Semi-Analytical Solutions of the Rayleigh Oscillator Using Laplace–Adomian Decomposition and Homotopy Perturbation Methods: Insights into Symmetric and Asymmetric Dynamics

Abstract

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica’s NDSolve function. We perform a detailed error analysis across different damping parameter values ε and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions.

1. Introduction

Nonlinear oscillators have drawn significant attention due to their relevance in numerous physical, biological, and engineering systems. Among these, the Rayleigh oscillator stands out as a classical model that captures the effects of nonlinear, asymmetric damping. Originally introduced by Lord Rayleigh in their seminal work The Theory of Sound [1], the Rayleigh oscillator describes systems in which the damping force depends nonlinearly on velocity, thereby breaking the time-reversal symmetry often found in linear oscillatory systems. This asymmetry introduces rich dynamical behaviors, such as self-sustained oscillations and limit cycles, making the Rayleigh model a key representative in the study of nonlinear and non-conservative systems. These behaviors include the emergence of stable limit cycles, which represent self-sustained oscillations largely independent of initial conditions—a phenomenon crucial for understanding rhythmic processes in nature and engineering. The departure from time-reversal symmetry, a direct consequence of the velocity-dependent damping, leads to irreversible energy dissipation patterns and complex phase-space trajectories that are not captured by simpler linear models or conservative nonlinear systems. Its status as a key representative stems from its ability to model systems where energy is dynamically balanced—supplied to sustain oscillations at a certain amplitude, counteracting dissipative forces in a nonlinear fashion—a characteristic observed in fields ranging from electronics to biomechanics. Understanding the mechanisms governing these dynamics, particularly the conditions that lead to symmetric or asymmetric oscillations, is crucial for both theoretical comprehension and practical application.

The general form of the Rayleigh oscillator equation is given by

$$\ddot{x}+\epsilon (1-\alpha {\dot{x}}^2)\dot{x}+{\omega }^{2}x=0,$$

(1)

where $\ddot{x}$ represents acceleration, $\dot{x}$ the velocity, x the displacement, $\epsilon$ a small positive parameter representing the strength of the nonlinear damping, $\alpha$ a constant determining the nature of the nonlinearity, and $\omega$ the natural frequency of the system. This general form encompasses the classical Rayleigh oscillator, commonly expressed as

$$\ddot{x}+\epsilon \left({\displaystyle \frac{1}{3}}{\dot{x}}^3-\dot{x}\right)+x=0.$$

(2)

or equivalently

$$\ddot{x}-\epsilon \dot{x}+x+{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3=0,0<\epsilon \ll 1$$

(3)

Rayleigh’s early work [1] was the starting point for later studies on nonlinear oscillators. Because of the nonlinear nature of these systems, finding exact solutions was very difficult in most cases. This challenge led researchers to develop many different methods, both analytical and numerical, over the years. Nayfeh and Mook [2] expanded on this by providing a comprehensive treatment of nonlinear oscillators, including both the Rayleigh and Van der Pol models. Their work, along with contributions by Strogatz [3] and Jordan and Smith [4] employing further analytical and numerical methods, significantly advanced the understanding of these complex systems. A recent study by Zhang et al. [5] proposed an improved Rayleigh–Plesset-type bubble dynamics equation that incorporates the effects of surface tension and viscosity to model the oscillation and collapse behavior of cavitation bubbles within droplets. Their work combines high-speed photography experiments with numerical simulations to analyze bubble collapse modes and associated droplet splash phenomena. While exact solutions are often intractable, semi-analytical methods such as the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM) have proven effective for obtaining approximate solutions [6]. Recent studies, such as that by Alqahtani et al. [7], further demonstrate the robustness of the HPM and its flexibility in addressing complex dynamical systems. Other modern techniques such as the Adomian Decomposition Method (ADM) [8] and the Variational Iteration Method (VIM) [9] have also enabled researchers to derive approximate solutions to these complex equations, each offering unique advantages for particular classes of problems.

The Rayleigh oscillator finds applications in a wide range of disciplines, underscoring its fundamental importance and the practical necessity of robust solution methodologies. For instance, in electrical engineering, it models the behavior of nonlinear electrical circuits, particularly those exhibiting self-oscillations like multivibrators and feedback oscillators [2]. Historically, in early electronics, it was applied to understand the dynamics of vacuum tube oscillators, crucial components in the development of radio and telecommunications [1]. Beyond electronics, its framework is instrumental in describing aeroelastic flutter in mechanical structures, a dangerous phenomenon in aerospace engineering where aerodynamic forces interact with structural vibrations, potentially leading to catastrophic failure [3]. Furthermore, it also plays an essential role in mathematical biology, for example, in understanding limit cycles in biological systems such as neural firing patterns or rhythmic physiological processes [4]. Moreover, the oscillator has been used to investigate combustion instabilities in engines and power plants and to design control systems that regulate or harness complex nonlinear behaviors [2]. The diverse applicability of the Rayleigh oscillator, particularly its capacity to exhibit both symmetric and asymmetric responses depending on system parameters and excitation, highlights the critical need for versatile solution methods capable of capturing its nuanced dynamics across these varied physical contexts.

Despite its wide applicability and long history of study, obtaining accurate analytical or semi-analytical solutions for the Rayleigh oscillator across all parameter regimes remains a significant undertaking. Solving the Rayleigh oscillator typically involves either analytical perturbation techniques or numerical schemes. While regular perturbation methods, as discussed by Nayfeh and Mook [2], have been instrumental, they often rely on the assumption of a small parameter (like $\epsilon$ in Equation (3)). This dependence can limit their accuracy and range of validity, especially when the nonlinear effects are not weak or when the long-term behavior of the system is of primary interest. Furthermore, such methods might struggle to fully capture global dynamical features or the intricacies of asymmetric responses. Numerical methods, on the other hand, can provide highly accurate solutions for specific initial conditions and parameter values. However, they may offer limited direct insight into the general parametric dependence of the solution’s structure and can be computationally intensive for extensive parameter sweeps or for achieving high precision in potentially stiff systems. Moreover, while precise, numerical solutions may not always directly reveal the underlying mathematical mechanisms leading to specific dynamical features such as bifurcations or the emergence of particular symmetries in the oscillations. Therefore, the development and application of powerful semi-analytical methods continue to be crucial for bridging the gap between purely numerical results and the often-restrictive assumptions of classical perturbation theory, offering a balance between analytical insight and broad applicability.

In recent years, many researchers have applied various semi-analytical methods to solve nonlinear differential equations, such as those describing heat transfer and oscillatory phenomena, demonstrating their power and versatility. In this work, we focus on two such prominent approaches to solve the Rayleigh oscillator equation: the Laplace–Adomian Decomposition Method (LADM) [6,10] and the Homotopy Perturbation Method (HPM) [11]. These methods are particularly well-suited for the Rayleigh oscillator because they do not strictly require the presence of a small parameter, offering potential advantages in regimes where traditional perturbation methods may struggle. The HPM, introduced by He in 1998, constructs a homotopy with an embedding parameter $p\in [0, 1]$, deforming a difficult problem into a series of simpler, solvable problems. The solution is then expressed as a power series in p, which often converges rapidly to an accurate approximation of the true solution. This method has been shown to be highly effective across a range of linear and nonlinear problems, including the nonlinear Schrödinger Equation [12,13,14,15,16]. A key advantage of the HPM is its ability to provide analytical expressions that can explicitly elucidate the influence of system parameters on the solution structure, which is particularly valuable for understanding how parameters govern the symmetric or asymmetric nature of the oscillations predicted by the Rayleigh model.

