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Abstract

A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by constructing a homotopy equation with one or more auxiliary parameters embedding in the linear term with a clear advantage in accelerating and controlling the approximation convergence speed. Moreover, a revision of a recent amplitude-period approximation formula is presented providing an answer to an open problem related to the optimal approximation along with a new universal formula. Duffing equation is used as an example to illustrate the solution process for the homotopy perturbation method, and only one or few iterations are needed in practical applications, making the method much attractive. From the side of amplitude-period formulation, the nonlinear pendulum, the Duffing equation and an oscillator with discontinuity are analyzed providing an asymptotic exact equivalence for bigger parameter values in the case of Duffing’s system. This mini review gives a tutorial guideline for practical applications of the homotopy perturbation method, the references are not exhaustive.

Keywords

Homotopy perturbation method auxiliary parameter parameter-expansion method He’s polynomials variational iteration method amplitude-period formula Duffing oscillator

Introduction

Nonlinear oscillations occur in many and diverse application’s fields (see for instance Cveticanin¹ and Kovacic and Brennan²). The ODE’s nonlinear nature of the dynamical modeling for these oscillators makes it impossible to derive exact closed-form solutions (except for a few particular cases³).

To overcome this shortcoming, several approximations were proposed in the past to count (see for instance Constantin⁴ and the references therein): Homotopy perturbation, harmonic balance, a domain decomposition, variational formulation, variational iteration, pseudospectral method, Rayleigh-balance, energy-balance, max-min approach, amplitude-frequency formulation, homotopy analysis, optimal homotopy asymptotic method.

Some of these methods evidenced recent advances in the last few years. This is the case of the homotopy perturbation method proposed in 1999,⁵ that has been improved by the owner through 2000 to the present.^6–18

The theory has been extensively studied for decades by numerous authors, for examples Ganji and Sadighi,¹⁹ Odibat and Momani,²⁰ Cveticanin,²¹ Khan and Mohyud-Din,²² to mention a few. Using Clarivate Analytics’ web of science, there are 4076 records by searching for the topic of “homotopy perturbation or HPM” until 26 December 2017.

The method has now matured into a full-fledged mathematical normative theory due to its effectiveness and simplicity. When Davood Domiri Ganji talked with ScienceWatch.com on February 2008, he said “Wherever a nonlinear equation is found, Dr He’s HPM will be the primary tool of discovery”, and “He’s perturbation method itself is mathematically beautiful and extremely accessible to non-mathematicians.”

There are many modifications of the method,^23–28 according to web of science there are 338 hits by search for the topic of “modified homotopy perturbation method” and there are 113 articles on sciencedirect.com using a modified homotopy perturbation method until 26 December 2017.

The main development of the homotopy perturbation method was to adopt the parameter-expansion technology^29–33 to effectively deal with the zero-order approximate and He’s polynomials to deal with the nonlinear terms.^34–40

Now there is a trend to combination of the homotopy perturbation method with other methods,¹⁸ for examples combination of the method with variational theory⁴¹ and least squares technology.⁴² The prevailing trend is to adopt the Laplace transform in the homotopy perturbation method.⁴³ The Laplace transform is extremely simple to deal with the linear differential equations, and the homotopy perturbation method is to change a nonlinear equation into a series of linear equations. This effective modification has been widely used in open literature with different names: Laplace transform-homotopy perturbation method,^44,45 homotopy perturbation method with Laplace transform,⁴⁶ homotopy perturbation transform method,⁴⁷ homotopy perturbation Padé transform method;³⁶ all the above methods are actually the same method with different names. Mishra and Nagar⁴⁸ suggested this method should be called as He-Laplace method, which is extremely effective for fractional calculus.^49–55

For a comprehensive review on the homotopy perturbation method, from its basic ideas, concept, skills and applications, the reader is referred to a few review articles and elementary introductions published in International Journal of Modern Physics B,^8,9 Abstract and Applied Analysis¹⁶ and International Journal of Theoretical Physics.¹⁸

Though it becomes matured, the method has still some space for further improvement. In this paper, we will introduce an auxiliary parameter in the linear term instead of the nonlinear term to accelerate the convergence.

