DeepAI AI Chat
Log In Sign Up

Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

10/10/2022
by   Vittorino Pata, et al.
0

For 0<ν_2<ν_1≤ 1, we analyze a linear integro-differential equation on the space-time cylinder Ω×(0,T) in the unknown u=u(x,t) 𝐃_t^ν_1(ϱ_1u)-𝐃_t^ν_2(ϱ_2 u)-ℒ_1u-𝒦*ℒ_2u =f where 𝐃_t^ν_i are the Caputo fractional derivatives, ϱ_i=ϱ_i(x,t) with ϱ_1≥μ_0>0, ℒ_i are uniform elliptic operators with time-dependent smooth coefficients, 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

08/02/2018

Machine Learning of Space-Fractional Differential Equations

Data-driven discovery of "hidden physics" -- i.e., machine learning of d...
06/24/2021

Optimal Control, Numerics, and Applications of Fractional PDEs

This article provides a brief review of recent developments on two nonlo...
05/31/2022

Computational Wavelet Method for Multidimensional Integro-Partial Differential Equation of Distributed Order

This article provides an effective computational algorithm based on Lege...
08/29/2023

Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations

We prove linear convergence for a new family of modified Dirichlet–Neuma...
03/01/2022

Nonlocal diffusion models with consistent local and fractional limits

For some spatially nonlocal diffusion models with a finite range of nonl...
01/10/2020

A new reproducing kernel approach for nonlinear fractional three-point boundary value problems

In this article, a new reproducing kernel approach is developed for obta...