The LADM, on the other hand, works by combining the Laplace transform with the Adomian decomposition technique [8]. First, the Laplace transform changes the differential equation into an algebraic one, which makes it easier to deal with the linear parts and the initial conditions. Then, the Adomian decomposition method is used to handle the nonlinear parts by expressing them as a series of special polynomials, called Adomian polynomials. This step-by-step approach helps to build the solution as a series. One of the main benefits of the LADM is that it can be used directly on many types of nonlinear equations without needing to simplify them or turn them into numerical form. It also tends to give solutions that converge quickly [6,10,17,18,19].

In this work, we apply both the LADM and HPM to find approximate solutions for the Rayleigh oscillator. Our goal is to compare the two methods and show how effective they are in solving this kind of nonlinear problem. We also aim to highlight how these methods can offer useful insights into how the oscillator’s behavior changes with different parameters. These insights are important for gaining a deeper understanding of the system and for using it in various science and engineering applications. In the analysis of nonlinear oscillators, numerical methods serve as indispensable benchmarks for validating analytical approximations. We employ NDSolve from Mathematica, which utilizes adaptive step-size control and advanced algorithms (including Runge–Kutta and backward differentiation formulae) to generate high-precision numerical solutions. This approach follows established validation methodologies in nonlinear dynamics [2].

The structure of this paper is designed to provide a clear and systematic presentation of our work, drawing parallels with established methodologies in nonlinear dynamics (e.g., ref. [4]). Section 2 and Section 3 are dedicated to outlining the mathematical foundations of the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM), respectively. Subsequently, Section 4 and Section 5 present our detailed derivations of the analytical solutions for the Rayleigh oscillator using the LADM and HPM. A comprehensive error analysis, including comparisons with numerical results (NDSolve) and discussions on physical interpretations, is provided in Section 6. The crucial aspect of solution convergence for the obtained series is addressed in Section 7. Finally, Section 8 summarizes the key findings and conclusions of this study.

2. The Laplace Decomposition Approach

This section provides a brief review of the standard Laplace–Adomian Decomposition Method (LADM), outlining its established procedural steps for solving general nonlinear differential equations. The purpose is to present the foundational methodology, as described in the existing literature [17], which will subsequently be applied to the specific problem of the Rayleigh oscillator. This review serves as a basis for understanding our application and is not a new derivation of the method itself.

To illustrate the fundamental principles of this technique, we examine a general non-homogeneous partial differential equation accompanied by an initial condition, expressed as [17]

$$\mathcal{D}u(x,t)+\mathcal{R}u(x,t)+\mathcal{N}u(x,t)=f(x,t),\mathrm{with}u(x,0)=\varphi \left(x\right),$$

(4)

where

• $\mathcal{D}$ denotes the linear differential operator ${\displaystyle \frac{\partial }{\partial t}}$;
• $\mathcal{R}$ represents the remaining linear operator of lower order compared to $\mathcal{D}$;
• $\mathcal{N}$ stands for the nonlinear differential operator;
• $f(x,t)$ is the source term.

Applying the Laplace transform $\mathcal{L}$ to (4) yields

$$\mathcal{L}\left\{\mathcal{D}u(x,t)\right\}+\mathcal{L}\left\{\mathcal{R}u(x,t)\right\}+\mathcal{L}\left\{\mathcal{N}u(x,t)\right\}=\mathcal{L}\left\{f(x,t)\right\}.$$

(5)

Rearranging terms, we obtain

$$\mathcal{L}\left\{u(x,t)\right\}={\displaystyle \frac{\varphi \left(x\right)}{s}}-{\displaystyle \frac{1}{s}}\mathcal{L}\left\{\mathcal{R}u(x,t)\right\}+{\displaystyle \frac{1}{s}}\mathcal{L}\left\{f(x,t)\right\}-{\displaystyle \frac{1}{s}}\mathcal{L}\left\{\mathcal{N}u(x,t)\right\}.$$

(6)

Assuming the solution can be represented as an infinite series expansion,

$$u(x,t)=\sum _{k=0}^{\infty }{u}_k(x,t),$$

(7)

the nonlinear term $\mathcal{N}u(x,t)$ is decomposed into Adomian polynomials $A_k$:

$$\mathcal{N}u(x,t)=\sum _{k=0}^{\infty }{A}_k\left(u\right),$$

(8)

where the Adomian polynomials are defined by

$$A_k={\displaystyle \frac{1}{k!}}{\left.{\displaystyle \frac{d^k}{d{\lambda }^k}}\left[\mathcal{N}\left(\sum _{m=0}^{\infty }{\lambda }^m{u}_m\right)\right]\right|}_{\lambda =0},k=0,1,2,\dots .$$

(9)

Substituting (7) and (8) into (6), we derive

$$\sum _{k=0}^{\infty }\mathcal{L}\left\{u_k(x,t)\right\}={\displaystyle \frac{\varphi \left(x\right)}{s}}-{\displaystyle \frac{1}{s}}\mathcal{L}\left\{\mathcal{R}u(x,t)\right\}+{\displaystyle \frac{1}{s}}\mathcal{L}\left\{f(x,t)\right\}-{\displaystyle \frac{1}{s}}\sum _{k=0}^{\infty }\mathcal{L}\left\{A_k\left(u\right)\right\}.$$

(10)

By matching terms on both sides of (10), we obtain the following recursive relations:

$$\begin{array}{cc}\hfill \mathcal{L}\left\{u_0(x,t)\right\}& ={\displaystyle \frac{\varphi \left(x\right)}{s}}+{\displaystyle \frac{1}{s}}\mathcal{L}\left\{f(x,t)\right\},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \mathcal{L}\left\{u_{k+1}(x,t)\right\}& =-{\displaystyle \frac{1}{s}}\mathcal{L}\left\{\mathcal{R}{u}_k(x,t)\right\}-{\displaystyle \frac{1}{s}}\mathcal{L}\left\{A_k\left(u\right)\right\},k\ge 0.\hfill \end{array}$$

(12)

Applying the inverse Laplace transform ${\mathcal{L}}^{-1}$ to these relations gives

$$u_0(x,t)=\Psi (x,t),$$

(13)

and for $k\ge 0$:

$$u_{k+1}(x,t)=-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{1}{s}}\mathcal{L}\left\{\mathcal{R}{u}_k(x,t)\right\}+{\displaystyle \frac{1}{s}}\mathcal{L}\left\{A_k\left(u\right)\right\}\right\},$$

(14)

where $\Psi (x,t)$ incorporates the initial condition and the source term [17].

3. The Homotopy Perturbation Technique

This section revisits the established framework of the Homotopy Perturbation Method (HPM), detailing its core principles and standard formulation for addressing nonlinear equations, drawing from existing works such as [9,20]. The objective is to clearly present the theoretical underpinnings of the HPM before its application to the Rayleigh oscillator in subsequent sections.

