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

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

References

1. [1] Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M., “The fractional-order governing equation of Lévy motion”, Water Resources Research 36(6) (2000) : 1413–1423.
2. [2] Berkowitz, B., Cortis, A., Dentz, M., Scher, H., “Modeling non-Fickian transport in geological formations as a continuous time random walk”, Reviews of Geophysics 44(2) (2006).
3. [3] Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X. et al., “What is the fractional Laplacian? A comparative review with new results”, Journal of Computational Physics 404 (2020) : 109009.
4. [4] Hrizi, M., Hajji, F., Prakash, R., Novotny, A.A., “Reconstruction of a singular source in a fractional subdiffusion problem from a single point measurement”, Applied Mathematics & Optimization 90.2(40) (2024).
5. [5] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
6. [6] Metzler, R., Klafter, J., “The random walk’s guide to anomalous diffusion: a fractional dynamics approach”, Physics Reports 339(1) (2000) : 1–77.
7. [7] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier 198 (1998).
8. [8] Tarasov, V.E., “Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media”, Springer Science & Business Media 2011.

9. [9] Lin, Y.H., Railo, J., Zimmermann, P., “The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients”, Revista Matemática Iberoamericana 41(3) (2025) : 1129-1172.
10. [10] Stein, E.M., “Singular Integrals and Differentiability Properties of Functions”, Princeton University Press 30 (1970).
11. [11] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media 44 (2012).
12. [12] Arfaoui, H., Makhlouf, A.B., “Stability of a time fractional advection–diffusion system”, Chaos, Solitons & Fractals 157 (2022) : 111949.
13. [13] Agarwal, R.P., O'Regan, D., Shakhmurov, V.B., “Separable anisotropic differential operators in weighted abstract spaces and applications”, Journal of mathematical analysis and applications 338(2) (2008) : 970-983.
14. [14] Arshad, S., Baleanu, D., Huang, J., Al Qurashi, M.M., Tang, Y., Zhao, Y., “Finite difference method for time–space fractional advection–diffusion equations with Riesz derivative”, Entropy 20(5) (2018) : 321.
15. [15] Albritton, D., Armstrong, S., Mourrat, J.C., Novack, M., “Variational methods for the kinetic Fokker–Planck equation”, Analysis & PDE 17(6) (2024) : 1953–2010.
16. [16] Di Nezza, E., Palatucci, G., Valdinoci, E., “Hitchhiker’s guide to the fractional Sobolev spaces”, Bulletin des Sciences Mathématiques 136(5) (2012) : 521–573.