On the other hand, the recent method of amplitude-period approximations (see literature^56–58 for instance) has witnessed great achievements. In particular, Dan and Zhi⁵⁹ present a recent review in this direction. To figure out these advances, it is worth mentioning: He’s frequency formulation (see for instance He⁶⁰), Max-Min approach to nonlinear oscillators (see for instance He⁶¹), Taylor series (see for instance He⁶²), the simplest Amplitude-frequency formulation.⁶³

In this paper, two methods are reviewed: the homotopy perturbation method^5–18 and the amplitude-period formula presented in García⁶⁴ with complement in Suárez Antola.⁶⁵

In both cases new research is also presented: in the homotopy case, a new way to construct a homotopy is presented, thus opening the door to multiple scales oscillators or even forced oscillators with (possible) multiple parameters; for the amplitude-period formula, the open problem in García⁶⁴ is answered using optimization for small-amplitude oscillations, but also a necessary condition to provide explicit universal amplitude-period approximation formulas with exact asymptotic behavior.

Homotopy perturbation method

Consider the following general differential equation

L (u) + N (u) = 0

(1)

where L is the linear operator and N is the nonlinear operator. The simplest approach to construction of a homotopy equation is as follows⁸

L (u) + p N (u) = 0

(2)

L (u) - L (u_{0}) + p [L (u) + N (u)] = 0

(3)

where p is a homotopy parameter

p \in [0, 1]

and

u_{0}

is the initial solution.

The homotopy analysis method^66,67 introduces an auxiliary parameter in the form

(1 - p) [L (u) - L (u_{0})] + p h [L (u) + N (u)] = 0

(4)

where h is a non-zero parameter used to control the convergence. When p = 1, equation (4) becomes

h [L (u) + N (u)] = 0

(5)

For any non-zero h, equation (5) is equivalent to the original one. It requires skill to optimally choose the value of h because h can be any real number except zero. Though it can be freely chosen for an infinite series solution, e.g. h = 0.0001 or h = –0.0001 or h = 10,000, the choice of h will greatly affect the asymptotic property of a finite series solution. There are many publications on how to identify the value of h, and the prevailing method is the h-curve method.^66,67

For nonlinear oscillator, equation (2) does not work. As an example, we consider the Duffing equation

u^{″} + u + ε u^{3} = 0, u (0) = A, u^{'} (0) = 0

(6)

If a homotopy equation is constructed in the form

u^{″} + u + p ε u^{3} = 0

(7)

its zero-th order approximate equation is

{u^{″}}_{0} + u_{0} = 0

(8)

and only a 2π-periodic solution can be obtained.

For absence of the linear term in an equation, for example He⁹

u^{″} + u^{3} = 0, u (0) = A, u^{'} (0) = 0

(9)

We can only obtain a series of solution instead of a periodic solution if the following homotopy equation is constructed

u^{″} + p u^{3} = 0

(10)

To solve the problem, the parameter-expansion method^29–33 can be used. We re-write equation (9) in the form

u^{″} + 0 \cdot u + p u^{3} = 0

(11)

The coefficient, zero, of u in the missing linear term can be expanded in a series of p^29–33

0 = a_{0} + a_{1} p + a_{2} p^{2} + \dots

(12)

where

a_{i}

can be determined in view of no secular term in

u_{i}

. The zero-order equation is obtained by setting p = 0

{u^{″}}_{0} + a_{0} u_{0} = 0

(13)

When p = 1, equation (11) returns to be the original one. So, the solution process is to deform a linear oscillator with an unknown frequency when p = 0 to the original one when p = 1. This process converges very fast, and only one iteration is always enough for most problems, though iteration can continue without any difficulty to obtain higher order approximate solutions.

