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Open AccessArticle

Fractional Vertical Infiltration

1
Mexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec 62550, Morelos, Mexico
2
Department of Mathematics, Faculty of Science, National Autonomous University of México, Av. Universidad 3000, Circuito Exterior S/N, Delegación Coyoacán 04510, Ciudad de México, Mexico
3
Water Research Center, Department of Irrigation and Drainage Engineering, Autonomous University of Querétaro, Cerro de las Campanas SN, Col. Las Campanas 76010, Querétaro, Mexico
*
Author to whom correspondence should be addressed.
Academic Editor: Duarte Valério
Mathematics 2021, 9(4), 383; https://doi.org/10.3390/math9040383
Received: 28 January 2021 / Revised: 10 February 2021 / Accepted: 10 February 2021 / Published: 14 February 2021
(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)
The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as independent variables and space as a dependent variable, and found the solution as a power series in t1/2 that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in tν/2, where ν(0,2) is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil. View Full-Text
Keywords: asymptotic solution; parlange equations; Darcy’s law; fractional Caputo derivative asymptotic solution; parlange equations; Darcy’s law; fractional Caputo derivative
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MDPI and ACS Style

Fuentes, C.; Alcántara-López, F.; Quevedo, A.; Chávez, C. Fractional Vertical Infiltration. Mathematics 2021, 9, 383. https://doi.org/10.3390/math9040383

AMA Style

Fuentes C, Alcántara-López F, Quevedo A, Chávez C. Fractional Vertical Infiltration. Mathematics. 2021; 9(4):383. https://doi.org/10.3390/math9040383

Chicago/Turabian Style

Fuentes, Carlos; Alcántara-López, Fernando; Quevedo, Antonio; Chávez, Carlos. 2021. "Fractional Vertical Infiltration" Mathematics 9, no. 4: 383. https://doi.org/10.3390/math9040383

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