# A New Approach to Approximate Solutions for Nonlinear Differential Equation

Authors: Safia Meftah
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.


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## Body

## 1. Introduction

In the last few years, the study of approximations methods for systems of differential equations has been extensively developed; see, for example, [1]. This technique, known as the perturbation method (see [2]), has many applications in the theory of fractional differentiation operators (see [3]), in reaction-diffusion equations, stochastic stability, and asymptotic stability (see [4–9]), and for some numerical considerations (see, for example, [10–12]).In current applications, some considerations require only the use of a small number of terms in the perturbation expansion, but the simple application of the perturbation is problematic if we want to calculate a uniformly valid solution.Therefore, to structure a uniformly valid solution, one must look for an approximation that eliminates the terms causing the problem (secular terms). A technique to avoid the presence of these terms has been developed by Lindstedt. The principle of the Lindstedt method is to find approximations for periodic solutions, by convergent series using the expansion theorem and the periodicity of the solution [13, 14]. This method has various applications and properties; see, for example, [15]. Later, Poincaré proved that the expansion obtained by the Lindstedt technique is both asymptotic and uniformly valid.The aim of this work is to present an analytical approximation study of periodic solutions for systems of second-order nonlinear differential equations. Although our analysis is based on the Lindstedt method, nevertheless the chosen development is according to a different approach from the one usually used. Thus, we recover an improvement in the process of the approximation.Our paper consists of three sections. In the first section we present the general framework of our study. In the second, we recall most of the preliminary notions and the necessary definitions, and we prove the third approximation in the general case. Finally, in Section3 we define and study the approximations of a new nonclassical equation.


## 2. Preliminaries and Definitions

In this section, we present an approximation method, based on the expansion of a solution of a differential equation in a series in a small parameter. It is used to construct uniformly valid periodic solutions to second-order nonlinear differential equations in the form(1)d2ydt2t,ϵ+yt,ϵ=ϵFyt,ϵ,dydtt,ϵ,0<ϵ<<1,with y0,ϵ=A,dy/dt0,ϵ0=0, where 0<ϵ<<1 means that the positive parameter ϵ is small enough to be close to zero and F is supposed to be an analytical function of yt,ϵ and dy/dtt,ϵ.Ifϵ=0 we obtain the following nonperturbed problem:(2)d2ydt2t,0+yt,0=0.Before we discuss our subject, we present some basic concepts concerning the perturbation theory. Then we introduce the Lindstedt method, which we use to determine uniformly valid solutions, in order to find a closer approximate solution for (2) (y∗∗ is closer to y than y∗ means that y-y∗∗<y-y∗). For further developments concerning the Lindstedt method see [16, 17].


### 2.1. Approximation Technique

We assume that then+1th approximate solution of (1) can be written as(3)yt,ϵ=∑m=0nϵmymt,0+Oϵn+1.The general procedure of the simple approximation is to substitute (3) into (1), develop in powers of ϵ, and put all coefficients of the powers of ϵ equal to zero. This gives a system of linear nonhomogeneous differential equations that we can solve recursively.But the simple approximation takes us on a problem, if we need to calculate an analytical approximations of periodic solutions of nonlinear differential equations in the form given by (1). We illustrate this type of difficulty in the following example.


#### 2.1.1. Example

We apply the simple approximation to the following equation:(4)d2ydt2t,ϵ+ϵdydtt,ϵ2+yt,ϵ=0,0<ϵ<<1,with the initial values y0,ϵ=A,dy/dt0,ϵ=0.The fourth approximate solution of (4) is yt,ϵ=y0t,0+ϵy1t,0+ϵ2y2t,0+ϵ3y3t,0+Oϵ4. After substituting and calculating, we find(5)yt,ϵ=Acos⁡t+ϵA26-3+4cos⁡t-cos⁡2t+ϵ2A372-48+61cos⁡t-16cos⁡2t+12tsin⁡t-3cos⁡3t+ϵ3A4-2324+659540cos⁡t-13cos⁡2t+112cos⁡3t-131080cos⁡4t+13tsin⁡t-118tsin⁡2t+Oϵ4.We remark that the termsy2t,0 and y3t,0 are nonperiodic and unbounded as t→+∞. This leads to the notion of secular terms.


