A new modification of the homotopy perturbation method

Abstract

This study proposes a new modification of the homotopy perturbation method. A new parameter alpha is introduced into the homotopy equation in order to improve the results and accuracy. An optimal analysis identifies the parameter alpha, aimed at improving the solutions. A comparative analysis of the proposed method reveals that the new method presents results with higher degree of accuracy and precision than the classic homotopy perturbation method. Absolute error analysis shows the convenience of the proposed method, providing much smaller errors. Two examples are presented: Duffing and Van der pol’s nonlinear oscillators to demonstrate the efficiency, accuracy, and applicability of the new method.

Keywords

Homotopy perturbation method homotopy equation nonlinear oscillator alpha parameter Duffing oscillator Van der Pol oscillator

Introduction

Nonlinear oscillators occupy an important part of mathematics and have diverse applications in different fields of science and engineering. Despite the wide applicability, obtaining exact solutions for such oscillators have proved difficult for researchers. Efforts to overcome these difficulties have led to formulation of several analytical and approximate techniques. Some of these techniques include non-perturbative method,^1–5 weighted linearization,⁶ Adomian decomposition,^7–14 variational iteration,^15–20 energy balance,^21–23 tanh,^24–26 F-expansion,^27,28 and homotopy perturbation methods.^29–34 Most of the methods have recorded recent advancements and improvements, as evidenced in the homotopy perturbation method. Since its inception in 1998,^35,36 the homotopy perturbation has passed many forms of modifications and improvements.^37,38 The method has been studied widely by lot of researchers such as Adamu and Ogenyi,³¹ Ganji and Sadighi,³⁹ Odibat and Momnni,⁴⁰ Cveticanin,⁴¹ and Khan and Mohyud-Din.⁴² In a recent development, the parameter expansion technique⁴³ has employed homotopy perturbation⁴⁴ to effectively handle the zero-order approximate and He’s polynomials to deal with the nonlinear term.⁴⁵ Presently, the method is used in combination with other emerging methods.⁴⁶ Homotopy perturbation method combines least square technique⁴⁶ and variational theory.⁴⁷ Laplace transform has also adopted the homotopy perturbation method.^48,49 These coupled methods are simple in theory and application.

Homotopy perturbation method has developed quickly and well, but as often, leaves space for additional improvements and modifications. Accordingly, new modification of the homotopy perturbation method is proposed. The method embeds new parameter, alpha, to the traditional homotopy perturbation technique for fast and improved solutions.

Homotopy perturbation method

Consider the following general oscillator equation

L (u) + N (u) = 0

(1)

where L is the linear operator and N is the nonlinear operator. To construct the homotopy equation, consider⁸

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

(2)

Equivalently

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

(3)

where p is a homotopy parameter

p \in [0,1]

and

u_{0}

is the initial solution.

For better results, the homotopy analysis method^36,37 introduces an auxiliary parameter

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

(4)

Here, h is a nonzero parameter used to control the convergence. Equations (3) and (4) lead

\begin{array}{l} L (u) + p N (u) = 0, \\ h [L (u) + p N (u)] = 0 \end{array}

(5)

When p = 1, for any nonzero h, equation (5) is equivalent to the original problem. Optimal identification of h requires skill and care; the choice of h can influence the asymptotic behavior of a finite series solution.³⁶

At $p = 0$ , both equations (3) and (4) transform

L (u) - L (u_{0}) = 0

(6)

Then, the homotopy perturbation method presents its solutions in the form of summation of finite series

u = u_{0} + P u_{1} + P^{2} u_{2} + P^{3} u_{3} \dots = \sum_{i}^{\infty} p^{i} u_{i}

(7)

With p set as p = 1, equation (7) yields

u = \lim_{p \to 1} u u_{0} + u_{1} + u_{2} + u_{3} + \dots = \sum_{i}^{\infty} u_{i}

(8)

Proposed method

The homotopy perturbation method has recorded rapid development. To keep up with the current reality of homotopy’s modifications and improvements, this study suggests alternative reconstruction of the homotopy equation with the introduction of a new parameter, $α$ .

Consequently, equation (1) can be written as

L (u) + N (u) + ω^{2} α u - ω^{2} α u = 0

(9)

by introducing the

α

parameter along with the angular frequency

ω

. The choice of

ω

is predetermined on the fact that we are dealing with oscillatory systems. However, only the alpha and the dependent functions

α u

are introduced, and the angular frequency is ignored in the new homotopy equation.