For a nonlinear oscillator with a negative linear term, for example

u^{″} - u + ε u^{3} = 0

(14)

the parameter expansion technology^29–33 can be also effectively used. The coefficient of u can be expanded into the form

- 1 = a_{0} + a_{1} p + a_{2} p^{2} + \dots

(15)

The above remarks reflect the development of the homotopy perturbation method. The most important step in the homotopy perturbation method is to construct a suitable homotopy equation with a possible one free parameter. For equation (1) the general construction of the homotopy perturbation method is^11–13

\tilde{L} (u) + p [L (u) + N (u)] = 0

(16)

where

\tilde{L} (u) = 0

with the given boundary/initial condition can describe the basic property of the equation. It can be a linear equation with a free parameter or a nonlinear equation easy to be solved. For nonlinear oscillator, we always assume that

\tilde{L} (u) = u^{″} + ω^{2} u

(17)

where

ω

is the frequency to be further determined.

An auxiliary parameter in the homotopy perturbation method

An auxiliary parameter was introduced in the variational iteration method^68–71 and the homotopy perturbation method⁷² to accelerate convergence. Some special technologies have been developed to optimally determine the auxiliary parameter by the least square method or other collocation methods, and modifications are called as the optimal variational iteration method^73–77 and optimal homotopy perturbation method.^25,78–82 This paper suggests an alternative construction of homotopy equation with an auxiliary parameter. This is an extension of the homotopy perturbation method with an auxiliary term¹⁵ and homotopy perturbation method with two expanding parameters.¹⁷

We re-write equation (1) in the form

L_{1} u + c u + N (u) = 0

(18)

by decomposing the linear operation into two parts:

L u = L_{1} u + c u

, where c can be any values

c \in R

. We can construct a homotopy equation in the form

L_{1} u + [a + (1 - a) p] c u + p N (u) = 0

(19)

L_{1} u + [a + (c - a) p] u + p N (u) = 0

(20)

where a is a free parameter to be further determined.

For the well-known Duffing equation, the homotopy equation is constructed in the form

u^{″} + (a + (1 - a) p) u + p ε u^{3} = 0

(21)

We assume that the solution can be expanded into a series of p in the form

u = u_{0} + p u_{1} + p^{2} u_{2} + \dots

(22)

Submitting equation (22) into equation (21), and manipulating as the classic perturbation method does, we have

{u^{″}}_{0} + a u_{0} = 0, u_{0} (0) = A, {u^{'}}_{0} (0) = 0

(23)

{u^{″}}_{1} + a u_{1} + (1 - a) u_{0} + ε u_{0}^{3} = 0, u_{1} (0) = 0, {u^{'}}_{1} (0) = 0

(24)

Solving equation (23), we have

u_{0} (t) = A \cos (a^{1 / 2} t)

(25)

Substituting equation (25) into equation (24) results in

{u^{″}}_{1} + a u_{1} + (1 - a) A \cos (a^{1 / 2} t) + ε A^{3} \cos^{3} (a^{1 / 2} t) = 0

(26)

By a simple operation, we have

{u^{″}}_{1} + a u_{1} + (1 - a + \frac{3}{4} ε A^{2}) A \cos (a^{1 / 2} t) + \frac{1}{4} ε A^{3} \cos (3 a^{1 / 2} t) = 0

(27)

No secular term in $u_{1} (t)$ requires that

1 - a + \frac{3}{4} ε A^{2} = 0

(28)

Solving a from equation (28) yields the result

a = 1 + \frac{3}{4} ε A^{2}

(29)

We, therefore, obtain the following approximate solution

u_{0} (t) = A \cos {{(1 + \frac{3}{4} ε A^{2})}^{1 / 2} t}

(30)

When $ε = 0$ , equation (30) is the exact solution; while when $ε \to \infty$ , the exact frequency is $ω_{e x} = 0.9318 \sqrt{ε A^{2}}$ , the relative error of equation (30) is 7%.^8,9

Amplitude-period formulation: García’s formula and improvements

As discussed in previous sections, the homotopy equation given in Equation (2) is not working for nonlinear oscillators. As described, several methods including the novel way to construct the homotopy using an auxiliary parameter in this paper overcome this shortcoming.