### 2.2. Secular Terms

The conservation of a finite numbers of terms on the right-side of expansion (5) determines a function that is not only nonperiodic, but also unbounded as t→+∞.Definition 1.
Terms such astmcos⁡pt or tmsin⁡nt where m,n∈N∗,p∈N are called secular terms.These expressions appear because expansion (5) is not uniformly valid. The existence of such expressions destroys the periodicity of expansion (5) when only a finite number of terms is conserved. Therefore, to obtain a uniformly valid solution, we must look for an approximation that eliminates secular terms. A technique to avoid the presence of secular terms and allows for an approximation that is valid for all time has been developed by Lindstedt-Poincaré as described above in what follows.


### 2.3. Lindstedt-Poincaré Method

The substance of this method is to introduce a new independent variable linearly linked to the old independent variable. This transformation completely eliminates the secular terms. The basic idea came from the astronomer Lindstedt, based on the change of variableθ=ωϵt with ω0=w(0)=1,ωϵ≠1, and both yθ,ϵ and ωϵ are expanded in powers of ϵ as follows:(6)yθ,ϵ=y0θ,0+ϵy1θ,0+…+ϵnynθ,0+…,ωϵ=1+ϵω1+…+ϵnωn+…,and we note that, in this step, ωj are unknowns; we obtain them by elimination of the secular terms.First, we introduce the following notations:(7)y˙≡dydθθ,ϵ,y¨≡d2ydθ2θ,ϵ,Fyy,ωy˙≡∂Fyθ,ϵ,y˙∂yθ,ϵ,Fy˙y,y˙≡∂Fyθ,ϵ,y˙∂y˙,and (1) becomes(8)ω2y¨+y=ϵFy,ωy˙,0<ϵ<<1,with y0,ϵ=A,y˙0,ϵ=0. When we substitute expansion (6) into (8) we have (9)1+ϵω1+ϵ2ω2+ϵ3ω3+…2y0¨+ϵy1¨+ϵ2y2¨+ϵ3y3¨+…+y0+ϵy1+ϵ2y2+ϵ3y3+…=ϵFy0,y0˙+ϵ2∂Fy0,y0˙∂ϵ+ϵ32∂2Fy0,y0˙∂ϵ2+….Then we put all different powers of ϵ to zero, and we obtain (10), (11), (12), (13), and (15), such that(10)y0¨+y0=0,(11)y1¨+y1=-2ω1y0¨+Fy0,y0˙≕G1y0θ,0,y0˙θ,0=G1θ,(12)y2¨+y2=-2ω1y1¨-ω12+2ω2y0¨+Fyy0,y0˙y1+Fy˙y0,y0˙ω1y0˙+y1˙≕G2y0θ,0,y1θ,0,y0˙θ,0,y1˙θ,0=G2θ,(13)y3¨+y3≕G3y0θ,0,y1θ,0,y2θ,0;y0˙θ,0,y1˙θ,0,y2˙θ,0=G3θ,(14)⋯⋯⋯(15)yn¨+yn=Gny0θ,0,y1θ,0,…,yn-1θ,0;y0˙θ,0,y1˙θ,0,…,y˙n-1θ,0=Gnθ;note here that Gi,i=1,..,n, is also an analytical function of y0,y1,…,yi-1;y0˙,y1˙,…,y˙i-1.To calculate an approximate periodic solutions of (8), we must solve (11), (12), (13), and (15). The following proposition gives the general formula of periodic solutions. Although these results are in [17], they are not detailed.Proposition 2.
We consider the following equation:(16)y¨+y=Gθ,y0=0,y˙0=0,withGθ≠0,and the solution of problem (16) is(17)yθ=∫0θsin⁡θ-τGτdτ.Moreover, problem (16) has a periodic solution y1θ,0 if and only if(18)∫02πFAcos⁡θ,-Asin⁡θsin⁡θdθ=0,2πω1A+∫02πFAcos⁡θ,-Asin⁡θcos⁡θdθ=0.Proof.
We know that the solution of (16) is yθ=C1cos⁡θ+C2sin⁡θ+ypθ such that ypθ=C1θcos⁡θ+C2θsin⁡θ. By variation of constants we find (19)C1′θcos⁡θ+C2′θsin⁡θ=0,-C1′θsin⁡θ+C2′θcos⁡θ=Gθ⇓-C1′θ=-sin⁡θGθ⟹C1θ=-∫0θsin⁡τGτdτ,C10=0,-C2′θ=cos⁡θGθ⟹C1θ=∫0θcos⁡τGτdτ,C20=0⟹ypθ=-∫0θsin⁡τGτdτcos⁡θ+∫0θ‍cos⁡τGτdτsin⁡θ=∫0θ‍(-sin⁡τcos⁡θ+cos⁡τsin⁡θ)Gτdτ.
⟹ y p θ = ∫ 0 θ sin ⁡ θ - τ G τ d τ ⟹ y θ = C 1 cos ⁡ θ + C 2 sin ⁡ θ + ∫ 0 θ sin ⁡ θ - τ G τ d τ with the initial values y0=0,y˙0=0; we have C1=C2=0, so we deduce that problem (16) admits (17) as a solution.
Moreover, (16) gives (20)y˙1=y2y˙2=-y1+Gτ.
On the other hand, the condition of periodicity for the new variableθ can be expressed as yθ=yθ+2π, so the corresponding conditions for ynθ are ynθ=ynθ+2π,n=1,2,…(21)⟹y12π=y10=0y22π=y20=0 which yields to the periodicity condition ∫θθ+2π‍sin⁡θ-τGτdτ=0,(22)⟹∫02πcos⁡θGθdθ=0,∫02πsin⁡θGθdθ=0.According to (11) we have Gθ=-2ω1y0¨+Fy0,y0˙,y0=Acos⁡θ; we rewrite (22) as (23)⟹∫02πcos⁡θ2ω1Acos⁡θ+FAcos⁡θ,-Asin⁡θdθ=0,∫02πsin⁡θ2ω1Acos⁡θ+FAcos⁡θ,-Asin⁡θdθ=0,⟹2ω1πA+∫02πcos⁡θFAcos⁡θ,-Asin⁡θdθ=0,∫02πsin⁡θFAcos⁡θ,-Asin⁡θdθ=0, which is required.


