Fractional powers of 3*3 block operators matrices: Application to PDES
Hatem Baloudi, Mohamed Ali Dbeibia, Aref Jeribi
TL;DR
This work addresses the problem of defining and computing fractional powers $\Lambda^{-\alpha}$ of $3\times3$ block operator matrices arising in coupled PDE systems. It develops a rigorous framework based on the holomorphic functional calculus for positive block operators, using the integral representation $\Lambda^{-\alpha}=\frac{\sin(\pi\alpha)}{\pi}\int_{0}^{\infty}s^{-{\alpha}}(s+\Lambda)^{-1} ds$ and deriving explicit block-wise formulas. The authors present three computational strategies—alternative formulas, change of variables, and the second resolvent identity—to obtain tractable expressions and connect them to fractional powers of diagonal blocks like $A^{-\alpha}$. Through PDE-inspired examples, such as reaction-diffusion and diffusion-advection systems, the paper demonstrates how fractional block-operator calculus captures memory and non-local effects, with potential extensions to quaternionic settings and higher-dimensional block matrices.
Abstract
In this paper, we investigate the fractional powers of block operator matrices, with a particular focus on their applications to partial differential equations (PDEs). We develop a comprehensive theoretical framework for defining and calculating fractional powers of positive operators and extend these results to block operator matrices. Various methods, including alternative formulas, change of variables, and the second resolvent identity, are employed to obtain explicit expressions for fractional powers. The results are applied to systems of PDEs, demonstrating the relevance and effectiveness of the proposed approaches in modeling and analyzing complex dynamical systems. Examples are provided to illustrate the theoretical findings and their applicability to concrete problems