Approximate analytical solution for the one-dimensional nonlinear Boussinesq equation

Abstract

Purpose – The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation. Design/methodology/approach – The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions. Findings – The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences. Originality/value – The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.