#### 2.3.1. Example

We apply the method of Lindstedt to (4) with the initial values y0,ϵ=A,y˙0,ϵ=0, and we calculate yi(θ,0),i=1,2,3 according to Proposition 2. Thus, we find that the fourth approximation of the periodic solution of (4) is(24)yθ,ϵ=Acos⁡θ+ϵA26-3+4cos⁡θ-cos⁡2θ+ϵ2A33-2+6124cos⁡θ-23cos⁡2θ-18cos⁡3θ+ϵ3-2324A4+659540A4cos⁡θ-13A4cos⁡2θ+A412cos⁡3θ-131080A4cos⁡4θ+Oϵ4,with θ=ωϵt such that ωϵ=1-ϵ2A2/6-ϵ32/9A3+Oϵ4.Remark 3.
Although the calculation ofy3θ is very long, usually in applications, the fourth approximation is among the high orders that are often useful. For this reason, we give its equation in the next proposition.Remark 4.
The Lindstedt method gives only periodic solutions.


## 2.1. Approximation Technique

We assume that then+1th approximate solution of (1) can be written as(3)yt,ϵ=∑m=0nϵmymt,0+Oϵn+1.The general procedure of the simple approximation is to substitute (3) into (1), develop in powers of ϵ, and put all coefficients of the powers of ϵ equal to zero. This gives a system of linear nonhomogeneous differential equations that we can solve recursively.But the simple approximation takes us on a problem, if we need to calculate an analytical approximations of periodic solutions of nonlinear differential equations in the form given by (1). We illustrate this type of difficulty in the following example.


### 2.1.1. Example

We apply the simple approximation to the following equation:(4)d2ydt2t,ϵ+ϵdydtt,ϵ2+yt,ϵ=0,0<ϵ<<1,with the initial values y0,ϵ=A,dy/dt0,ϵ=0.The fourth approximate solution of (4) is yt,ϵ=y0t,0+ϵy1t,0+ϵ2y2t,0+ϵ3y3t,0+Oϵ4. After substituting and calculating, we find(5)yt,ϵ=Acos⁡t+ϵA26-3+4cos⁡t-cos⁡2t+ϵ2A372-48+61cos⁡t-16cos⁡2t+12tsin⁡t-3cos⁡3t+ϵ3A4-2324+659540cos⁡t-13cos⁡2t+112cos⁡3t-131080cos⁡4t+13tsin⁡t-118tsin⁡2t+Oϵ4.We remark that the termsy2t,0 and y3t,0 are nonperiodic and unbounded as t→+∞. This leads to the notion of secular terms.


