. Moreover, the corresponding inverse integral transformation is formally established to guarantee the equivalence and well-posedness of the solutions of these models. Finally, as an example, the solution of a fractional order diffusion model <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, obtained through the solution of its integer order counterpart and the proposed transformation, is compared with the solution derived by using the Fourier transform.https://www.mdpi.com/2227-7390/10/5/753integer order diffusion modelfractional calculusintegral transformationfractional order diffusion model |
| spellingShingle | Claudia A. Pérez-Pinacho Cristina Verde A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model Mathematics integer order diffusion model fractional calculus integral transformation fractional order diffusion model |
| title | A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model |
| title_full | A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model |
| title_fullStr | A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model |
| title_full_unstemmed | A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model |
| title_short | A Note on an Integral Transformation for the Equivalence between a Fractional and Integer Order Diffusion Model |
| title_sort | note on an integral transformation for the equivalence between a fractional and integer order diffusion model |
| topic | integer order diffusion model fractional calculus integral transformation fractional order diffusion model |
| url | https://www.mdpi.com/2227-7390/10/5/753 |
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