Definition 1 

([20]). Let $\mathcal{X}$ and $\mathcal{Y}$ be topological spaces. Two continuous maps f and g are called homotopic relative to a subset $\mathcal{A}\subseteq \mathcal{X}$ if there exists a continuous homotopy $\mathcal{H}:\mathcal{X}\times [0,1]\to \mathcal{Y}$ satisfying

$$\begin{array}{cc}\hfill \mathcal{H}(x,0)& =f\left(x\right),\forall x\in \mathcal{X},\hfill \\ \hfill \mathcal{H}(x,1)& =g\left(x\right),\forall x\in \mathcal{X},\hfill \\ \hfill \mathcal{H}(a,p)& =f\left(a\right)=g\left(a\right),\forall p\in [0,1],\forall a\in \mathcal{A}.\hfill \end{array}$$

To illustrate the core methodology of the Homotopy Perturbation Method (HPM), consider the following nonlinear equation:

$$\mathcal{A}\left(u\right)=f\left(x\right),x\in \Omega ,$$

(15)

where $\mathcal{A}$ is a general operator and f is a known function. The operator $\mathcal{A}$ can be split into a linear component $\mathcal{L}$ and a nonlinear component $\mathcal{N}$, yielding

$$\mathcal{L}\left(u\right)+\mathcal{N}\left(u\right)-f\left(x\right)=0.$$

(16)

Following [9], we define a homotopy $vs.:\Omega \times [0,1]\to \mathbb{R}$ that satisfies

$$\mathcal{H}(v,p)=(1-p)\left[\mathcal{L}\left(v\right)-\mathcal{L}\left(u_0\right)\right]+p\left[\mathcal{A}\left(v\right)-f\left(x\right)\right]=0,$$

(17)

which expands to

$$\mathcal{H}(v,p)=\mathcal{L}\left(v\right)-\mathcal{L}\left(u_0\right)+p\mathcal{L}\left(u_0\right)+p\mathcal{N}\left(v\right)-pf\left(x\right)=0,p\in [0,1].$$

(18)

Here, $u_0$ is an initial guess for the solution of (15). The homotopy satisfies

$$\begin{array}{cc}\hfill \mathcal{H}(v,0)& =\mathcal{L}\left(v\right)-\mathcal{L}\left(u_0\right)=0,\hfill \\ \hfill \mathcal{H}(v,1)& =\mathcal{A}\left(v\right)-f\left(x\right)=0.\hfill \end{array}$$

As p evolves from 0 to 1, the solution $v(x,p)$ continuously deforms from $u_0\left(x\right)$ to $u\left(x\right)$, implying

$$\begin{array}{cc}\hfill \mathcal{L}\left(v\right)-\mathcal{L}\left(u_0\right)& \sim \mathcal{A}\left(v\right)-f\left(x\right),x\in \Omega ,\hfill \\ \hfill u_0\left(x\right)& \sim u\left(x\right),x\in \Omega .\hfill \end{array}$$

Assuming the solution can be expanded as a power series in the embedding parameter p:

$$v(x,p)=\sum _{k=0}^{\infty }{p}^k{v}_k\left(x\right),$$

(19)

the exact solution to (15) is obtained by setting $p=1$:

$$u\left(x\right)=\underset{p\to 1}{lim}v(x,p)=\sum _{k=0}^{\infty }{v}_k\left(x\right).$$

(20)

The convergence of this series is guaranteed under specific conditions, as detailed in [9].

The Homotopy Perturbation Method (HPM) merges the principles of perturbation theory with homotopy analysis, effectively overcoming the constraints inherent in classical perturbation techniques [12,13,14].

The analytical solutions of the nonlinear Rayleigh oscillator equation are presented below, utilizing the two different methods: the Laplace–Adomian Decomposition Method and the Homotopy Perturbation Method.

4. Analytical Solution of the Nonlinear Rayleigh Oscillator Using the Laplace–Adomian Decomposition Method

This section outlines the main contribution of this part of the study. It applies the Laplace–Adomian Decomposition Method (LADM) to solve the nonlinear Rayleigh oscillator using the given initial conditions. The process includes forming the equation in the Laplace domain and computing the first three terms ($x_0,x_1,x_2$) of the Adomian series. The resulting three-term approximation is then compared to numerical results, showing how the LADM works for this nonlinear system. Consider the Rayleigh oscillator Equation (3):

$$\ddot{x}-\epsilon \dot{x}+x+{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3=0.$$

(21)

To solve the Rayleigh oscillator equation using the Laplace–Adomian decomposition method, we apply the steps mentioned in the previous section. Assume $x\left(0\right)=0$ and $\dot{x}\left(0\right)=1$ as initial conditions. Applying the Laplace transform to both sides yields

$$\mathcal{L}\left\{\ddot{x}\right\}-\epsilon \mathcal{L}\left\{\dot{x}\right\}+\mathcal{L}\left\{x\right\}+{\displaystyle \frac{\epsilon }{3}}\mathcal{L}\left\{{\dot{x}}^3\right\}=\mathcal{L}\left\{0\right\}.$$

(22)

Using the properties of the Laplace transform for derivatives, we obtain

$$s^2\mathcal{L}\left\{x\right\}-sx\left(0\right)-\dot{x}\left(0\right)-\epsilon (s\mathcal{L}\left\{x\right\}-x\left(0\right))+\mathcal{L}\left\{x\right\}+{\displaystyle \frac{\epsilon }{3}}\mathcal{L}\left\{{\dot{x}}^3\right\}=0.$$

(23)

Applying the initial conditions:

$$s^2\mathcal{L}\left\{x\right\}-1-\epsilon s\mathcal{L}\left\{x\right\}+\mathcal{L}\left\{x\right\}+{\displaystyle \frac{\epsilon }{3}}\mathcal{L}\left\{{\dot{x}}^3\right\}=0,$$

(24)

gathering similar terms:

$$(s^2-\epsilon s+1)\mathcal{L}\left\{x\right\}=1-{\displaystyle \frac{\epsilon }{3}}\mathcal{L}\left\{{\dot{x}}^3\right\},$$

(25)

rearranging the equation,

$$\mathcal{L}\left\{x\right\}={\displaystyle \frac{1}{s^2-\epsilon s+1}}-{\displaystyle \frac{\epsilon }{3(s^2-\epsilon s+1)}}\mathcal{L}\left\{{\dot{x}}^3\right\}.$$

(26)

Finally, computing Inverse Laplace Transform and making $x\left(t\right)$ the subject, we get

$$x\left(t\right)={\displaystyle \frac{e^{t\left(\frac{\epsilon }{2}-\frac{\sqrt{{ϵ}^2-4}}{2}\right)}-e^{t\left(\frac{\sqrt{{\epsilon }^2-4}}{2}+\frac{\epsilon }{2}\right)}}{\sqrt{{\epsilon }^2-4}}}-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{{\dot{x}}^3\right\}\right\}.$$

(27)

Alternatively, by applying the definition of the hyperbolic sine function, $sinh\left(B\right)={\displaystyle \frac{e^B-e^{-B}}{2}}$, and the identity $sinh\left(A\right)+cosh\left(A\right)=e^A$, the expression for $x\left(t\right)$ becomes

$$x\left(t\right)={\displaystyle \frac{2sinh\left(\frac{1}{2}\sqrt{{\epsilon }^2-4}t\right)\left(sinh\left(\frac{\epsilon t}{2}\right)+cosh\left(\frac{\epsilon t}{2}\right)\right)}{\sqrt{{\epsilon }^2-4}}}-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{{\dot{x}}^3\right\}\right\}.$$

(28)