## 2.1.1. Example

We apply the simple approximation to the following equation:(4)d2ydt2t,ϵ+ϵdydtt,ϵ2+yt,ϵ=0,0<ϵ<<1,with the initial values y0,ϵ=A,dy/dt0,ϵ=0.The fourth approximate solution of (4) is yt,ϵ=y0t,0+ϵy1t,0+ϵ2y2t,0+ϵ3y3t,0+Oϵ4. After substituting and calculating, we find(5)yt,ϵ=Acos⁡t+ϵA26-3+4cos⁡t-cos⁡2t+ϵ2A372-48+61cos⁡t-16cos⁡2t+12tsin⁡t-3cos⁡3t+ϵ3A4-2324+659540cos⁡t-13cos⁡2t+112cos⁡3t-131080cos⁡4t+13tsin⁡t-118tsin⁡2t+Oϵ4.We remark that the termsy2t,0 and y3t,0 are nonperiodic and unbounded as t→+∞. This leads to the notion of secular terms.


## 2.2. Secular Terms

The conservation of a finite numbers of terms on the right-side of expansion (5) determines a function that is not only nonperiodic, but also unbounded as t→+∞.Definition 1.
Terms such astmcos⁡pt or tmsin⁡nt where m,n∈N∗,p∈N are called secular terms.These expressions appear because expansion (5) is not uniformly valid. The existence of such expressions destroys the periodicity of expansion (5) when only a finite number of terms is conserved. Therefore, to obtain a uniformly valid solution, we must look for an approximation that eliminates secular terms. A technique to avoid the presence of secular terms and allows for an approximation that is valid for all time has been developed by Lindstedt-Poincaré as described above in what follows.


