Lagrangian Homotopy Analysis Method using the Least Action Principle

TL;DR

This paper introduces the Lagrangian Homotopy Analysis Method (LHAM), which uses the Least Action Principle to optimize the HAM parameter $c_0$ for Lagrangian systems, aiming to improve convergence and reduce computation time. By extremizing the approximated action $S_{approx}$ and selecting $c_0$ via energy conservation, LHAM yields faster convergence than the standard residual-error-square HAM, particularly for strongly nonlinear problems. The authors validate the approach on the harmonic oscillator, quartic and cubic anharmonic oscillators, and the KdV equation, demonstrating improved energy conservation and accuracy at given approximation orders. The method offers a robust variational framework with potential extensions to fractional equations and broader Hamiltonian systems, enhancing the applicability of HAM to complex nonlinear dynamics.

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.

Figures (7)

• Figure 1: The derivative of the action with respect to $c_0$, $\frac{\partial \,S}{\partial \,c_0}$, at the $6th$-order HAM approximation using LAP for the harmonic oscillator (\ref{['App2']}) with $t=1$ (and $\omega=1$). In the panels (b)-(c)-(d), we progressively zoom into the flat region of panel (a) in order to look for minima and maxima.
• Figure 2: Panel (a) shows the derivative of the action with respect to $c_0$, $\frac{\partial \,S}{\partial \,c_0}$, at different orders of HAM approximation with LAP with $t=1$ (and $\omega=1$). Panel (b) reports the energy vs. time showing energy conservation up to a time $t$ which increases with the orders of HAM approximation. Panel (c) shows the HAM approximation of the solution at increasing order. In panel (d) we compare the exact solution with the $12^{th}$-order of approximation. Plotting the difference between the exact solution and the approximate one at the $12^{th}$-order of the approximation, one sees that for $t$ between $-\pi$ and $\pi$ the error is smaller than $10^{-12}$. $\omega$ is taken as one.
• Figure 3: Panels (a), (b) and (c) show $|\Delta E|$ as a function of $c_0$ at different orders of HAM approximation. Panels (d), (e) and (f) show the derivative of the action with respect to $c_0$, $\frac{\partial \,S}{\partial \,c_0}$, at different orders of HAM approximation with LAP, respectively. $\omega$ and the time $t$ are taken as one.
• Figure 4: Panel (a) shows the derivative the action with respect to $c_0$, $\frac{\partial\,S}{\partial\,c_0}$, at different orders of HAM approximation with LAP with $t=1$. Panel (b) reports the energy vs. time showing energy conservation up to a time $t$ which increases with the orders of HAM approximation. Panel (c) shows the HAM approximation of the solution at increasing order. In panel (d) we compare the exact solution with the $8^{th}$-order of approximation. $k$ and $\gamma$ are constants of order unity, and $a=\frac{1}{\sqrt{2}}$.
• Figure 5: Numerical solution of the cubic anharmonic oscillator problem. The thick dashed plot represents the exact solution with zero velocity $(\dot{x}_0=0)$, while the dashed plot gives the solution with nonzero velocity $(\dot{x}_0=\frac{1}{4})$.