For the nonlinear oscillator, we construct the following homotopy

L (u) - L (u_{0}) + ω^{2} α u + p [L (u_{0}) + N (u) - ω^{2} α u] = 0, p ϵ [0,1]

(10)

The transition of “p” from 0 to 1 signifies continuous deformation of the problem from linear to nonlinear. Following the classical homotopy solution, the new method suggests its solution in the form of series

u = u_{0} + P u_{1} + P^{2} u_{2} + P^{3} u_{3} \dots = \sum_{i}^{\infty} p^{i} u_{i}

(11)

With p = 1, equation (11) transforms

u = \lim_{p \to 1} u = u_{0} + u_{1} + u_{2} + u_{3} + \dots = \sum_{i}^{\infty} u_{i}

(12)

as the approximate solution.

Equation (10) works for any kind of differential equations involving both implicit and explicit functions. As an example, consider the first-order nonlinear ordinary differential equation

\ddot{u} (t) + u^{2} (t) = 0, | t | < 1

(13)

This equation is implicitly formulated. Thus, the constructed homotopy equation will appear in the following way

\ddot{u} (t) - {\dot{u}}_{0} + α p ({\dot{u}}_{0} + u^{2} (t)) = 0, p ϵ [0,1]

(14)

Applicability of the proposed method is illustrated in section 2, and the results are presented in the subsequent section.

Applications

Example 1

Consider the Duffing oscillator

\ddot{u} + u + ε u^{3} = 0, u (0) = A, \dot{u} (0) = 0

(15)

Homotopy perturbation method

To solve equation (15) via the homotopy perturbation method, we rewrite equation (15) as follows

\ddot{u} + u + ω^{2} u - ω^{2} u + ε u^{3} = 0, u (0) = A, \dot{u} (0) = 0

(16)

The following homotopy equation is constructed for equation (16)

\ddot{u} + ω^{2} u + p [(1 - ω^{2}) u + ε u^{3}] = 0, u (0) = A, \dot{u} (0) = 0 p \in [0,1]

(17)

With p = 1, equations (15) and (17) are equivalent. Using p, we have

u = u_{0} + P u_{1} + P^{2} u_{2} + P^{3} u_{3} + \dots

(18)

In addition, for p = 1, equation (18) transforms as follows

u = \lim_{p \to 1} u = u_{0} + u_{1} + u_{2} + u_{3} + \dots

(19)

Substituting equation (18) into equation (17) and equating the identical powers of p, we have

P^{0} : {\ddot{u}}_{0} + ω^{2} u_{0} = 0, u_{0} (0) = A, {\dot{u}}_{0} (0) = 0

(20)

P^{1} : {\ddot{u}}_{1} + ω^{2} u_{1} + (1 - ω^{2}) u_{0} + ε u_{0}^{3} = 0, u_{1} (0) = A, {\dot{u}}_{1} (0) = 0

(21)

The solution for equation (20) using the boundary condition leads to

u_{0} = A cos ω t

(22)

Substituting equation (22) into equation (21) and using the expansion of $u_{0}^{3}$ , we obtain

{\ddot{u}}_{1} + ω^{2} u_{1} + [(1 - ω^{2}) A + \frac{3}{4} ε A^{3}] cos ω t + \frac{1}{4} ε A^{3} cos 3 ω t = 0

(23)

Removing the secular term in equation (23) gives

(1 - ω^{2}) A + \frac{3}{4} ε A^{3} = 0

(24)

ω_{H} = \sqrt{\frac{3}{4} ε A^{2} + 1}

(25)

The approximate solution of equation (15) using the homotopy perturbation method is given by

u_{H} (t) = A cos ω_{H} t

(26)

where

u_{H}

and

ω_{H}

denote the approximate solution and angular frequency via the homotopy perturbation method.

Proposed method

For the application of the proposed method to equation (15), we rewrite the equation in the following form

\ddot{u} + u + ω^{2} α u - ω^{2} α u + ε u^{3} = 0, u (0) = A, \dot{u} (0) = 0

(27)

From equation (27), we construct the following homotopy

\ddot{u} + ω^{2} α u + p [(1 - ω^{2} α) u + ε u^{3}] = 0, u (0) = A, \dot{u} (0) = 0 p \in [0,1]

(28)

Equation (15) coincides with equation (28) in the region where p = 1 but changes to the second-order linear differential equation when p = 0. Substituting equation (18) into the homotopy equation and equating the same powers of p, we have the following