## 2.3. Lindstedt-Poincaré Method

The substance of this method is to introduce a new independent variable linearly linked to the old independent variable. This transformation completely eliminates the secular terms. The basic idea came from the astronomer Lindstedt, based on the change of variableθ=ωϵt with ω0=w(0)=1,ωϵ≠1, and both yθ,ϵ and ωϵ are expanded in powers of ϵ as follows:(6)yθ,ϵ=y0θ,0+ϵy1θ,0+…+ϵnynθ,0+…,ωϵ=1+ϵω1+…+ϵnωn+…,and we note that, in this step, ωj are unknowns; we obtain them by elimination of the secular terms.First, we introduce the following notations:(7)y˙≡dydθθ,ϵ,y¨≡d2ydθ2θ,ϵ,Fyy,ωy˙≡∂Fyθ,ϵ,y˙∂yθ,ϵ,Fy˙y,y˙≡∂Fyθ,ϵ,y˙∂y˙,and (1) becomes(8)ω2y¨+y=ϵFy,ωy˙,0<ϵ<<1,with y0,ϵ=A,y˙0,ϵ=0. When we substitute expansion (6) into (8) we have (9)1+ϵω1+ϵ2ω2+ϵ3ω3+…2y0¨+ϵy1¨+ϵ2y2¨+ϵ3y3¨+…+y0+ϵy1+ϵ2y2+ϵ3y3+…=ϵFy0,y0˙+ϵ2∂Fy0,y0˙∂ϵ+ϵ32∂2Fy0,y0˙∂ϵ2+….Then we put all different powers of ϵ to zero, and we obtain (10), (11), (12), (13), and (15), such that(10)y0¨+y0=0,(11)y1¨+y1=-2ω1y0¨+Fy0,y0˙≕G1y0θ,0,y0˙θ,0=G1θ,(12)y2¨+y2=-2ω1y1¨-ω12+2ω2y0¨+Fyy0,y0˙y1+Fy˙y0,y0˙ω1y0˙+y1˙≕G2y0θ,0,y1θ,0,y0˙θ,0,y1˙θ,0=G2θ,(13)y3¨+y3≕G3y0θ,0,y1θ,0,y2θ,0;y0˙θ,0,y1˙θ,0,y2˙θ,0=G3θ,(14)⋯⋯⋯(15)yn¨+yn=Gny0θ,0,y1θ,0,…,yn-1θ,0;y0˙θ,0,y1˙θ,0,…,y˙n-1θ,0=Gnθ;note here that Gi,i=1,..,n, is also an analytical function of y0,y1,…,yi-1;y0˙,y1˙,…,y˙i-1.To calculate an approximate periodic solutions of (8), we must solve (11), (12), (13), and (15). The following proposition gives the general formula of periodic solutions. Although these results are in [17], they are not detailed.Proposition 2.
We consider the following equation:(16)y¨+y=Gθ,y0=0,y˙0=0,withGθ≠0,and the solution of problem (16) is(17)yθ=∫0θsin⁡θ-τGτdτ.Moreover, problem (16) has a periodic solution y1θ,0 if and only if(18)∫02πFAcos⁡θ,-Asin⁡θsin⁡θdθ=0,2πω1A+∫02πFAcos⁡θ,-Asin⁡θcos⁡θdθ=0.Proof.
We know that the solution of (16) is yθ=C1cos⁡θ+C2sin⁡θ+ypθ such that ypθ=C1θcos⁡θ+C2θsin⁡θ. By variation of constants we find (19)C1′θcos⁡θ+C2′θsin⁡θ=0,-C1′θsin⁡θ+C2′θcos⁡θ=Gθ⇓-C1′θ=-sin⁡θGθ⟹C1θ=-∫0θsin⁡τGτdτ,C10=0,-C2′θ=cos⁡θGθ⟹C1θ=∫0θcos⁡τGτdτ,C20=0⟹ypθ=-∫0θsin⁡τGτdτcos⁡θ+∫0θ‍cos⁡τGτdτsin⁡θ=∫0θ‍(-sin⁡τcos⁡θ+cos⁡τsin⁡θ)Gτdτ.
⟹ y p θ = ∫ 0 θ sin ⁡ θ - τ G τ d τ ⟹ y θ = C 1 cos ⁡ θ + C 2 sin ⁡ θ + ∫ 0 θ sin ⁡ θ - τ G τ d τ with the initial values y0=0,y˙0=0; we have C1=C2=0, so we deduce that problem (16) admits (17) as a solution.
Moreover, (16) gives (20)y˙1=y2y˙2=-y1+Gτ.
On the other hand, the condition of periodicity for the new variableθ can be expressed as yθ=yθ+2π, so the corresponding conditions for ynθ are ynθ=ynθ+2π,n=1,2,…(21)⟹y12π=y10=0y22π=y20=0 which yields to the periodicity condition ∫θθ+2π‍sin⁡θ-τGτdτ=0,(22)⟹∫02πcos⁡θGθdθ=0,∫02πsin⁡θGθdθ=0.According to (11) we have Gθ=-2ω1y0¨+Fy0,y0˙,y0=Acos⁡θ; we rewrite (22) as (23)⟹∫02πcos⁡θ2ω1Acos⁡θ+FAcos⁡θ,-Asin⁡θdθ=0,∫02πsin⁡θ2ω1Acos⁡θ+FAcos⁡θ,-Asin⁡θdθ=0,⟹2ω1πA+∫02πcos⁡θFAcos⁡θ,-Asin⁡θdθ=0,∫02πsin⁡θFAcos⁡θ,-Asin⁡θdθ=0, which is required.


### 2.3.1. Example

We apply the method of Lindstedt to (4) with the initial values y0,ϵ=A,y˙0,ϵ=0, and we calculate yi(θ,0),i=1,2,3 according to Proposition 2. Thus, we find that the fourth approximation of the periodic solution of (4) is(24)yθ,ϵ=Acos⁡θ+ϵA26-3+4cos⁡θ-cos⁡2θ+ϵ2A33-2+6124cos⁡θ-23cos⁡2θ-18cos⁡3θ+ϵ3-2324A4+659540A4cos⁡θ-13A4cos⁡2θ+A412cos⁡3θ-131080A4cos⁡4θ+Oϵ4,with θ=ωϵt such that ωϵ=1-ϵ2A2/6-ϵ32/9A3+Oϵ4.Remark 3.
Although the calculation ofy3θ is very long, usually in applications, the fourth approximation is among the high orders that are often useful. For this reason, we give its equation in the next proposition.Remark 4.
The Lindstedt method gives only periodic solutions.