On the other hand and recently developed, simple amplitude-period formulas can be also considered to study nonlinear oscillators (see for instance literature^{58,63–65,83–85})

\begin{array}{l} \ddot{x} (t) = f (x (t)), \frac{d x}{d t} (0) = A, x (T) = A \\ T (aprox) = 2 \cdot π \cdot \sqrt{\frac{\int_{0}^{A} W (u) \cdot d u}{(N + 2) \cdot \int_{0}^{A} W (u) \cdot f (u) \cdot d u}, N \in ℕ} \end{array}

(31)

where A is the amplitude of the oscillations and T the period, with T(aprox) an approximation to the true amplitude-period relationship. It turns out that

f (0) = 0

and

\frac{f (x)}{x} < 0, x \neq 0

When W(x) takes the form $W (x) = x^{N}$ , the approximation formula presented at García⁶⁴ arises

T (aprox) = 2 \cdot π \cdot \sqrt{\frac{A^{N + 2}}{(N + 2) \cdot \int_{0}^{A} u^{N} \cdot f (u) \cdot d u}}, N \in ℕ

(32)

In García⁶⁴ an open problem was posed to prove or disprove whether or not equation (32) is an optimal approximation to the true amplitude-period relationship with N = 3. To provide an answer to this problem when $f (x)$ is a polynomial, a more tractable expression in equation (32) other than integrating with weighted functions will be developed.

Taking the integration $\int_{0}^{A} u^{N} \cdot f (u) \cdot d u$ and integrating by parts

\int_{0}^{A} u^{N} \cdot f (u) \cdot d u = A^{N} \cdot \int_{0}^{A} f (σ) \cdot d σ - N \cdot \int_{0}^{A} u^{N - 1} \cdot \int_{0}^{u} f (σ) \cdot d σ \cdot d u

(33)

Naming

f_{1} (u) = \int_{0}^{u} f (σ) \cdot d σ

(34)

Equation (33) becomes

\int_{0}^{A} u^{N} \cdot f (u) \cdot d u = A^{N} \cdot f_{1} (A) - N \cdot \int_{0}^{A} u^{N - 1} \cdot f_{1} (u) \cdot d u

(35)

Integrating again by parts

\int_{0}^{A} u^{N} \cdot f (u) \cdot d u = A^{N} \cdot f_{1} (A) - N \cdot (A^{N - 1} \cdot f_{2} (A) + - (N - 1) \cdot \int_{0}^{A} u^{N - 2} \cdot f_{2} (u) \cdot d u)

(36)

With $f_{2} (u) = \int_{0}^{u} f_{1} (σ) \cdot d σ$ . Iteratively

\begin{array}{l} \int_{0}^{A} u^{N} \cdot f (u) \cdot d u = \sum_{i = 0}^{N} A^{N - i} \cdot f_{i + 1} (A) \cdot {(- 1)}^{i} \cdot i! \\ f_{i} (u) = \int_{0}^{u} f_{i - 1} (u) \cdot d u), f_{0} (u) = f (u) \end{array}

(37)

where i! means the classical factorial function for integers. Clearly equation (37) requires to iteratively integrate only the function f(x) without weightings.

Utilizing equation (37) and assuming that $f (x)$ is a polynomial, the following theorem can be proved.

Theorem 1 (small-amplitude oscillations): Given a nonlinear oscillator $\ddot{x} (t) = f (x (t)), \frac{d x}{d t} (0) = A, x (T) = A$ , with $f (x) = \sum_{j = 1}^{m} b_{j} \cdot x^{j}$ , the amplitude-period approximation formula (31) with N = 3 approximates the true amplitude-period relationship with a local minimum error when $A \to 0$ .

