Growth Equation of the General Fractional Calculus
We consider the Cauchy problem <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/7/615 |
| _version_ | 1819018621715742720 |
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| author | Anatoly N. Kochubei Yuri Kondratiev |
| author_facet | Anatoly N. Kochubei Yuri Kondratiev |
| author_sort | Anatoly N. Kochubei |
| collection | DOAJ |
| description | We consider the Cauchy problem <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> <mi>u</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>u</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> </inline-formula> is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory <b>71</b> (2011), 583−600), <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. The solution is a generalization of the function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>↦</mo> <msub> <mi>E</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <msup> <mi>t</mi> <mi>α</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>E</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula> is the Mittag−Leffler function. The asymptotics of this solution, as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, are studied. |
| first_indexed | 2024-12-21T03:22:20Z |
| format | Article |
| id | doaj.art-7a8beb66d6d74800b87158f9715fd409 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-21T03:22:20Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7a8beb66d6d74800b87158f9715fd4092022-12-21T19:17:41ZengMDPI AGMathematics2227-73902019-07-017761510.3390/math7070615math7070615Growth Equation of the General Fractional CalculusAnatoly N. Kochubei0Yuri Kondratiev1Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, 01024 Kyiv, UkraineDepartment of Mathematics, University of Bielefeld, D-33615 Bielefeld, GermanyWe consider the Cauchy problem <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> <mi>u</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>u</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> </inline-formula> is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory <b>71</b> (2011), 583−600), <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. The solution is a generalization of the function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>↦</mo> <msub> <mi>E</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <msup> <mi>t</mi> <mi>α</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>E</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula> is the Mittag−Leffler function. The asymptotics of this solution, as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, are studied.https://www.mdpi.com/2227-7390/7/7/615generalized fractional derivativesgrowth equationMittag–Leffler function |
| spellingShingle | Anatoly N. Kochubei Yuri Kondratiev Growth Equation of the General Fractional Calculus Mathematics generalized fractional derivatives growth equation Mittag–Leffler function |
| title | Growth Equation of the General Fractional Calculus |
| title_full | Growth Equation of the General Fractional Calculus |
| title_fullStr | Growth Equation of the General Fractional Calculus |
| title_full_unstemmed | Growth Equation of the General Fractional Calculus |
| title_short | Growth Equation of the General Fractional Calculus |
| title_sort | growth equation of the general fractional calculus |
| topic | generalized fractional derivatives growth equation Mittag–Leffler function |
| url | https://www.mdpi.com/2227-7390/7/7/615 |
| work_keys_str_mv | AT anatolynkochubei growthequationofthegeneralfractionalcalculus AT yurikondratiev growthequationofthegeneralfractionalcalculus |