## 2.3.1. Example

We apply the method of Lindstedt to (4) with the initial values y0,ϵ=A,y˙0,ϵ=0, and we calculate yi(θ,0),i=1,2,3 according to Proposition 2. Thus, we find that the fourth approximation of the periodic solution of (4) is(24)yθ,ϵ=Acos⁡θ+ϵA26-3+4cos⁡θ-cos⁡2θ+ϵ2A33-2+6124cos⁡θ-23cos⁡2θ-18cos⁡3θ+ϵ3-2324A4+659540A4cos⁡θ-13A4cos⁡2θ+A412cos⁡3θ-131080A4cos⁡4θ+Oϵ4,with θ=ωϵt such that ωϵ=1-ϵ2A2/6-ϵ32/9A3+Oϵ4.Remark 3.
Although the calculation ofy3θ is very long, usually in applications, the fourth approximation is among the high orders that are often useful. For this reason, we give its equation in the next proposition.Remark 4.
The Lindstedt method gives only periodic solutions.


## 3. Our Results

### 3.1. General Formula

Practically, for many considerations we are forced to use a small number of terms in the perturbation expansion. We note here that the second and third terms are determined by (11) and (12) in [17]. In the following proposition, (13) which determines the fourth term is explicitly stated.Proposition 5.
The general formula of (13) is (25)y3¨+y3=G3θ=-2ω1y2¨-ω12+2ω2y1¨-2ω3+2ω1ω2y0¨+y2Fyy0,y0˙+y122Fyyy0,y0˙++y1ω1y0˙+y1˙Fyy˙y0,y0˙+ω2y0˙+ω1y1˙+y2˙Fy˙y0,y0˙+12ω1y0˙+y1˙2Fy˙y˙y0,y0˙.Proof.
First, (9) gives (26)ϵ3y3¨+y3+2ω1y2¨+ω12+2ω2y1¨+2ω3+2ω1ω2y0¨=ϵ32∂2Fy0,y0˙∂ϵ2⟹y3¨+y3=-2ω1y2¨-ω12+2ω2y1¨-2ω3+2ω1ω2y0¨+12∂2Fy0,y0˙∂ϵ2,such that (27)∂2Fy,ωy˙∂ϵ2=∂∂ϵ∂F∂y∂y∂ϵ+∂F∂y˙∂ωy˙∂ϵ=∂y∂ϵ∂∂ϵ∂F∂y+∂2y∂ϵ2∂F∂y+∂ωy˙∂ϵ∂∂ϵ∂F∂y˙+∂2ωy˙∂ϵ2∂F∂y˙=∂y∂ϵ∂2F∂y2∂y∂ϵ+∂2F∂y∂y˙∂ωy˙∂ϵ+∂2y∂ϵ2Fy+∂2ωy˙∂ϵ2Fy˙+∂2ωy˙∂ϵ∂2F∂y∂y˙∂y∂ϵ+∂2F∂y2∂ωy˙∂ϵ=∂y∂ϵ2Fyy+2∂y∂ϵ∂ωy˙∂ϵFyy˙+∂2y∂ϵFy+∂2ωy˙∂ϵ2Fy˙+∂ωy˙∂ϵ2Fy˙y˙, with (28)Fyy≡∂2F∂y2,Fyy˙≡∂2F∂y∂y˙,Fy˙y˙≡∂2F∂y˙∂y˙. On the other hand, in the third order we have (29)y=y0+ϵy1+ϵ2y2+ϵ3y3⟹∂y∂ϵ=y1+2ϵy2+3ϵ2y3⟹∂2y∂ϵ2=2y2+6ϵy3, and (30)∂ωy˙∂ϵ=∂ω∂ϵy˙+∂y˙∂ϵω=ω1+2ϵω2+3ϵ2ω3y0˙+ϵy1˙+ϵ2y2˙+ϵ3y3˙+y1˙+2ϵy2˙+3ϵ2y3˙1+ϵω1+ϵ2ω2+ϵ3ω3, and also (31)∂2ωy˙∂ϵ2=2ω2+6ϵω3y0˙+ϵy1˙+ϵ2y2˙+ϵ3y3˙+2y1˙+2ϵy2˙+3ϵ2y3˙ω1+2ϵω2+3ϵ2ω3+2y2˙+6ϵy3˙1+ϵω1+ϵ2ω2+ϵ3ω3⟹12∂2Fy0,y0˙∂ϵ2=y122Fyyy0,y0˙+ω1y0˙+y1˙y1Fyy˙y0,y0˙+y2Fyy0,y0˙+ω2y0˙+ω1y1˙+y2˙Fy˙y0,y0˙+ω1y0˙+y1˙2Fy˙y˙y0,y0˙,and when we substitute (31) into (26), we get (25).


