Spectral and Coercivity Analysis of a Time-Space Fractional Advection-Diffusion System

Year 2025, Volume: 10 Issue: 3, 129 - 136, 30.12.2025

Mehmet Fatih Karaaslan

https://doi.org/10.30931/jetas.1700514

Abstract

In this paper, we consider a time–space fractional advection–diffusion equation that models complex transport phenomena in heterogeneous media. The equation involves a Caputo fractional derivative and a fractional Laplacian. A detailed mathematical analysis of the proposed model is presented. The spectral properties of the corresponding operator are examined and a uniform coercivity condition is obtained under certain assumptions. It is also shown that the operator is sectorial, which allows using semigroup theory to prove existence and uniqueness of mild solutions. In contrast to most existing works that mainly focus on numerical approximations or particular cases, we provide a unified functional analytic framework for the fractional advection–diffusion model, clarifying its stability and solvability. The proposed approach gives us strong theoretical guarantees but may involve challenges for numerical implementation due to the nonlocal nature of the operators.

Keywords

Time-space fractional advection-diffusion equation , fractional Laplacian , coercivity estimate , spectral analysis

Zaman-Mekan Kesirli Adveksiyon-Difüzyon Sisteminin Spektral ve Zorlayıcılık Analizi

Abstract

In this paper, we consider a time–space fractional advection–diffusion equation that models complex transport phenomena in heterogeneous media. The equation involves a Caputo fractional derivative and a fractional Laplacian. A detailed mathematical analysis of the proposed model is presented. The spectral properties of the corresponding operator are examined and a uniform coercivity condition is obtained under certain assumptions. It is also shown that the operator is sectorial, which allows using semigroup theory to prove existence and uniqueness of mild solutions. In contrast to most existing works that mainly focus on numerical approximations or particular cases, we provide a unified functional analytic framework for the fractional advection–diffusion model, clarifying its stability and solvability. The proposed approach gives us strong theoretical guarantees but may involve challenges for numerical implementation due to the nonlocal nature of the operators.

Keywords

Zaman-mekân kesirli adveksiyon-difüzyon denklemi , kesirli Laplace operatörü , zorlayıcılık kestirimi , spektral analiz

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