Proof: Since we are interested in evaluating N = 3 with minimum error, let us consider the following nonlinear programming problem

\min_{x \in ℜ} \frac{1}{2} \cdot {(F (x) - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}})}^{2}

(38)

where

F (x) = (x + 2) \cdot \sum_{i = 0}^{N} \frac{f_{i + 1} (A)}{A^{i + 2}} \cdot {(- 1)}^{i} \cdot \leq x \cdot (x - 1) \dots (x - i + 1)

and T_true is the period of the nonlinear oscillator. Taking derivative in equation (38)

\frac{\partial}{\partial x} [(x + 2) \cdot \sum_{i = 0}^{N} \frac{f_{i + 1} (A)}{A^{i + 2}} \cdot {(- 1)}^{i} \cdot x \cdot (x - 1) \dots (x - i + 1)] = 0

(39)

It is important to remark that the optimal problem requests for a possible local extremum satisfying equation (39) in the interval: $x \in [N - 0.5, N + 0.5]$ . It is enough to prove that such a local optimal exists for a certain N taking into consideration that $f (x) = \sum_{j = 1}^{m} b_{j} \cdot x^{j}$

I f N = 3, A \to 0 \approx {\begin{matrix} \begin{matrix} f_{0} (x) = b_{1} \cdot x \\ f_{1} (x) = b_{1} \cdot \frac{x^{2}}{2} \end{matrix} \\ f_{2} (x) = b_{1} \cdot \frac{x^{3}}{3!} \\ \begin{matrix} f_{3} (x) = b_{1} \cdot \frac{x^{4}}{4!} \\ f_{4} (x) = b_{1} \cdot \frac{x^{5}}{5!} \end{matrix} \end{matrix}

(40)

Then

\begin{matrix} \frac{\partial}{\partial x} [(x + 2) \cdot (b_{1} \cdot \frac{A^{2}}{2 \cdot A^{2}} - b_{1} \cdot \frac{A^{3}}{3! \cdot A^{3}} \cdot x + b_{1} \cdot \frac{A^{4}}{4! \cdot A^{4}} \cdot x \cdot (x - 1) \\ - b_{1} \cdot \frac{A^{5}}{5! \cdot A^{5}} \cdot x \cdot (x - 1) \cdot (x - 2))] = 0 \end{matrix}

(41)

Simplifying

- 4 \cdot x^{3} + 18 \cdot x^{2} - 22 \cdot x + 6 = 0

(42)

With roots

x_{1} = 0.381966, x_{2} = 1.5, x_{3} = 2.61803

(43)

Clearly, the integer approximation of the third roots gives the desired result. This completes the proof.

Notice that the theorem is valid even for infinite series representation: $m \to \infty$ . In this way, Theorem 1 encourages the use of N = 3

Theorem 2 (universal bounds): Given a nonlinear oscillator $\ddot{x} (t) = f (x (t)), \frac{d x}{d t} (0) = A, x (T) = A$ , such that

| \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} | \leq ρ, ρ \in ℝ^{+}

(44)

Then

\underline{A} \leq A \leq \bar{A}

(45)

where

{\underline{A}, \bar{A}}

are the minimum and the maximum roots of the equation given by

| \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} | = ρ

(46)

Proof: Assuming the condition (equation (44)) and defining a linear programming problem with constraints

\begin{array}{l} \min / \max_{A \in ℝ} A \\ s u c h t h a t : \\ | \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} | \leq ρ, ρ \in ℝ^{+} \end{array}

(47)

The Karush-Kuhn-Tucker conditions read

\begin{array}{l} 1 + μ_{1} \cdot \frac{\partial}{\partial A} (\frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} - ρ) + μ_{2} \cdot \frac{\partial}{\partial A} (\frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} + ρ) = 0 \\ μ_{1} \cdot (\frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} - ρ) = 0 \\ μ_{2} \cdot (\frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} + ρ) = 0 \end{array}

(48)