### 3.2. Main Result

In this essential part of our work, we deal with some nonclassical equations, more general than (1), and also different from the equation studied in [2]. We consider equations in the following form:(32)d2y~dt2t,ϵ+y~t,ϵ=gϵFy~t,ϵ,dy~dtt,ϵ,0<ϵ<<1,with y~0,ϵ=A,dy~/dt0,ϵ=0, where ϵ is a small positive parameter and F is supposed to be an analytical function of y~t,ϵ and dy~/dtt,ϵ.To compute an uniformly approximate periodic solution, a new variableθ~=ω~t is introduced, and both y~ and ω~ are expanded in powers of ϵ as follows:(33)y~θ~,ϵ=y~0θ~,0+ϵy~1θ~,0+…+ϵny~nθ~,0+…,with(34)ω~ϵ=1+ϵω~1+…+ϵnω~n+….We note that, in this step, ω~j are unknowns, and we obtain them by elimination of the secular terms.To use the uniformly approximate periodic solution (33), we give firstly the general formula of y~0θ~,0,y~1θ~,0, and y~2θ~,0 in the following proposition.Proposition 6.
The termsy~0θ~,0,y~1θ~,0andy~2θ~,0 are, respectively, solutions of (35), (36), and (37) such that(35)y~¨0+y~0=0,(36)y~¨1+y~1=G~1θ~,(37)y~¨2+y~2=G~2θ~,with G~1θ~=-2ω~1y~¨0+c1Fy~0θ~,0,y~˙0θ~,0,(38)G~2θ~=-2ω~1y~¨0+c1Fy~0θ~,0,y~˙0θ~,0-2ω~1y~¨1-ω~12+2ω~2y~¨0++c1Fy~y~0θ~,0,y~˙0θ~,0y~1+Fy~˙y~0θ~,0,y~˙0θ~,0ω1~y~˙0+y~˙1+c2Fy~0θ~,0,y~˙0θ~,0.Proof.
Equation (32) will be written as (39)d2y~θ~,ϵdt2+y~θ~=Σk≥1ϵkckFy~θ~,ϵ,dy~θ~,ϵdt=ϵΣk≥0ϵkck+1Fy~θ~,ϵ,dy~θ~,ϵdt=ϵKy~θ~,ϵ,dy~θ~,ϵdt, where K is an analytical function of y and dy/dt.
Substituting (33) into (32), we have (40)1+ϵω~1+ϵ2ω~2+ϵ3ω~3+…2y~¨0+ϵy~¨1+ϵ2y~¨2+ϵ3y~¨3+…+y~0+ϵy~1+ϵ2y~2+ϵ3y~3+…=ϵKy~0,y~˙0+ϵ2∂Ky~0,y~˙0∂ϵ+ϵ32∂2Ky~0,y~˙0∂ϵ2+…,=ϵc1Fy~0,y~˙0+ϵ2c1∂Fy~0,y~˙0∂ϵ+c2Fy~0,y~˙0+…,=ϵc1Fy~0,y~˙0+ϵ2c1Fy~y~0,y~˙0y~1+Fy~˙y~0,y~˙0ω1~y~˙0+y~˙1+c2Fy~0,y~˙0+⋯, then we put all different powers of ϵ to zero, and we obtain (35), (36), (37), and so on.The aim of this study is to construct a new approach to (2), which gives a closer approximate solution of (2) more than an approximate solution of (1). The relations between an approximate solution of (32) and that of (1) are determined by the following lemma.Lemma 7.
If the functiongϵ is expanded in powers of ϵ with g0=0i.egϵ=∑k≥1ϵkck, where ck are real constants, we have
( 1 )  y ~ 0 θ ~ , 0 = y 0 θ ~ , 0.
( 2 )  y ~ 1 θ ~ , 0 = c 1 y 1 θ ~ , 0.
( 3 )  y ~ 2 θ ~ , 0 = c 1 2 y 2 θ ~ , 0 + c 2 y 1 θ ~ , 0.Proof.