P^{0} : {\ddot{u}}_{0} + ω^{2} α u_{0} = 0, u_{0} (0) = A, {\dot{u}}_{0} (0) = 0

(29)

P^{1} : {\ddot{u}}_{1} + ω^{2} α u_{1} + (1 - ω^{2} α) u_{0} + ε u_{0}^{3} = 0, u_{1} (0) = 0, {\dot{u}}_{1} (0) = 0

(30)

The new method assumes the following initial solution for equation (15)

u_{0} = A cos ω t

(31)

Using the expansion of $u_{0}^{3}$ and the substitution of equation (31) in equation (30), we obtain

{\ddot{u}}_{1} + ω^{2} α u_{1} + [(1 - ω^{2} α) A + \frac{3}{4} ε A^{3}] cos ω t + \frac{1}{4} ε A^{3} cos 3 ω t = 0

(32)

Elimination of the secular term in equation (32) yields

(1 - ω^{2} α) A + \frac{3}{4} ε A^{3} = 0

(33)

ω_{N} = \sqrt{\frac{3 ε A^{2}}{4 α_{o p t}} + \frac{1}{α_{o p t}}}

(34)

where

α_{o p t}

denotes the optimal value of the alpha.

The method on this study provides a solution of equation (15) in the form

u_{N} (t) = A cos ω_{N} t

(35)

where

u_{N}

and

ω_{N}

denote the approximate solution and the angular frequency via this method. Equation (15) has the following exact solution³¹

ω_{E x} (t) = \frac{2 π}{6.743} \sqrt{ε A^{2}} a s ε \to \infty

(36)

u_{E x} (t) = A cos ω_{E x} t

(37)

where

u_{E x}

and

ω_{E x}

denote exact solution and angular frequency, respectively.

To identify $α_{o p t}$ , we let

ω_{N} = ω_{E x} \Leftrightarrow \sqrt{\frac{3 ε A^{2}}{4 α_{o p t}} + \frac{1}{α_{o p t}}} = \frac{2 π}{6.743} \sqrt{ε A^{2}}

(38)

Obtaining the following solution

α_{o p t} = 11.367 \frac{(2 ε A^{2} + 1)}{ε {(π A)}^{2}}

(39)

Example 2

Consider the classical fractional Van der Pol damped nonlinear oscillator equation

\ddot{u} + u^{1 / 3} + ε (1 - u^{2}) \dot{u} = 0, u (0) = A, \dot{u} (0) = 0

(40)

Homotopy perturbation method

To solve equation (40) through the homotopy perturbation method, we rewrite the equation as

\ddot{u} + u^{1 / 3} + ω^{2} u - ω^{2} u + ε (1 - u^{2}) \dot{u} = 0, u (0) = A, \dot{u} (0) = 0

(41)

Formulating homotopy equation for equation (41), we have

\ddot{u} + ω^{2} u + P [u^{1 / 3} - ω^{2} u + ε (1 - u^{2}) \dot{u}] = 0, P \in [0, 1]

(42)

Equation (42) turns from second-order linear to the nonlinear differential equation as p changes from 0 to 1. Without loss of generality, we substitute equation (18) into equation (42) and solve for identical powers of p, to obtain

P^{0} : {\ddot{u}}_{0} + ω^{2} u_{0} = 0, u_{0} (0) = A, {\dot{u}}_{0} (0) = 0

(43)

P^{1} : {\ddot{u}}_{1} + ω^{2} u_{1} + u_{0}^{1 / 3} - ω^{2} u_{0} + ε (1 - u_{0}^{2}) {\dot{u}}_{0} = 0, u_{1} (0) = 0, {\dot{u}}_{1} (0) = 0

(44)

Consequently

u_{0} = A cos ω t

(45)

Ref. [34,35] gives the Fourier series of ${cos}^{1 / 3} ω t$ as

{cos}^{1 / 3} ω t = b_{1} cos ω t + b_{2} cos 3 ω t

(46)

Finally, determine $b_{1}$ and $b_{2}$ as 1.15960 and −0.231919, respectively.

The substitution of equations (45) and (46) into equation (44), we obtain

{\ddot{u}}_{1} + ω^{2} u_{1} + \frac{1}{4} [4 b_{1} A^{1 / 3} - 4 A ω^{2}] cos ω t + b_{2} A^{1 / 3} cos 3 ω t + \frac{1}{4} [A^{3} εω - 4 A εω] sin ω t + \frac{1}{4} A^{3} εω sin 3 ω t = 0

(47)

Eliminating secular term yields

4 b_{1} A^{1 / 3} - 4 {A ω}^{2} = 0, A^{2} - 4 = 0

(48)

From equation (46),⁴⁸ it follows directly that

ω_{H} = \frac{\sqrt{b_{1}}}{A^{2 / 3}}, A = 2

(49)

ω_{H} = \frac{\sqrt{1.15960}}{2^{2 / 3}} = 0.6784

(50)

Solution of equation (40) is given as defined in equation (26).