Since (see for instance García⁶⁴)

T_{t r u e} = 2 \cdot \sqrt{2} \cdot \int_{0}^{A} \frac{d x}{\sqrt{\int_{x}^{A} f (u) \cdot d u}}

(49)

It is clear that $\frac{4 \cdot π^{2}}{T_{t r u e}^{2}}$ is in fact a $C^{\infty} (ℝ)$ function, then equation (48) yields

\begin{matrix} μ_{1} \neq 0 \Rightarrow \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} - ρ = 0 \\ μ_{2} \neq 0 \Rightarrow \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} + ρ = 0 \end{matrix}

(50)

In other words

| \frac{5 \cdot \int_{0}^{A} u^{3} \cdot f (u) \cdot d u}{A^{5}} - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} | = ρ

(51)

This completes the proof, taking the minimum/maximum roots of this equation.

The important remark is about the universality nature of equation (44) providing amplitude-period approximations with a given error bound. Moreover, when the nonlinear oscillator is polynomial: $f (x) = \sum_{j = 1}^{m} b_{j} \cdot x^{j}$ , explicit amplitude-period relations can be constructed from Theorem 2:

Corollary 1: Given a nonlinear oscillator $\ddot{x} (t) = f (x (t)), \frac{d x}{d t} (0) = A, x (T) = A$ , such that $f (x) = \sum_{j = 1}^{m} b_{j} \cdot x^{j}$

| 5 \cdot \sum_{i = 0}^{3} \frac{f_{i + 1} (A)}{A^{i + 2}} \cdot {(- 1)}^{i} \cdot - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} | \leq ρ, ρ \in ℝ^{+}

(52)

Then

\underline{A} \leq A \leq \bar{A}

(53)

where

{\underline{A}, \bar{A}}

are given by

\begin{array}{l} \underline{A} = \max {1 + \frac{| c_{m - 2} |}{| c_{m - 1} |}, \frac{| c_{m - 3} |}{| c_{m - 1} |}, \dots, \frac{| c_{0} |}{| c_{m - 1} |}} \\ \bar{A} = \frac{1}{\bar{A}} \end{array}

(54)

where

5 \cdot \sum_{i = 0}^{3} \frac{f_{i + 1} (A)}{A^{i + 2}} \cdot {(- 1)}^{i} \cdot - \frac{4 \cdot π^{2}}{T_{t r u e}^{2}} = \pm ρ

is written as

\sum_{k = 0}^{m - 1} c_{k} (\frac{4 \cdot π^{2}}{T_{t r u e}^{2}} \pm ρ) \cdot A^{k}

(55)

Proof:

According to Theorem 2 and the proof in Theorem 1 it is enough to consider bounds for the maximum and minimum positive roots of a polynomial (see for instance Cohen⁸⁶). Finally performing the change: $A \to \frac{1}{A}$ , the result follows. This completes the proof.

Results and discussion

Homotopy method with auxiliary parameters

The homotopy perturbation method always stops before second iteration; however, iterations can be continued without any difficulties. If higher order approximates solutions are needed, equation (19) can be extended to more than one auxiliary parameter, for example if a second-order approximate solution is searched for, the homotopy equation can be written in the form

L_{1} u + (a_{0} + a_{1} p + a_{2} p^{2}) u + p N (u) = 0

(56)

where the auxiliary parameters

a_{1}

and

a_{2}

are determined, respectively, by the request of no secular terms in

u_{1}

and

u_{2}

, and

a_{0}

is identified by the following equation

a_{0} + a_{1} + a_{2} = c

(57)

For N-th order approximate solution, we have

L_{1} u + (\sum_{n = 0}^{N} a_{n} p^{n}) u + p N (u) = 0

(58)

and

\sum_{n = 0}^{N} a_{n} = c

(59)

where the auxiliary parameter

a_{n}

can be similarly identified by the request of no secular term in

u_{n}

The present technology can be easily extended to the multiple scales homotopy perturbation method⁸⁷ to deal with forced oscillators.

