Can We Remove Secular Terms for Analytical Solution of Groundwater Response under Tidal Influence?

TL;DR

This paper introduces a homotopy perturbation method to analytically solve nonlinear groundwater flow equations under tidal influence, effectively removing secular terms and providing solutions valid across all parameter ranges.

Contribution

It develops a secular term removal technique using homotopy perturbation, enabling accurate analytical solutions for tidal groundwater response without pre-specified parameters.

Findings

Analytical solutions up to third-order are derived.

The method effectively eliminates secular terms.

Solutions are valid for all ranges of A/D values.

Abstract

This paper presents a secular term removal methodology based on the homotopy perturbation method for analytical solutions of nonlinear problems with periodic boundary condition. The analytical solution for groundwater response to tidal fluctuation in a coastal unconfined aquifer system with the vertical beach is provided as an example. The non-linear one-dimensional Boussinesq's equation is considered as the governing equation for the groundwater flow. An analytical solution is provided for non-dimensional Boussinesq's equation with cosine harmonic boundary condition representing tidal boundary condition. The analytical solution is obtained by using homotopy perturbation method with a virtual embedding parameter. The present approach does not require pre-specified perturbation parameter and also facilitates secular terms elimination in the perturbation solution. The solutions starting from zeroth-order up to third-order are obtained. The non-dimensional expression, $A/D_{\infty}$ emerges as an implicit parameter from the homotopy perturbation solution. The non-dimensional solution is valid for all ranges of $A/D$ values. Higher order solution reveals the characteristics of the tidal groundwater table fluctuations.