[Paper Review] Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

Vittorino Pata, Sergii V. Siryk+1 more|arXiv (Cornell University)|Oct 10, 2022

Fractional Differential Equations Solutions4 citations

TL;DR

This paper proposes a novel analytical and numerical framework for multi-term fractional subdiffusion equations modeling oxygen transport in capillaries, using Caputo derivatives and time-dependent coefficients. It establishes global classical solvability under general conditions by introducing a specialized regularizer technique, overcoming the need for nonnegative convolution kernels, and validates the approach with high-accuracy finite-difference schemes and numerical examples.

ABSTRACT

For $0<ν_2<ν_1\leq 1$, we analyze a linear integro-differential equation on the space-time cylinder $Ω imes(0,T)$ 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.

Motivation & Objective

• To address the well-posedness of initial-boundary value problems for multi-term time-fractional subdiffusion equations with variable coefficients.
• To overcome the standard restriction requiring nonnegative convolution kernels in fractional derivative models.
• To develop a robust numerical scheme for solving such equations with high accuracy and stability.
• To provide theoretical and computational evidence for the classical solvability of the proposed model in realistic biological contexts.
• To lay the foundation for future work on inverse problems and nonlinear extensions of the model.

Proposed method

• The authors employ a generalized regularizer technique adapted from parabolic PDE theory to handle the integro-differential structure of the multi-term fractional equation.
• The model uses Caputo fractional derivatives of order $\nu_1$ and $\nu_2$ with $0 < \nu_2 < \nu_1 \leq 1$, allowing for memory effects in oxygen subdiffusion.
• The equation incorporates time-dependent diffusion coefficients $\varrho_1, \varrho_2$, uniform elliptic operators $\mathcal{L}_1, \mathcal{L}_2$, and a summable convolution kernel $\mathcal{K}$ to represent memory effects.
• A finite-difference scheme is constructed using a Crank-Nicolson-type discretization in time and central differences in space, with convolution quadrature for fractional derivatives.
• The scheme is validated using manufactured solutions with known analytical forms, ensuring consistency and convergence.
• Theoretical a priori estimates are derived to prove global classical solvability under smoothness and ellipticity conditions on the coefficients.

Experimental results

Research questions

• RQ1 Under what conditions is the initial-boundary value problem for the multi-term fractional subdiffusion equation classically solvable?
• RQ2 Can the classical solvability be established without assuming nonnegativity of the convolution kernel in the fractional derivative term?
• RQ3 How accurately can the proposed finite-difference scheme approximate the solution of the multi-term fractional subdiffusion equation?
• RQ4 What is the convergence behavior of the numerical scheme for varying fractional orders $\nu_1$ and $\nu_2$?
• RQ5 Can the model be extended to inverse problems or nonlinear variants with degenerate coefficients?

Key findings

• The proposed regularizer technique successfully establishes global classical solvability for the initial-boundary value problem without requiring the convolution kernel to be nonnegative.
• The finite-difference scheme achieves high accuracy, with the discrete $L^2$-norm error $\gimel$ reaching as low as $1.8727 \times 10^{-4}$ for $\nu_1 = 0.9$.
• For $\nu_1 = 0.5$, the numerical solution closely matches the manufactured analytical solution, with $\gimel = 3.1859 \times 10^{-4}$, confirming scheme reliability.
• The method is robust across different fractional orders, with the error decreasing from $6.4793 \times 10^{-4}$ at $\nu_1 = 0.1$ to $1.8727 \times 10^{-4}$ at $\nu_1 = 0.9$, indicating improved convergence for higher orders.
• The numerical results confirm the theoretical convergence rate, demonstrating the scheme's stability and efficiency.
• The model successfully captures complex subdiffusive dynamics in oxygen transport, with the full system solved accurately in 2D space-time domains.