Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries
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arXiv
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| Natura: | Preprint |
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2022
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| author | Pata, Vittorino Siryk, Sergii Vasylyeva, Nataliya |
| author_facet | Pata, Vittorino Siryk, Sergii Vasylyeva, Nataliya |
| contents | For $0<ν_2<ν_1\leq 1$, we analyze a linear integro-differential equation on the space-time cylinder $Ω\times(0,T)$ in the unknown $u=u(x,t)$ $$\mathbf{D}_{t}^{ν_1}(\varrho_{1}u)-\mathbf{D}_{t}^{ν_2}(\varrho_2 u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u =f$$ where $\mathbf{D}_{t}^{ν_i}$ are the Caputo fractional derivatives, $\varrho_i=\varrho_i(x,t)$ with $\varrho_1\geq μ_0>0$, $\mathcal{L}_{i}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel, and $f$ is an external force. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under suitable conditions on the given data, the global classical solvability of the associated initial-boundary value problems is addressed. To this end, a special technique is needed, adapting the concept of a regularizer from the theory of parabolic equations. This allows us to remove the usual assumption about the nonnegativity of the kernel representing fractional derivatives. The problem is also investigated from the numerical point of view. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_05009 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries Pata, Vittorino Siryk, Sergii Vasylyeva, Nataliya Analysis of PDEs Numerical Analysis 35R11, 35C15, 65M06 (Primary) 45N05, 26A33, 35B30 (Secondary) For $0<ν_2<ν_1\leq 1$, we analyze a linear integro-differential equation on the space-time cylinder $Ω\times(0,T)$ in the unknown $u=u(x,t)$ $$\mathbf{D}_{t}^{ν_1}(\varrho_{1}u)-\mathbf{D}_{t}^{ν_2}(\varrho_2 u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u =f$$ where $\mathbf{D}_{t}^{ν_i}$ are the Caputo fractional derivatives, $\varrho_i=\varrho_i(x,t)$ with $\varrho_1\geq μ_0>0$, $\mathcal{L}_{i}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel, and $f$ is an external force. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under suitable conditions on the given data, the global classical solvability of the associated initial-boundary value problems is addressed. To this end, a special technique is needed, adapting the concept of a regularizer from the theory of parabolic equations. This allows us to remove the usual assumption about the nonnegativity of the kernel representing fractional derivatives. The problem is also investigated from the numerical point of view. |
| title | Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries |
| topic | Analysis of PDEs Numerical Analysis 35R11, 35C15, 65M06 (Primary) 45N05, 26A33, 35B30 (Secondary) |
| url | https://arxiv.org/abs/2210.05009 |