Proposed method

For the solution of equation (40), using the proposed method, we write

\ddot{u} + u^{1 / 3} + ω^{2} α u - ω^{2} α u + ε (1 - u^{2}) \dot{u} = 0, u (0) = A, \dot{u} (0) = 0

(51)

Constructing the homotopy equation for equation (51), we have

\ddot{u} + ω^{2} α u + P [u^{1 / 3} - ω^{2} α u + ε (1 - u^{2}) \dot{u}] = 0, P \in [0, 1]

(52)

Following the previous steps, we have

P^{0} : {\ddot{u}}_{0} + ω^{2} α u_{0} = 0, u_{0} (0) = A, {\dot{u}}_{0} (0) = 0

(53)

P^{1} : {\ddot{u}}_{1} + ω^{2} α u_{1} + u_{0}^{1 / 3} - ω^{2} α u_{0} + ε (1 - u_{0}^{2}) {\dot{u}}_{0} = 0, u_{1} (0) = 0, {\dot{u}}_{1} (0) = 0

(54)

u_{0} = A cos ω t

(55)

Using Fourier series decomposition

{cos}^{1 / 3} ω t = b_{1} cos ω t + b_{2} cos 3 ω t

(56)

b_{1} and b_{2}

are defined as 1.15960 and −0.231919, respectively^34,35

Substituting equations (55) and (56) into (54), we obtain

{\ddot{u}}_{1} + ω^{2} α u_{1} + \frac{1}{4} [4 b_{1} A^{1 / 3} - 4 A α ω^{2}] cos ω t + b_{2} A^{1 / 2} cos 3 ω t + \frac{1}{4} [A^{3} α εω - 4 A α εω] sin ω t + \frac{1}{4} A^{3} ε^{3 / 2} ω sin 3 ω t = 0

(57)

The elimination of the secular term from equation (57) leads

4 b_{1} A^{1 / 3} - 4 A α ω^{2} = 0, A^{3} α εω - 4 A α εω = 0

(58)

b_{1} A^{1 / 3} - A {αω}^{2} = 0, A^{2} - 4 = 0

(59)

ω_{N} = \frac{\sqrt{b_{1}}}{A^{2 / 3} \sqrt{α_{o p t}}}, A = 2

(60)

As previously stated, the new method gives its solution in equation (35)

Equation (40) has the following exact solution³⁴

u_{E x} (t) = \frac{2 A}{\sqrt{A^{2} + (4 - A^{2}) e^{- ε t}}} cos ω_{E x} t

(61)

ω_{E x} = \frac{\sqrt{π} Γ (5 / 4)}{A^{\frac{1}{3}} Γ (7 / 4) \sqrt{(8 / 3)}}

(62)

Following the same procedure as outlined in the previous example, we identify the optimal value of α

α_{o p t} = \frac{8 b_{1}}{3 π {(\sqrt[3]{A})}^{2}} \frac{{[Γ (7 / 4)]}^{2}}{Γ (5 / 4)}

(63)

Results and discussion

Efficiency of the proposed method

A comparative analysis of the results presented by both methods: HPM and the method in this study show the potentiality of the proposed method, offering better results with higher degree of accuracy than the homotopy perturbation method. And as a salient property, the new method possesses also fewer deviations to arrive a solution.

Error analysis

Careful checks on Tables 1 and 2 and Figures 1–5 indicate that the method present in this study has a smaller absolute error than the classic homotopy perturbation. Figures 2 and 3 show the errors of the proposed method are sufficiently small and close to the exact solution as a result of the high precision of the new method.

Table 1.

Numerical comparison between the new technique, homotopy perturbation method and the exact solution of equation (15) for $ε = 0.01, A = 1, α_{o p t} = 117.59$ .

