Lagrangian Homotopy Analysis Method using the Least Action Principle

TL;DR

This paper introduces a variational enhancement to the Homotopy Analysis Method using the Least Action Principle, improving convergence and accuracy for solving Lagrangian systems and nonlinear differential equations.

Contribution

It proposes a novel variational approach based on the Least Action Principle to optimize the HAM parameter, enhancing solution accuracy and convergence speed.

Findings

Accelerates convergence of the HAM parameter to the exact solution.

Performs better than standard HAM when the exact solution is unknown.

Shows improved accuracy with higher approximation order and stronger nonlinearity.

Abstract

The Homotopy Analysis Method (HAM) is a powerful technique which allows to derive approximate solutions of both ordinary and partial differential equations. We propose to use a variational approach based on the Least Action Principle (LAP) in order to improve the efficiency of the HAM when applied to Lagrangian systems. The extremization of the action is achieved by varying the HAM parameter, therefore controlling the accuracy of the approximation. As case studies we consider the harmonic oscillator, the cubic and the quartic anharmonic oscillators, and the Korteweg-de Vries partial differential equation. We compare our results with those obtained using the standard approach, which is based on the residual error square method. We see that our method accelerates the convergence of the HAM parameter to the exact value in the cases in which the exact solution is known. When the exact solution is not analytically known, we find that our method performs better than the standard HAM for the cases we have analyzed. Moreover, our method shows better performance when the order of the approximation is increased and when the nonlinearity of the equations is stronger.