A Theoretical Counterexample of Solutions to Unsaturated Infiltration Equations

DOI

10.2991/978-94-6463-398-6_16How to use a DOI?

Keywords

Unsaturated infiltration; Richards equation; Laplace transform; Nonlinear differential equation; Convergence of functions

Abstract

A common method for obtaining the analytical solution of the unsaturated infiltration Richards equation under the first type of boundary condition is the Laplace transform. By linearizing the differential equation through the Laplace transform, solving it, and then applying the inverse transform, the result can be obtained. In this process, the traditional calculation scheme assumes that the original function converges, which is correct in most engineering cases, but theoretically, counterexamples may exist. This article presents a classic counterexample: when the unsaturated infiltration curve has a power-law form, the original integral diverges, resulting in the non-existence of the Laplace transform and the solution of the equation cannot represent the real infiltration situation. Based on this, this article suggests discussing the convergence of the original function before using the Laplace transform to solve the unsaturated infiltration equation to ensure that the results have mathematical significance. The content discussed in this article can be regarded as an improvement and supplement to the traditional unsaturated infiltration calculation scheme.

Copyright

© 2024 The Author(s)

Open Access

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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TY  - CONF
AU  - Yuelu Zhu
AU  - Yingxing Chen
PY  - 2024
DA  - 2024/04/24
TI  - A Theoretical Counterexample of Solutions to Unsaturated Infiltration Equations
BT  - Proceedings of the 2023 5th International Conference on Hydraulic, Civil and Construction Engineering (HCCE 2023)
PB  - Atlantis Press
SP  - 155
EP  - 165
SN  - 2589-4943
UR  - https://doi.org/10.2991/978-94-6463-398-6_16
DO  - 10.2991/978-94-6463-398-6_16
ID  - Zhu2024
ER  -