By the Adomian decomposition method, let $x\left(t\right)$ be an infinite series:

$$x\left(t\right)=\sum _{n=0}^{\infty }{x}_n\left(t\right),$$

(29)

and the nonlinear term ${\dot{x}}^3$ is represented as an infinite series of $A_n$ called Adomian polynomials given by (9), then Equation (28) becomes

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{x}_n\left(t\right)& ={\displaystyle \frac{2sinh\left(\frac{1}{2}\sqrt{{\epsilon }^2-4}t\right)\left(sinh\left(\frac{\epsilon t}{2}\right)+cosh\left(\frac{\epsilon t}{2}\right)\right)}{\sqrt{{\epsilon }^2-4}}}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & -{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{\sum _{n=0}^{\infty }{A}_n\right\}\right\}.\hfill \end{array}$$

(31)

Therefore,

$$x_0\left(t\right)={\displaystyle \frac{2sinh\left(\frac{1}{2}\sqrt{{\epsilon }^2-4}t\right)\left(sinh\left(\frac{\epsilon t}{2}\right)+cosh\left(\frac{\epsilon t}{2}\right)\right)}{\sqrt{{\epsilon }^2-4}}},$$

(32)

and

$$x_n\left(t\right)=-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{A_n\right\}\right\},n\ge 1.$$

(33)

To evaluate the Adomian polynomials $A_n$, we use (9), which gives

$$A_0={\dot{x_0}}^3,$$

(34)

$$A_1=3\left({\dot{x_0}}^2\dot{x_1}\right),$$

(35)

$$A_2=3\left({\dot{x_0}}^2\dot{x_2}\right)+3\left(\dot{x_0}{\dot{x_1}}^2\right),$$

(36)

$$A_3=3\left({\dot{x_0}}^2\dot{x_3}\right)+6\left(\dot{x_0}\dot{x_1}\dot{x_2}\right)+{\dot{x_1}}^3,$$

(37)

$$A_4=3{\dot{x_0}}^2\dot{x_4}+3{\dot{x_1}}^2\dot{x_2}+3{\dot{x_2}}^2\dot{x_0}+6\dot{x_0}\dot{x_1}\dot{x_3}.$$

(38)

In the subsequent steps, we derive the next two terms of the solution $x\left(t\right)$. Using Equation (33), we obtain

$$x_1\left(t\right)=-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{A_0\right\}\right\},n\ge 1,$$

(39)

where

$$\begin{array}{cc}\hfill A_0& ={\dot{x_0}}^3\hfill \\ \hfill & ={\left[{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2sinh\left(\frac{1}{2}\sqrt{{\epsilon }^2-4}t\right)\left(sinh\left(\frac{\epsilon t}{2}\right)+cosh\left(\frac{\epsilon t}{2}\right)\right)}{\sqrt{{\epsilon }^2-4}}}\right)\right]}^3\hfill \\ \hfill & =\left(cosh\left({\displaystyle \frac{1}{2}}\sqrt{{\epsilon }^2-4}t\right)\left(sinh\left({\displaystyle \frac{\epsilon t}{2}}\right)+cosh\left({\displaystyle \frac{\epsilon t}{2}}\right)\right)\right.\hfill \\ \hfill & +{\left.{\displaystyle \frac{2sinh\left(\frac{1}{2}\sqrt{{\epsilon }^2-4}t\right)}{\sqrt{{\epsilon }^2-4}}}\left({\displaystyle \frac{1}{2}}\epsilon sinh\left({\displaystyle \frac{\epsilon t}{2}}\right)+{\displaystyle \frac{1}{2}}\epsilon cosh\left({\displaystyle \frac{\epsilon t}{2}}\right)\right)\right)}^3.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill x_1=& {\displaystyle \frac{\left(sinh\left(\frac{1}{2}\left(\epsilon -3\sqrt{{\epsilon }^2-4}\right)t\right)+cosh\left(\frac{1}{2}\left(\epsilon -3\sqrt{{\epsilon }^2-4}\right)t\right)\right)}{12{\left({\epsilon }^2-4\right)}^{3/2}\left(3{\epsilon }^2-16\right)}}\hfill \\ \hfill & \times [\left(96-18{\epsilon }^2\right)\left(sinh\left(\left(\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)+cosh\left(\left(\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)\right)\hfill \\ \hfill & +6\left(3{\epsilon }^2-16\right)\left(sinh\left(\left(2\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)+cosh\left(\left(2\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)\right)\hfill \\ \hfill & +\left({\epsilon }^2-4\right)\left(5{\epsilon }^2+\sqrt{{\epsilon }^2-4}\epsilon -24\right)\left(sinh\left(2\sqrt{{\epsilon }^2-4}t\right)+cosh\left(2\sqrt{{\epsilon }^2-4}t\right)\right)\hfill \\ \hfill & +\left({\epsilon }^2-4\right)\left(-5{\epsilon }^2+\sqrt{{\epsilon }^2-4}\epsilon +24\right)\left(sinh\left(\sqrt{{\epsilon }^2-4}t\right)+cosh\left(\sqrt{{\epsilon }^2-4}t\right)\right)\hfill \\ \hfill & -\epsilon \left({\epsilon }^3+\sqrt{{\epsilon }^2-4}{\epsilon }^2-4\sqrt{{\epsilon }^2-4}-6\epsilon \right)\hfill \\ \hfill & \times \left(sinh\left(\left(3\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)+cosh\left(\left(3\sqrt{{\epsilon }^2-4}+\epsilon \right)t\right)\right)\hfill \\ \hfill & +\epsilon \left({\epsilon }^3-\sqrt{{\epsilon }^2-4}{\epsilon }^2+4\sqrt{{\epsilon }^2-4}-6\epsilon \right)\left(sinh\left(\epsilon t\right)+cosh\left(\epsilon t\right)\right)].\hfill \end{array}$$

Similarly, we calculate $x_2$:

$$x_2\left(t\right)=-{\mathcal{L}}^{-1}\left\{{\displaystyle \frac{\epsilon }{3(s^2-\epsilon -1)}}\mathcal{L}\left\{A_1\right\}\right\}, n\ge 1.$$

(40)

By substituting the expression of $A_1$ from Equation (35) and applying the inverse Laplace transform, we obtain

