Reducción de problemas de difusión con condiciones de contorno complejas a problemas de Riemann–Hilbert y su aplicación en ingeniería civil

Jaime Adrián Peralta-Delgado; Ronald Pastor Delgado-Alvia; Edgar Mauro Caicedo-Álvarez; Edgar Mauricio Caicedo-Tapia

Authors

Peralta-Delgado Jaime Adrián Universidad Estatal del Sur de Manabí. Jipijapa, Ecuador. https://orcid.org/0000-0003-3830-9719
Delgado-Alvia Ronald Pastor Universidad Estatal del Sur de Manabí. Jipijapa, Ecuador. https://orcid.org/0000-0001-9098-2922
Caicedo-Álvarez Edgar Mauro Universidad Estatal del Sur de Manabí. Jipijapa, Ecuador. https://orcid.org/0000-0002-6354-3307
Caicedo-Tapia Edgar Mauricio Universidad Politécnica Salesiana. Quito, Ecuador. https://orcid.org/0009-0001-8931-4326

Keywords:

Riemann–Hilbert problem; Fourier transform; contaminant transport; porous media; diffusion modeling; boundary-value problems

Abstract

DOI: https://doi.org/10.46296/ig.v9i17.0334

Abstract

Diffusion modeling is fundamental in civil and environmental engineering, particularly for describing contaminant transport in soils and porous media. In many practical situations, complex boundary conditions hinder analytical solutions and encourage reliance on numerical models, which may introduce discretization errors when reference solutions are unavailable. This study proposes a methodological framework to reduce parabolic partial differential equations with complex boundary conditions to Riemann–Hilbert problems using the Fourier transform. The boundary-value problem is reformulated in the transformed domain, where solvability is examined through analytic function theory. Solutions are obtained in quadrature form and reconstructed by inverse transforms. The approach provides a systematic pathway to derive analytical or semi-analytical solutions in semi-infinite domains. From an engineering standpoint, these solutions serve as benchmarks for validating numerical simulations in contaminant diffusion and related transport processes in porous materials. Although direct application is limited in highly heterogeneous media, the framework offers value as a reference methodology linking theoretical analysis and applied modeling.

Keywords: Riemann–Hilbert problem; Fourier transform; contaminant transport; porous media; diffusion modeling; boundary-value problems.

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References

. Mwakilama, E., Gathungu, D., & Magagula, V. (2023). Use of the repeated integral transformation method to describe the transport of solute in soil. Heliyon, 9(1), e12774. https://doi.org/10.1016/j.heliyon.2022.e12774

. Radha, R., Singh, R. K., & Singh, M. K. (2022). Contaminant transport analysis under non-linear sorption in a heterogeneous groundwater system. Applied Mathematics in Science and Engineering, 30(1), 736–761. https://doi.org/10.1080/27690911.2022.2138867

. Kim, M. (2024). Modeling pollutant diffusion in the ground using conformal fractional time derivative. Symmetry, 16(10), 1358. https://doi.org/10.3390/sym16101358

. Hwang, G. (2021). Analytical solution for the two-dimensional linear advection-dispersion equation in porous media via the Fokas method. Journal of Applied Analysis & Computation, 11(5), 2334–2354. https://doi.org/10.11948/20200383

. Singh, D. K., Goswami, A., & Paul, T. (2023). Contaminant dispersion with axial input sources in soil media under non-linear sorption. Environmental Technology, 44(13), 1903-1915. https://doi.org/10.1080/09593330.2021.2016992

. Dentz, M., Le Borgne, T., Englert, A., & Bijeljic, B. (2011). Mixing, spreading and reaction in heterogeneous media: A brief review. Journal of Contaminant Hydrology, 120–121, 1–17. https://doi.org/10.1016/j.jconhyd.2010.05.002

. Pino Tarragó, J. C., et al. (2025). Impacto de los residuos de construcción y demolición en la producción de hormigón: un enfoque hacia la sostenibilidad. Revista Ciencia y Construcción, 6(4).

. Pino Tarragó, J. C., et al. (2024). Integrated wastewater treatment technology: Efficiency of lime and Eichhornia crassipes for agricultural irrigation. Revista Salud, Ciencia y Tecnología. https://doi.org/10.56294/saludcyt2024.1313

. Leij, F. J., Toride, N., & van Genuchten, M. T. (1991). Analytical solutions for non-equilibrium solute transport in three-dimensional porous media. Journal of Hydrology, 151(2–4), 193–228. https://doi.org/10.1016/0022-1694(93)90218-E

. Berkowitz, B., Cortis, A., Dentz, M., & Scher, H. (2006). Modeling non-Fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics, 44(2), RG2003. https://doi.org/10.1029/2005RG000178

. Zhang, Y., Benson, D. A., & Reeves, D. M. (2009). Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources, 32(4), 561–581. https://doi.org/10.1016/j.advwatres.2009.01.008

. Zeng, X., Zhan, L.-T., & Chen, Y.-M. (2017). Applicability of boundary conditions for analytical modelling of advection–dispersion transport in low-permeability clay column tests. Chinese Journal of Geotechnical Engineering, 39(4), 636–644. https://doi.org/10.11779/CJGE201704007

. Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O. A., & Grathwohl, P. (2009). Enhancement of dilution and transverse reactive mixing in porous media: Experiments and model-based interpretation. Journal of Contaminant Hydrology, 110(3–4), 130–142. https://doi.org/10.1016/j.jconhyd.2009.10.003