IJCOPE Journal

UGC Logo DOI / ISO Logo

International Journal of Creative and Open Research in Engineering and Management

A Peer-Reviewed, Open-Access International Journal Supporting Multidisciplinary Research, Digital Publishing Standards, DOI Registration, and Academic Indexing.
Journal Information
ISSN: 3108-1754 (Online)
Crossref DOI: Available
ISO Certification: 9001:2015
Publication Fee: 599/- INR
Compliance: UGC Journal Norms
License: CC BY 4.0
Peer Review: Double Blind
Volume 02, Issue 10

Published on: October 2026

FRACTIONAL-ORDER ADVECTION–DIFFUSION EQUATIONS: A CRITICAL REVIEW OF PHYSICAL ORIGINS, ANALYTICAL AND NUMERICAL METHODS, AND APPLICATIONS

Weiguang Huang

DrHuang.com, Sydney, NSW 2033, Australia
DOI:https://doi.org/10.55041/ijcope.v2i10.088

Article Status

Plagiarism Passed Peer Reviewed Open Access

Available Documents

Download PDF Review Report

Abstract

The advection–diffusion equation is the canonical continuum description of transport, yet it fails for a broad class of systems in which particle motion is non-Fickian. Fractional-order advection–diffusion equations replace integer derivatives with operators of non-integer order and, in doing so, encode long-range spatial jumps and long-memory waiting times that classical models discard. This review provides a critical, evidence-based synthesis of the field from its foundations in continuous-time random walks and Lévy motion through the mathematical and numerical landscape. We first examine the family of fractional operators and the physical meaning of their kernels. We then trace the micro–macro derivations that connect fractional advection–diffusion equations to anomalous transport, mobile–immobile systems, tempered and variable-order processes. Analytical methods and the now-mature toolbox of finite difference, finite element, spectral, meshless and lattice-Boltzmann discretizations are appraised for accuracy, stability and computational cost. Inverse problems and parameter estimation, which decide whether such models earn their extra complexity, are reviewed next. Applications in hydrology, porous media, plasmas, space physics, heat and mass transfer, biology and finance illustrate both success and unresolved tension. Throughout, we identify trends—rising use of non-singular-kernel operators, machine-learning solvers and realistic benchmarking—and we flag persistent gaps: inconsistent operator conventions, sparse validation against real data, and scarce higher-dimensional and multiscale results. We close with a concrete research agenda for making fractional advection–diffusion models predictive rather than descriptive.

Keywords: fractional calculus; anomalous diffusion; advection–diffusion equation; Lévy motion; Caputo derivative; mobile–immobile transport; finite difference methods; inverse problems.

How to Cite this Paper

Huang, W. (2026). Fractional-Order Advection–Diffusion Equations: A Critical Review of Physical Origins, Analytical and Numerical Methods, and Applications. International Journal of Creative and Open Research in Engineering and Management, <i>02</i>(10), 1-9. https://doi.org/10.55041/ijcope.v2i10.088

Huang, Weiguang. "Fractional-Order Advection–Diffusion Equations: A Critical Review of Physical Origins, Analytical and Numerical Methods, and Applications." International Journal of Creative and Open Research in Engineering and Management, vol. 02, no. 10, 2026, pp. 1-9. doi:https://doi.org/10.55041/ijcope.v2i10.088.

Huang, Weiguang. "Fractional-Order Advection–Diffusion Equations: A Critical Review of Physical Origins, Analytical and Numerical Methods, and Applications." International Journal of Creative and Open Research in Engineering and Management 02, no. 10 (2026): 1-9. https://doi.org/https://doi.org/10.55041/ijcope.v2i10.088.

Search & Index

Google Scholar Crossref DOI ResearchGate

References


  1. Benson DA, Wheatcraft SW, Meerschaert MM. Application of a fractional advection-dispersion equation. Water Resources Research. 2000;36(6):1403-1412. doi:10.1029/2000wr900031

  2. Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports. 2000;339(1):1-77. doi:10.1016/s0370-1573(00)00070-3

  3. Benson DA, Schumer R, Meerschaert MM, Wheatcraft SW. Fractional dispersion, Lévy motion, and the MADE tracer tests. Transport in Porous Media. 2001;42(1-2):211-240. doi:10.1023/a:1006733002131

  4. Baeumer B, Benson DA, Meerschaert MM, Wheatcraft SW. Subordinated advection-dispersion equation for contaminant transport. Water Resources Research. 2001;37(6):1543-1550. doi:10.1029/2000wr900409

  5. Huang G, Huang Q, Zhan H. Evidence of one-dimensional scale-dependent fractional advection-dispersion. Journal of Contaminant Hydrology. 2006;85(1-2):53-71. doi:10.1016/j.jconhyd.2005.12.007

  6. del-Castillo-Negrete D, Carreras BA, Lynch VE. Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Physical Review Letters. 2005;94(6):065003. doi:10.1103/physrevlett.94.065003

  7. Zimbardo G, Perri S. On the fractional diffusion-advection equation for fluids and plasmas. Fluids. 2019;4(2):62. doi:10.3390/fluids4020062

  8. Zhang X, Lv M, Crawford JW, Young IM. The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: defining the fractional dispersive flux with the Caputo derivatives. Advances in Water Resources. 2007;30(5):1205-1217. doi:10.1016/j.advwatres.2006.11.002

  9. Lei D, Sun H, Zhang Y, Blaszczyk T, Yu Z. Upscaling solute transport in rough single-fractured media with matrix diffusion using a time fractional advection-dispersion equation. Journal of Hydrology. 2023;627:130280. doi:10.1016/j.jhydrol.2023.130280

  10. Metzler R, Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General. 2004;37(31):R161-R208. doi:10.1088/0305-4470/37/31/r01

Ethical Compliance & Review Process

  • •All submissions are screened under plagiarism detection.
  • •Review follows editorial policy.
  • •Authors retain copyright.
  • •Peer Review Type: Double-Blind Peer Review
  • •Published on: Oct 15 2026
CCBYNC

This article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. You are free to share and adapt this work for non-commercial purposes with proper attribution.

View License
Back to Volume 02, Issue 10View All Issues Next Article

← Previous Article

Fractional Calculus Meets Mathematics Education

Next Article →

Geometry of Lead-Time Spectra: Bullwhip as Curvature on a Constrained Manifold
Scroll to Top