( 1 ) Equation (35) gives y~0θ~,0=Acos⁡θ~=Acos⁡ω~0t=Acos⁡t=Acos⁡ω0t=Acos⁡θ=y0θ,0.
( 2 ) When we apply the periodicity condition (22) to (36), we have (41)∫02πcos⁡θ~G~1θ~dθ~=0,∫02πsin⁡θ~G~1θdθ~=0.⇓∫02πcos⁡θ~-2ω~1y~¨0+c1Fy~0,y~˙0dθ~=0,∫02πsin⁡θ~-2ω~1y~¨0+c1Fy~0,y~˙0dθ~=0.⇓ω~1=-c12πA∫02πcos⁡θ~Fy~0,y~˙0dθ~=c1ω1,∫02πsin⁡θ~Fy~0,y~˙0dθ~=0. On the other hand, according to (17) the solution y~1θ~,0 of (36) is given by (42)y~1θ~,0=∫0θ~sin⁡θ~-τ~G~1τ~dτ~=c1∫0θ~sin⁡θ~-τ~G1τ~dτ~=c1y1θ~,0.(3) When we apply the periodicity condition (22) to (12), we have(43)∫02πcos⁡θ~G~2θ~dθ~=0,∫02πsin⁡θ~G~2θdθ~=0.⇓∫02πcos⁡θ~c12G2θ~+c2Fy~0,y~˙0dθ~=0,∫02πsin⁡θ~c12G2θ~+c2Fy~0,y~˙0dθ~=0.⇓ω~2=c12ω2+c2ω1,∫02πsin⁡θ~Fy~0,y~˙0dθ~=0.
On the other hand, according to (17) the solution y~2θ~,0 of (11) is given by (44)y~2θ~,0=∫0θ~sin⁡θ~-τ~G~2τ~dτ~=c12∫0θ~sin⁡θ~-τ~G2τ~dτ~+c2∫0θ~sin⁡θ~-τ~-2ω~1y~¨0+c1Fy~0,y~˙0dτ~⟹y~2θ~,0=c12y2θ~,0+c2y1θ~,0.Theorem 8.
If the functiongϵ is expanded in powers of ϵ with g0=0i.egϵ=∑k≥1ϵkck, where ck are real constants, so one has the following:
(1) Ifc1≠0,withc1<1, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
(2) Ifc1=0 and c2<1,withc2≠0, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
Moreover, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1) where c1=0,c2≠0.Proof.
According to the results given by Lemma7, we have the following:
( 1 ) If c1<1 and c1≠0 we have (45)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵy~1θ~,0-oϵ=ϵy~1θ~,0+oϵ=c1ϵy1θ~,0+oϵ=c1ϵy1θ~,0+oϵ<ϵy1θ~,0+oϵ=y0θ,0-yθ~,ϵ.So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
( 2 ) If c1=0andc2<1withc2≠0, we have(46)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵ2y~2θ~,0-oϵ2=ϵ2y~2θ~,0+oϵ2=ϵ2c2y1θ~,0+oϵ2=c2ϵ2y1θ~,0+oϵ2=c2ϵ2y0θ,0-yθ~,ϵ+ϵ2y2θ~,0<ϵ2y0θ,0-yθ~,ϵ+ϵ2y2θ~,0<ϵ2y0θ,0-yθ~,ϵ+ϵ4y2θ~,0.Since y0θ,0-yθ~,ϵ=o(ϵ), for ϵ small enough, there exists a positive real constant C such that y0θ,0-yθ~,ϵ≤Cϵ.
Leth be a function defined by hϵ:=ϵ2-1C+ϵ3y2θ~,0. So, h is continuous with h0=-C<0, then ∃ϵ0>0suchthat∀ϵ∈0,ϵ0, we have hϵ<0.
Therefore, for allϵ∈0,ϵ0 we get(47)ϵ2-1y0θ,0-yθ~,ϵ+ϵ4y2θ~,0≤ϵϵ2-1C+ϵ3y2θ~,0=ϵhϵ<0which implies from what precedes that y0θ,0-y~θ~,ϵ<y0θ,0-yθ~,ϵ. So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
Moreover, ifc1=0andc2≠0, then ∃ϵ0<1/c2>0 such that, ∀ϵ∈0,ϵ0,(48)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵ2y~2θ~,0-oