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arXiv · 2405.14451

On fractional parabolic systems of vector order

Ravshan Ashurov·Ilyoskhuja Sulaymonov
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Abstract

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: $β_j=β$, $j=1,...,m$.

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Ravshan Ashurov, Ilyoskhuja Sulaymonov. 2024-05-23. On fractional parabolic systems of vector order. https://arxiv.org/abs/2405.14451

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