Amplitude-period formulation

Nonlinear pendulum

The nonlinear pendulum is always a source for testing approximations (see for instance Beléndez et al.³)

\ddot{x} (t) = - f (x (t)), f (x) = \frac{g}{L} \cdot \sin (x)

(60)

where g is the classical gravity constant and L the length of the pendulum.

Applying equation (37) and Theorem 1

{\begin{array}{l} \int_{0}^{A} u^{3} \cdot f (u) \cdot d u = \\ = A^{3} \cdot f_{1} (A) - 3 \cdot A^{2} \cdot f_{2} (A) + 6 \cdot A \cdot f_{3} (A) - 6 \cdot f_{4} (A) \\ f_{0} (u) = \frac{g}{L} \cdot \sin (x) \\ f_{1} (A) = \frac{g}{L} \cdot (- \cos (A) + 1) \\ f_{2} (A) = \frac{g}{L} \cdot (- \sin (A) + A) \\ f_{3} (A) = \frac{g}{L} \cdot (\cos (A) - 1 - \frac{A^{2}}{2}) \\ f_{4} (A) = \frac{g}{L} \cdot (\sin (A) - A - \frac{A^{3}}{6}) \end{array}

(61)

Finally

T (a p r o x) = 2 \cdot π \cdot \sqrt{\frac{A^{5}}{5 \cdot \frac{g}{L} \cdot (3 \cdot (A^{2} - 2) \cdot \sin (A) - A \cdot (A^{2} - 6) \cdot \cos (A))}}

(62)

Whereas the true amplitude-period relationship is (see Beléndez et al.³)

T (t r u e) = \frac{2}{π} \cdot 2 \cdot π \cdot \sqrt{\frac{L}{g}} \cdot K (k)

(63)

where K(k) is the complete elliptical integral and

k = \sin^{2} (\frac{A}{2})

To obtain a comparison with small amplitude oscillations)

l i m_{A \to 0} \frac{2 \cdot π \cdot \sqrt{\frac{L}{g}}}{2 \cdot π \cdot \sqrt{\frac{A^{5}}{5 \cdot \frac{g}{L} \cdot (3 \cdot (A^{2} - 2) \cdot \sin (A) - A \cdot (A^{2} - 6) \cdot \cos (A))}}} = 1

(64)

This means, zero error.

Duffing oscillator

To show the improved formula’s applicability, the classical Duffing oscillator is analyzed (see for instance Kovacic and Brennan²)

\ddot{x} (t) = - f (x (t)), f (x) = x + ϵ \cdot x^{3}

(65)

Applying Theorem 2

\int_{0}^{A} u^{3} \cdot f (u) \cdot d u = \sum_{i = 0}^{3} A^{3 - i} \cdot f_{i + 1} (A) \cdot {(- 1)}^{i} \cdot i! \cdot (\begin{matrix} 3 \\ 3 - i \end{matrix})

(66)

where

f_{i} (u) = \int_{0}^{u} f_{i - 1} (σ) \cdot d σ, f_{0} (u) = f (u)

(67)

Hence

{\begin{array}{l} \int_{0}^{A} u^{3} \cdot f (u) \cdot d u = \\ = A^{3} \cdot f_{1} (A) - 3 \cdot A^{2} \cdot f_{2} (A) + 6 \cdot A \cdot f_{3} (A) - 6 \cdot f_{4} (A) \\ f_{0} (u) = - u - ϵ \cdot u^{3} \\ f_{1} (A) = - \frac{A^{2}}{2} - ϵ \cdot \frac{A^{4}}{4} \\ f_{2} (A) = - \frac{A^{3}}{6} - ϵ \cdot \frac{A^{5}}{20} \\ f_{3} (A) = - \frac{A^{4}}{24} - ϵ \cdot \frac{A^{6}}{120} \\ f_{4} (A) = - \frac{A^{5}}{120} - ϵ \cdot \frac{A^{7}}{840} \end{array}