(t)	New method $(u_{N})$	HPM $(u_{H})$	Exact $(u_{Ex})$	$\| u_{N} - u_{Ex} \|$	$u_{H} - u_{Ex}$
0.1	0.99999998695033	0.99999846549016	0.99999998676489	1.85 × 10⁻¹⁰	2.11 × 10⁻⁹
0.2	0.99999994780134	0.99999386196533	0.99999994705954	7.42 × 10⁻¹⁰	6.09 × 10⁻⁶
0.3	0.99999988255301	0.99998618943966	0.99999988088397	1.67 × 10⁻⁹	1.37 × 10⁻⁵
0.4	0.99999979120535	0.99997544793668	0.99999978823818	2.97 × 10⁻⁹	2.43 × 10⁻⁵
0.5	0.99999967375836	0.99996163748936	0.99999966912216	4.64 × 10⁻⁹	3.80 × 10⁻⁵
0.6	0.99999953021205	0.99994475814009	0.99999952353593	6.68 × 10⁻⁹	5.48 × 10⁻⁵
0.7	0.99999936056642	0.99992480994067	0.99999935147947	9.09 × 10⁻⁹	7.45 × 10⁻⁵
0.8	0.99999916482148	0.99990179295232	0.99999915295281	1.19 × 10⁻⁸	9.74 × 10⁻⁵
0.9	0.99999894297722	0.99987570724568	0.99999892795594	1.50 × 10⁻⁸	1.23 × 10⁻⁴
1.0	0.99999869503366	0.99984655290081	0.99999896764887	2.72 × 10⁻⁷	1.52 × 10⁻⁴

Table 2.

Numerical comparison between the new technique, homotopy perturbation method and the exact solution of equation (40) for $ε = 0.01, A = 2$ , and $α_{o p t} = 0.6381$ .

(t)	New method $(u_{N})$	HPM $(u_{H})$	Exact $(u_{Ex})$	$\| u_{N} - u_{Ex} \|$	$u_{H} - u_{Ex}$
0.1	1.99999780328039	1.99999859806989	1.9999930864095	4.717 × 10⁻⁶	5.51 × 10⁻⁶
0.2	1.99999121312639	1.99999439228153	1.9999723458916	1.89 × 10⁻⁵	2.20 × 10⁻⁵
0.3	1.99998022955248	1.99998738264082	1.99993777843534	4.25 × 10⁻⁵	4.96 × 10⁻⁵
0.4	1.99996485258278	1.99997756915757	1.99988938433115	7.55 × 10⁻⁵	8.82 × 10⁻⁵
0.5	1.99994508225107	1.99996495184556	1.99982716391361	1.17 × 10⁻⁴	1.38 × 10⁻⁴
0.6	1.99992091860078	1.99994953072246	1.99975111761289	1.70 × 10⁻⁴	1.98 × 10⁻⁴
0.7	1.99989236168499	1.99993130580990	1.99966124595472	2.31 × 10⁻⁴	2.70 × 10⁻⁴
0.8	1.99985941156644	1.99991027713343	1.99955754956044	3.01 × 10⁻⁴	3.52 × 10⁻⁴
0.9	1.99982206831751	1.99988644472253	1.99944002914696	3.82 × 10⁻⁴	4.46 × 10⁻⁴
1.0	1.99978033202023	1.99985980861061	1.99930868552677	4.72 × 10⁻⁴	5.51 × 10⁻⁴

Figure 1.

Solution of equation (15) using the proposed and the HPM.

Figure 2.

Zoomed solution of equation (15) using the proposed and the HPM.

Figure 3.

Absolute errors of equation (15) using the proposed and the HPM.

Figure 4.

Solution of equation (40) using the proposed and the HPM.

Figure 5.

Absolute errors of equation (40) using the proposed and the HPM.

Optimization of the alpha parameter

The alpha parameter is an important part of the proposed method. For higher degree of accuracy, an optimized value of the parameter is recommended. In this study, we rely on the behavior of the exact solution to identify the value of the parameter. In a future study, we will develop an error bound analysis using Pontryagin’s maximum principle for optimal control which will present an acceptable approach on the identification of optimal value for the new method.

Conclusions

This study suggests a new modification of the homotopy perturbation via the introduction of a new parameter α. The method works well for the nonlinear oscillators and problems involving implicit functions, allowing also the extension of the idea to other nonlinear problems where the functions are explicitly presented. Numerical analysis of the proposed method shows more accurate results than the classic homotopy perturbation method and can be applied easily to any forms of differential equations without any difficulty.

Footnotes

Acknowledgment

Our acknowledgments go to Dr. Andres Garcia who has painstakingly contributed to the success of this study. Thank you for your tireless efforts.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Magaji Yunbunga Adamu

Peter Ogenyi

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