Discussion on Several Remaining Problems in the Laplace Transform Solution of Richards Equation
In this paper we addressed and elaborated on three critical issues concerning the Laplace transform solu-tion of the Richards equation:the convergence of the original function,the steps for inverse transform,and the ap-plicability of the equation.By presenting theoretical counterexamples,employing integral transformations and substi-tution schemes as well as real engineering cases,we demonstrated that assuming the convergence of the original function can lead to significant errors.Therefore,it is necessary to assess the convergence of the infiltration function before applying the Laplace transform.We also elucidated the steps for solving the Laplace inverse transform,theo-retically addressing a gap in geotechnical literature.Furthermore,we reveal an anomaly where,in certain applica-tions,the Laplace solution may show higher moisture content in lower soil layers than in upper layers.We analyzed this phenomenon from both physical and mathematical perspectives,highlighting inherent limitations in this solu-tion.The findings of this study may help to theoretically strengthen the framework of unsaturated soil theory.
soil mechanicsRichards Equationinverse Laplace transformunsaturated infiltrationsolution of dif-ferential equation
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