(68)

Finally

T (a p r o x) = 2 \cdot π \cdot \sqrt{\frac{A^{5}}{5 \cdot (\frac{A^{5}}{5} + ϵ \cdot \frac{A^{7}}{7})}}

(69)

Whereas the true amplitude-period relationship for $ϵ \to \infty$ is (see Dan and Zhi⁵⁹)

T (t r u e) = \frac{2 \cdot π}{0.9318 \cdot \sqrt{ϵ \cdot A^{2}}}

(70)

Moreover, equation (55) reads

\sum_{k = 0}^{m - 1} c_{k} (\frac{4 \cdot π^{2}}{T_{t r u e}^{2}} \pm ρ) \cdot A^{k} = 1 + ϵ \cdot \frac{5 \cdot A^{2}}{7} - (\frac{4 \cdot π^{2}}{T_{t r u e}^{2}} \pm ρ) = 0

(71)

Giving the following bounds

\begin{matrix} \bar{A} = \sqrt{\frac{7}{ϵ \cdot 5}} \sqrt{(ρ - 1) + (\frac{4 \cdot π^{2}}{T_{t r u e}^{2}})} \\ \underline{A} = \sqrt{\frac{7}{ϵ \cdot 5}} \sqrt{(- ρ - 1) + (\frac{4 \cdot π^{2}}{T_{t r u e}^{2}})} \end{matrix}

(72)

If $ϵ \to \infty$ , asymptotically

A \approx \sqrt{\frac{7}{ϵ \cdot 5}} \cdot \frac{2 \cdot π}{T_{t r u e}} (ϵ \to \infty)

(73)

The clear and important implication is that Theorem 2 provides an asymptotic equivalence between Garcia’s formula and the true amplitude-period relationship when the parameter $ϵ \to \infty$ , independent of the error bound ρ. This result is aligned with the homotopy method presented in previous sections.

An oscillator with discontinuity

To continue with the asymptotic equivalence, let us consider the following oscillator

\ddot{x} (t) = sign (x (t))

(74)

where

sign (x (t))

is the classical sign function. Applying Theorem 2 when

ρ \to 0

A \approx \frac{5}{16} \cdot \frac{T_{t r u e}^{2}}{π^{2}}

(75)

Comparing with the true amplitude-period relationship (see García⁶⁴)

A = \frac{T_{t r u e}^{2}}{32}

(76)

The asymptotic approximation is very precise (about 98.69%).

There are many alternative approaches to nonlinear oscillators, an answer was given to the open problem that appeared in García⁶⁴ about an amplitude-period formula for small amplitude period oscillations (Theorem 1). The technique can be extended to improve the theorem for non-small oscillations taking into account the universal bounds presented in Theorem 2 and Corollary 1.

It should also be pointed out that the nonlinear oscillation community finds a road to Rome by simple frequency-amplitude formulations discussed in Nonlinear Science Letters A^64,65 and literature.^63,83–86

Conclusions

In this paper, we suggest a novel way to construct a homotopy equation with one or more auxiliary parameters. The new formulation's salient property relies on the possibility to control or even accelerate the convergence of the approximations.

This idea can be extended to other nonlinear problems, and the present paper can be used as a paradigm for other applications.

Finally, the Garcia’s formula along with the universal bounds given in Theorem 1 makes such a formulation very promising from the computer implementation point of view. The methods discussed in this paper can be easily extended to fractional differential equations.^88–92

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is supported by Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), National Natural Science Foundation of China under grant No.11372205 and Universidad Tecnológica Nacional.

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Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

Abstract

Keywords

Introduction

Homotopy perturbation method

An auxiliary parameter in the homotopy perturbation method

Amplitude-period formulation: García’s formula and improvements

Results and discussion

Homotopy method with auxiliary parameters

Amplitude-period formulation

Nonlinear pendulum

Duffing oscillator

An oscillator with discontinuity

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References