$$\begin{array}{cc}\hfill x_2=& {\displaystyle \frac{\left(cosh\left(\frac{1}{2}t\left(\epsilon -5\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(\frac{1}{2}t\left(\epsilon -5\sqrt{{\epsilon }^2-4}\right)\right)\right)}{48{\left({\epsilon }^2-4\right)}^{5/2}\left(3{\epsilon }^2-16\right)\left(3{\epsilon }^2+4\right)\left(5{\epsilon }^2-36\right)}}\times \hfill \\ \hfill & [3\left(3{\epsilon }^2+4\right)\left(cosh\left(t\left(2\epsilon +5\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(2\epsilon +5\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left({\epsilon }^6+\sqrt{{\epsilon }^2-4}{\epsilon }^5-14{\epsilon }^4-12\sqrt{{\epsilon }^2-4}{\epsilon }^3+56{\epsilon }^2+34\sqrt{{\epsilon }^2-4}\epsilon -48\right){\epsilon }^2\hfill \\ \hfill & -3\left(3{\epsilon }^2+4\right)\left(cosh\left(2t\epsilon \right)+sinh\left(2t\epsilon \right)\right)\left({\epsilon }^6-\sqrt{{\epsilon }^2-4}{\epsilon }^5-14{\epsilon }^4\right.\hfill \\ \hfill & \left.+12\sqrt{{\epsilon }^2-4}{\epsilon }^3+56{\epsilon }^2-34\sqrt{{\epsilon }^2-4}\epsilon -48\right){\epsilon }^2\hfill \\ \hfill & -6\left(5{\epsilon }^2-36\right)\left(4{\epsilon }^5+4\sqrt{{\epsilon }^2-4}{\epsilon }^4-14{\epsilon }^3-15\sqrt{{\epsilon }^2-4}{\epsilon }^2-48\epsilon -16\sqrt{{\epsilon }^2-4}\right)\hfill \\ \hfill & \times \left(cosh\left(2t\left(\epsilon +2\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(2t\left(\epsilon +2\sqrt{{\epsilon }^2-4}\right)\right)\right)\epsilon \hfill \\ \hfill & -2\left(15{\epsilon }^6-148{\epsilon }^4+208{\epsilon }^2+576\right)\left({\epsilon }^3+\sqrt{{\epsilon }^2-4}{\epsilon }^2-5\epsilon -3\sqrt{{\epsilon }^2-4}\right)\hfill \\ \hfill & \times \left(cosh\left(t\left(\epsilon +4\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(\epsilon +4\sqrt{{\epsilon }^2-4}\right)\right)\right)\epsilon \hfill \\ \hfill & +2\left(15{\epsilon }^6-148{\epsilon }^4+208{\epsilon }^2+576\right)\left(cosh\left(t\left(\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left({\epsilon }^3-\sqrt{{\epsilon }^2-4}{\epsilon }^2-5\epsilon +3\sqrt{{\epsilon }^2-4}\right)\epsilon \hfill \\ \hfill & +6\left(5{\epsilon }^2-36\right)\left(cosh\left(t\left(2\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(2\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left(4{\epsilon }^5-4\sqrt{{\epsilon }^2-4}{\epsilon }^4-14{\epsilon }^3+15\sqrt{{\epsilon }^2-4}{\epsilon }^2-48\epsilon +16\sqrt{{\epsilon }^2-4}\right)\epsilon \hfill \\ \hfill & -{\left({\epsilon }^2-4\right)}^2\left(69{\epsilon }^6-21\sqrt{{\epsilon }^2-4}{\epsilon }^5-724{\epsilon }^4+122\sqrt{{\epsilon }^2-4}{\epsilon }^3+1296{\epsilon }^2\right.\hfill \\ \hfill & \left.+180\sqrt{{\epsilon }^2-4}\epsilon +2592\right)\left(cosh\left(2t\sqrt{{\epsilon }^2-4}\right)+sinh\left(2t\sqrt{{\epsilon }^2-4}\right)\right)\hfill \\ \hfill & -6\left(5{\epsilon }^2-36\right)\left(3{\epsilon }^6-3\sqrt{{\epsilon }^2-4}{\epsilon }^5+18{\epsilon }^4+12\sqrt{{\epsilon }^2-4}{\epsilon }^3-152{\epsilon }^2\right.\hfill \\ \hfill & \left.+8\sqrt{{\epsilon }^2-4}\epsilon -192\right)\left(cosh\left(2t\left(\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(2t\left(\epsilon +\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & +2\left(15{\epsilon }^6-148{\epsilon }^4+208{\epsilon }^2+576\right)\left(cosh\left(t\left(\epsilon +3\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(\epsilon +3\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left(15{\epsilon }^2+\sqrt{{\epsilon }^2-4}\epsilon -72\right)\hfill \\ \hfill & +2\left(15{\epsilon }^6-148{\epsilon }^4+208{\epsilon }^2+576\right)\left(cosh\left(t\left(\epsilon +2\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(\epsilon +2\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left(-15{\epsilon }^2+\sqrt{{\epsilon }^2-4}\epsilon +72\right)\hfill \\ \hfill & +6\left(5{\epsilon }^2-36\right)\left(cosh\left(t\left(2\epsilon +3\sqrt{{\epsilon }^2-4}\right)\right)+sinh\left(t\left(2\epsilon +3\sqrt{{\epsilon }^2-4}\right)\right)\right)\hfill \\ \hfill & \times \left(3{\epsilon }^6+3\sqrt{{\epsilon }^2-4}{\epsilon }^5+18{\epsilon }^4-12\sqrt{{\epsilon }^2-4}{\epsilon }^3-152{\epsilon }^2\right.\hfill \\ \hfill & \left.-8\sqrt{{\epsilon }^2-4}\epsilon -192\right)\hfill \\ \hfill & +{\left({\epsilon }^2-4\right)}^2\left(cosh\left(3t\sqrt{{\epsilon }^2-4}\right)+sinh\left(3t\sqrt{{\epsilon }^2-4}\right)\right)\hfill \\ \hfill & \times \left(69{\epsilon }^6+21\sqrt{{\epsilon }^2-4}{\epsilon }^5-724{\epsilon }^4-122\sqrt{{\epsilon }^2-4}{\epsilon }^3+1296{\epsilon }^2\right.\hfill \\ \hfill & \left.-180\sqrt{{\epsilon }^2-4}\epsilon +2592\right)].\hfill \end{array}$$

Recalling that $x_n\left(t\right)=x_0+x_1+\dots +x_n$, then the approximate solution will be

$$x\left(t\right)\simeq x_0+x_1+x_2.$$

(41)

5. Solving Nonlinear Rayleigh Oscillator Equation by Using Homotopy Perturbation Method (HPM)

This section presents another key contribution by applying the Homotopy Perturbation Method (HPM) to solve the nonlinear Rayleigh oscillator using the same initial conditions. It explains how the homotopy is constructed, outlines the iterative scheme, and computes the first three terms ($x_0,x_1,x_2$) of the solution. The resulting approximation is then compared with numerical results to highlight the effectiveness of the HPM and to allow comparison with the LADM approach. Recall the Rayleigh oscillator equation:

$$\ddot{x}-\epsilon \dot{x}+x+{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3=0,$$

(42)

with $x\left(0\right)=0$ and $\dot{x}\left(0\right)=1$ as initial conditions.

To solve the Rayleigh oscillator equation using the Homotopy Perturbation Method, consider a homotopy:

$$(1-p)(\ddot{x}+x)+p\left(\ddot{x}-\epsilon \dot{x}+x+{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3\right)=0.$$

(43)

Which simplifies to

$$\ddot{x}+x=p\left(\epsilon \dot{x}-{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3\right).$$

(44)

From the definition of the HPM, assume that the solution of Equation (44) can be expressed as a power series in p:

$$X(t,p)=p^0{x}_0+p^1{x}_1+p^2{x}_2+\dots +p^n{x}_n,$$

(45)

and by setting $p=1$, we obtain

$$x\left(x\right)=\underset{p\to 1}{lim}X(t,p)=\sum _{i=0}^{\infty }{x}_i\left(t\right),$$

(46)

using $x\left(0\right)=0$ and $\dot{x}\left(0\right)=1$ as initial conditions and Equation (45), one can find that

$x_0\left(0\right)=0$ and ${\dot{x}}_0\left(0\right)=1$ and that $x_i\left(0\right)=0$ and ${\dot{x}}_i\left(0\right)=0$ for all $i\ge 1$.

