On fractional parabolic systems of vector order
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910457795706880 |
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| author | Ashurov, Ravshan Sulaymonov, Ilyoskhuja |
| author_facet | Ashurov, Ravshan Sulaymonov, Ilyoskhuja |
| contents | The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators located on the diagonal are elliptic. The main distinctive feature of this system is that the vector-order $\mathcal{B}$ has different components $β_j\in (0,1]$, and $β_j$ are not necessarily rational. Sufficient conditions (in some cases they are necessary) on the initial function and the right-hand side of the equation are found to ensure the existence of a classical solution. Note that the existence of a classical solution to systems of fractional differential equations was studied by various authors, but in all these works the fractional order had the same components for each equation: $β_j=β$, $j=1,...,m$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On fractional parabolic systems of vector order Ashurov, Ravshan Sulaymonov, Ilyoskhuja Analysis of PDEs The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators located on the diagonal are elliptic. The main distinctive feature of this system is that the vector-order $\mathcal{B}$ has different components $β_j\in (0,1]$, and $β_j$ are not necessarily rational. Sufficient conditions (in some cases they are necessary) on the initial function and the right-hand side of the equation are found to ensure the existence of a classical solution. Note that the existence of a classical solution to systems of fractional differential equations was studied by various authors, but in all these works the fractional order had the same components for each equation: $β_j=β$, $j=1,...,m$. |
| title | On fractional parabolic systems of vector order |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.14451 |