Substituting

$$x=x_0+p{x}_1+p^2{x}_2+\dots$$

(47)

into the homotopy equation:

$$\ddot{x}+x=p\left(\epsilon \dot{x}-{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3\right),$$

(48)

where

$$\begin{array}{cc}\hfill \dot{x}& ={\dot{x}}_0+p{\dot{x}}_1+p^2{\dot{x}}_2,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \ddot{x}& ={\ddot{x}}_0+p{\ddot{x}}_1+p^2{\ddot{x}}_2.\hfill \end{array}$$

(50)

Thus,

$$\ddot{x}+x=({\ddot{x}}_0+x_0)+p({\ddot{x}}_1+x_1)+p^2({\ddot{x}}_2+x_2).$$

(51)

Now to expand the right side of (17), we first expand ${\dot{x}}^3$:

$$\begin{array}{cc}\hfill {\dot{x}}^3& ={({\dot{x}}_0+p{\dot{x}}_1+p^2{\dot{x}}_2)}^3\hfill \\ \hfill & ={\dot{x}}_0^3+p\left(3{\dot{x}}_0^2{\dot{x}}_1\right)+p^2(3{\dot{x}}_0^2{\dot{x}}_2+3{\dot{x}}_0{\dot{x}}_1^2)+\mathcal{O}\left(p^3\right),\hfill \end{array}$$

(52)

therefore,

$$\begin{array}{cc}\hfill p\left(\epsilon \dot{x}-{\displaystyle \frac{\epsilon }{3}}{\dot{x}}^3\right)& =p\epsilon ({\dot{x}}_0+p{\dot{x}}_1)\hfill \\ \hfill & -{\displaystyle \frac{p\epsilon }{3}}\left[{\dot{x}}_0^3+p\left(3{\dot{x}}_0^2{\dot{x}}_1\right)+p^2(3{\dot{x}}_0{\dot{x}}_1^2+3{\dot{x}}_0^2{\dot{x}}_2)\right]\hfill \\ \hfill & =p\left(\epsilon {\dot{x}}_0-{\displaystyle \frac{\epsilon }{3}}{\dot{x}}_0^3\right)\hfill \\ \hfill & +p^2\left(\epsilon {\dot{x}}_1-\epsilon {\dot{x}}_0^2{\dot{x}}_1\right)\hfill \\ \hfill & +p^3\left(\mathrm{higher}\mathrm{order}\mathrm{terms}\right).\hfill \end{array}$$

(53)

Equating (51) and (53) by powers of p:

$$\begin{array}{cc}\hfill \mathcal{O}\left(p^0\right):& {\ddot{x}}_0+x_0=0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \mathcal{O}\left(p^1\right):& {\ddot{x}}_1+x_1=\epsilon {\dot{x}}_0-{\displaystyle \frac{\epsilon }{3}}{\dot{x}}_0^3,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill \mathcal{O}\left(p^2\right):& {\ddot{x}}_2+x_2=\epsilon {\dot{x}}_1-\epsilon {\dot{x}}_0^2{\dot{x}}_1.\hfill \end{array}$$

(56)

Now we solve the above differential equations. For Equation (54), with $x_0\left(0\right)=0$ and ${\dot{x}}_0\left(0\right)=1$, we get

$$x_0=sint,$$

(57)

To solve differential Equation (55), with $x_1\left(0\right)=0$ and ${\dot{x}}_1\left(0\right)=0$, we start with

$$\begin{array}{cc}\hfill {\dot{x}}_0^3& ={cos}^{3}t={\displaystyle \frac{3}{4}}cost+{\displaystyle \frac{1}{4}}cos3t,\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill \mathrm{and}\mathrm{RHS}\mathrm{of}\left(55\right)& =\epsilon cost-{\displaystyle \frac{\epsilon }{3}}\left({\displaystyle \frac{3}{4}}cost+{\displaystyle \frac{1}{4}}cos3t\right)\hfill \\ \hfill & ={\displaystyle \frac{3\epsilon }{4}}cost-{\displaystyle \frac{\epsilon }{12}}cos3t.\hfill \end{array}$$

(59)

Substitute ${\dot{x}}_0=cost$:

$${\ddot{x}}_1+x_1=\epsilon cost-{\displaystyle \frac{\epsilon }{3}}{cos}^{3}t.$$

(60)

Therefore,

$${\ddot{x}}_1+x_1=\epsilon cost-{\displaystyle \frac{\epsilon }{3}}\left({\displaystyle \frac{3}{4}}cost+{\displaystyle \frac{1}{4}}cos3t\right)={\displaystyle \frac{3\epsilon }{4}}cost-{\displaystyle \frac{\epsilon }{12}}cos3t.$$

(61)

Solving the linear non-homogeneous differential equation using undetermined coefficients, we get

$$\begin{array}{cc}\hfill x_1\left(t\right)& ={\displaystyle \frac{1}{96}}(-39\epsilon cost+40\epsilon {cos}^{3}t-\epsilon costcos4t\hfill \\ \hfill & +36t\epsilon sint+16\epsilon sintsin2t\hfill \\ \hfill & -\epsilon sintsin4t).\hfill \end{array}$$

(62)

which can be simplified as

$$x_1=-{\displaystyle \frac{1}{48}}\epsilon sin\left(t\right)(sin\left(2t\right)-18t).$$

(63)

To compute $x_2$, we solve the differential Equation (56), which expands the term:

$${\dot{x}}_0^2{\dot{x}}_1={cos}^{2}t\left(-{\displaystyle \frac{1}{48}}\epsilon sin\left(t\right)(2cos\left(2t\right)-18)-{\displaystyle \frac{1}{48}}\epsilon (sin\left(2t\right)-18t)cos\left(t\right)\right)$$

(64)

Now we can solve the differential Equation (56), and we get

$$\begin{array}{cc}\hfill x_2\left(t\right)& ={\displaystyle \frac{1}{3072}}(72{\epsilon }^2{t}^{2}sin\left(t\right)+113{\epsilon }^{2}sin\left(t\right)\hfill \\ \hfill & -36{\epsilon }^{2}tsin\left(t\right)sin\left(4t\right)-456{\epsilon }^{2}tcos\left(t\right)\hfill \\ \hfill & +144{\epsilon }^{2}tcos\left(t\right)cos\left(2t\right)-36{\epsilon }^{2}tcos\left(t\right)cos\left(4t\right)\hfill \\ \hfill & -142{\epsilon }^{2}sin\left(t\right)cos\left(2t\right)+31{\epsilon }^{2}sin\left(t\right)cos\left(4t\right)\hfill \\ \hfill & -2{\epsilon }^{2}sin\left(t\right)cos\left(6t\right)+242{\epsilon }^{2}sin\left(2t\right)cos\left(t\right)\hfill \\ \hfill & -37{\epsilon }^{2}sin\left(4t\right)cos\left(t\right)+2{\epsilon }^{2}sin\left(6t\right)cos\left(t\right)),\hfill \end{array}$$

which can be simplified as

$$\begin{array}{cc}\hfill x_2\left(t\right)& ={\displaystyle \frac{{\epsilon }^2}{3072}}[2sin\left(t\right)\left(36t^2+15cos\left(2t\right)-cos\left(4t\right)+160\right)\hfill \\ \hfill & -384tcos\left(t\right)+36tcos\left(3t\right)].\hfill \end{array}$$

Combining terms up to the second order:

$$\begin{array}{cc}\hfill x\left(t\right)& =sin\left(t\right)-{\displaystyle \frac{1}{48}}\epsilon sin\left(t\right)(sin\left(2t\right)-18t)\hfill \\ \hfill & +{\displaystyle \frac{{\epsilon }^2}{3072}}[2sin\left(t\right)\left(36t^2+15cos\left(2t\right)-cos\left(4t\right)+160\right)\hfill \\ \hfill & -384tcos\left(t\right)+36tcos\left(3t\right)].\hfill \end{array}$$

6. Comparative Error Analysis of LADM and HPM

This section evaluates the accuracy of both the Laplace–Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) by comparing their solutions to the exact solution of the Rayleigh oscillator for $\epsilon =0.1$. Error metrics and visualizations are presented to systematically assess performance.

6.1. LADM Error Analysis

Table 1. Absolute error comparing exact solution to LADM ($\epsilon =0.1$).

t	${\mathit{x}}_{\mathbf{Exact}}\left(\mathit{t}\right)$	${\mathit{x}}_{\mathbf{LADM}}\left(\mathit{t}\right)$	$|{\mathit{x}}_{\mathbf{Exact}}-{\mathit{x}}_{\mathbf{LADM}}|$
0	0.0	$-2.313\times {10}^{-19}$	$2.313\times {10}^{-19}$
2	0.98193	0.981985	0.0000544
4	−0.867398	−0.867996	0.000598
6	−0.352271	−0.353092	0.000821
8	1.29522	1.30594	0.01072
10	−0.735753	−0.752756	0.01700
12	−0.797198	−0.835962	0.03876
14	1.49783	1.67163	0.17380
16	−0.428608	−0.526762	0.09815
18	−1.25543	−1.80498	0.54955
20	1.52001	2.90272	1.38271

and

Table 2. Error metrics for LADM ($\epsilon =0.1$).

Metric	Value
Maximum absolute error	1.38271 at $t=20$
Average absolute error	0.123354
Root mean square error (RMSE)	0.259754

summarize the absolute errors and key metrics for the LADM solution, while Figure 1 illustrates the error progression over time.

The results show that the error accumulates over time, with the maximum deviation reaching 1.38271 at $t=20$, as summarized in Table 2 and illustrated in Figure 1. However, for shorter time intervals ($t<15$), the method demonstrates reasonable accuracy, with an average error of 0.123 and a root mean square error (RMSE) of 0.260, indicating moderate agreement with the exact solution.

6.2. HPM Error Analysis

Table 3. Absolute error comparing the exact solution to the HPM ($\epsilon =0.1$).

t	${\mathit{x}}_{\mathbf{Exact}}\left(\mathit{t}\right)$	${\mathit{x}}_{\mathbf{HPM}}\left(\mathit{t}\right)$	$|{\mathit{x}}_{\mathbf{Exact}}-{\mathit{x}}_{\mathbf{HPM}}|$
0	0.0	0.0	0.0
2	0.98193	0.981936	$5.828\times {10}^{-6}$
4	−0.867398	−0.868720	0.001321
6	−0.352271	−0.352004	0.000267
8	1.29522	1.30438	0.009167
10	−0.735753	−0.749666	0.013913
12	−0.797198	−0.810574	0.013375
14	1.49783	1.55351	0.055676
16	−0.428608	−0.459958	0.031351
18	−1.25543	−1.33387	0.078441
20	1.52001	1.67028	0.150277

and

Table 4. Error metrics for HPM ($\epsilon =0.1$).

Metric	Value
Maximum absolute error	0.150277 at $t=20$
Average absolute error	0.024504
Root mean square error (RMSE)	0.040730

present the HPM error data, with Figure 2 showing its error trend. The Homotopy Perturbation Method (HPM) exhibits strong numerical stability, with a maximum error of only 0.150 at $t=20$, as shown in Table 4 and Figure 2. This is significantly lower than the error observed in the LADM results. Additionally, the method maintains consistent accuracy throughout the time domain, as reflected by the low average error of 0.0245 and a root mean square error (RMSE) of 0.0407.

6.3. Numerical Solution Parameters

The reference numerical solutions were obtained using Mathematica’s NDSolve function with the following parameters to ensure high accuracy:

• Method: StiffnessSwitching (combines explicit and implicit methods for optimal stability);
• PrecisionGoal: 12 (relative error tolerance);
• AccuracyGoal: 12 (absolute error tolerance);
• MaxSteps: 100,000 (sufficient for $t\in [0,20]$);
• WorkingPrecision: MachinePrecision (approximately 16 decimal digits).

These settings guarantee that numerical errors remain below ${10}^{-10}$ throughout the solution interval, making them negligible compared to the approximation errors of the semi-analytical methods being evaluated.

7. Convergence Analysis of the LADM and HPM Solutions

This section addresses a crucial aspect of employing series-based semi-analytical methods: the convergence of the obtained solutions. We investigate the theoretical conditions under which the series solutions derived from the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM) for the Rayleigh oscillator are expected to converge. The first subsection revisits established convergence criteria for the LADM, focusing on the Lipschitz condition relevant to the nonlinearity present in the Rayleigh equation. The subsequent subsection examines the convergence of the HPM, employing a fixed-point theorem approach to demonstrate convergence under suitable conditions on the system parameters. Together, these analyses provide theoretical support for the validity of the approximate solutions presented in this study.

7.1. LADM Convergence

The convergence of the Adomian decomposition method has been investigated by several researchers, notably Adomian himself, Cherruault, and Wazwaz [8,21,22,23]. A common sufficient condition for the convergence of the Adomian series is that the nonlinear operator N (in our case, $N\left(\dot{x}\right)=(\epsilon /3){\dot{x}}^3$) must be Lipschitz continuous. That is, there exists a constant $K>0$ such that for any $u,v$ in a suitable Banach space (e.g., space of continuous functions on an interval $[0,T]$),

$$\parallel N\left(u\right)-N\left(v\right)\parallel \le K\parallel u-v\parallel$$

Let us examine the nonlinearity $f\left(y\right)=y^3$.

$$|f\left(y_1\right)-f\left(y_2\right)|=|y_1^3-y_2^3|=|(y_1-y_2)(y_1^2+y_1{y}_2+y_2^2)|$$

If we consider a domain where $y_1,y_2$ are bounded, i.e., $|y_1|\le M$ and $|y_2|\le M$ for some $M>0$, then

$$|y_1^2+y_1{y}_2+y_2^2|\le |y_1{|}^2+|y_1\left|\right|y_2|+|y_2{|}^2\le M^2+M^2+M^2=3M^2$$

Thus,

$$|y_1^3-y_2^3|\le \left(3M^2\right)|y_1-y_2|$$

This shows that $y^3$ is locally Lipschitz continuous. The Lipschitz constant $K=3M^2$ depends on the bound M of the function $\dot{x}\left(t\right)$. Since ${\dot{x}}_0\left(t\right)$ (the derivative of $x_0\left(t\right)$ from Equation (32)) is bounded on any finite time interval $[0,T]$, and assuming that the sum $\dot{x}\left(t\right)=\sum {\dot{x}}_n\left(t\right)$ remains bounded on this interval, the LADM series is expected to converge.

Cherruault and Adomian [8,21] proved that if N is a Lipschitzian operator in a Hilbert space H, and if $x_0$ belongs to H, then the Adomian series $\sum x_n$ converges. The convergence is often quite rapid for practical problems, especially if the time interval T is not too large.

7.2. HPM Convergence

To apply the Homotopy Perturbation Method (HPM), we rewrite the equation in the following form:

$$\ddot{x}+x=\epsilon \left(\dot{x}-{\displaystyle \frac{1}{3}}{\dot{x}}^3\right).$$

We identify the linear operator, $L\left(x\right)=\ddot{x}+x$, and the nonlinear operator, $N\left(x\right)=\epsilon \left(\dot{x}-{\displaystyle \frac{1}{3}}{\dot{x}}^3\right)$.

We construct the homotopy:

$$H(x,p)=L\left(x\right)-L\left(x_0\right)+pN\left(x\right)=0,$$

and assume a perturbation series solution:

$$x\left(t\right)=x_0\left(t\right)+p{x}_1\left(t\right)+p^2{x}_2\left(t\right)+\cdots .$$

Let X be a Banach space of continuously differentiable functions on the interval $[0,T]$, equipped with the norm

$$\parallel x\parallel =\underset{t\in [0,T]}{max}\left(\left|x\left(t\right)\right|+|\dot{x}\left(t\right)|\right).$$

(65)

Define the operator $\mathcal{T}$ such that

$$\mathcal{T}\left(x\right)=x_0-L^{-1}\left[N\left(x\right)\right],$$

(66)

where $L^{-1}$ is the inverse operator corresponding to $L\left(x\right)=\ddot{x}+x$ and satisfies the initial conditions.

We seek to show that $\mathcal{T}$ is a contraction mapping. Let $x,y\in X$. Then

$$\begin{array}{cc}\hfill \parallel \mathcal{T}\left(x\right)-\mathcal{T}\left(y\right)\parallel & =\parallel L^{-1}[N\left(x\right)-N\left(y\right)]\parallel \hfill \\ \hfill & \le \parallel L^{-1}\parallel ·\parallel N\left(x\right)-N\left(y\right)\parallel .\hfill \end{array}$$

(67)

Now consider the nonlinear operator $N\left(x\right)=\epsilon \left(\dot{x}-{\displaystyle \frac{1}{3}}{\dot{x}}^3\right)$. The difference is

$$\begin{array}{cc}\hfill N\left(x\right)-N\left(y\right)& =\epsilon \left[\dot{x}-{\displaystyle \frac{1}{3}}{\dot{x}}^3-\left(\dot{y}-{\displaystyle \frac{1}{3}}{\dot{y}}^3\right)\right]\hfill \\ \hfill & =\epsilon \left[\dot{x}-\dot{y}-{\displaystyle \frac{1}{3}}({\dot{x}}^3-{\dot{y}}^3)\right].\hfill \end{array}$$

(68)

Using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$, we have

$$\parallel N\left(x\right)-N\left(y\right)\parallel \le \epsilon \left(\parallel \dot{x}-\dot{y}\parallel +{\displaystyle \frac{1}{3}}\parallel \dot{x}-\dot{y}\parallel ·(\parallel \dot{x}{\parallel }^2+\parallel \dot{x}\parallel \parallel \dot{y}\parallel +\parallel \dot{y}{\parallel }^2)\right).$$

(69)

Hence,

$$\parallel \mathcal{T}\left(x\right)-\mathcal{T}\left(y\right)\parallel \le \parallel L^{-1}\parallel ·\epsilon ·\parallel \dot{x}-\dot{y}\parallel \left(1+{\displaystyle \frac{1}{3}}(\parallel \dot{x}{\parallel }^2+\parallel \dot{x}\parallel \parallel \dot{y}\parallel +\parallel \dot{y}{\parallel }^2)\right).$$

(70)

If $\epsilon$ is sufficiently small and $x,y\in B_r\subset X$ with r small enough, then the entire right-hand side becomes less than $\parallel x-y\parallel$, ensuring that $\mathcal{T}$ is a contraction.

Therefore, by Banach’s fixed-point theorem, $\mathcal{T}$ has a unique fixed point in X, which implies that the HPM series solution converges to a unique function in the chosen function space. Thus, the solution obtained by the HPM is valid and convergent under suitable smallness conditions on $\epsilon$.

8. Conclusions

In this work, we addressed the analytical solution of the Rayleigh oscillator equation, a nonlinear second-order differential equation characterized by a velocity-dependent, asymmetric damping force. To tackle this problem, we employed two distinct semi-analytical approaches: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM).

Each method offers unique strengths. The LADM combines the Laplace transform with the Adomian decomposition technique, making it particularly effective for equations where a linear operator simplifies the structure of the problem. On the other hand, the HPM provides a flexible framework for handling nonlinearities without requiring prior linearization, which proves advantageous for strongly nonlinear systems.

We derived approximate solutions using both methods and compared their performance. Visualization of the results for $\epsilon =0.05,\epsilon =0.1$, and $\epsilon =0.2$ was carried out using Mathematica, with the corresponding plots presented in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. The graphical comparisons consistently show that the solutions obtained via the Homotopy Perturbation Method closely align with the expected behavior, demonstrating greater accuracy than those obtained via the Laplace–Adomian Decomposition Method.

Moreover, due to the absence of exact analytical solutions for the Rayleigh oscillator in the existing literature, a quantitative comparison was performed by evaluating the absolute errors between high-precision numerical solutions (produced by Mathematica’s NDSolve function) and the approximations derived from both the LADM and HPM. This benchmarking further confirmed that the Homotopy Perturbation Method yields superior accuracy across the considered parameter ranges, as detailed in Table 1, Table 2, Table 3 and Table 4.

Our analysis reveals that both the Laplace–Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) exhibit significant dependence on the damping parameter $\epsilon$ and temporal evolution. The numerical experiments demonstrate that solution accuracy deteriorates progressively with increasing time t, with error accumulation becoming particularly pronounced for $t>10$ across all tested values of $\epsilon$. Furthermore, both methods show heightened sensitivity to the damping parameter, where larger $\epsilon$ values ($\epsilon >0.2$) lead to substantially degraded performance. While the HPM maintains better accuracy for moderate time scales ($t<15$) and smaller $\epsilon$ values ($\epsilon \le 0.1$), as shown in Figure 2, the LADM demonstrates more systematic error reduction through higher-order approximations.

It is worth noting that the accuracy of the LADM solution may deteriorate over large time intervals due to series truncation. Nevertheless, for applications that require high precision over extended time intervals, incorporating additional terms—or applying techniques such as the Padé approximant to enhance convergence—could be a viable strategy to improve the accuracy of the solution.

Although alternative methods such as the Variational Iteration Method (VIM) [24], the Homotopy Analysis Method (HAM) [25], and Padé-based techniques [26] have demonstrated strong performance in nonlinear settings, our implementations of the LADM and HPM offer an effective compromise between accuracy and computational simplicity. Note the following: (i) the LADM generates its series without the need for variational principles (unlike VIM); and (ii) the HPM achieves a level of accuracy comparable to HAM without introducing multiple auxiliary parameters. Future work could extend this study by integrating Padé approximants to broaden the time range of validity.

Overall, this study highlights the effectiveness of the HPM in solving nonlinear differential equations like the Rayleigh oscillator while also emphasizing the contexts where the LADM remains a powerful tool.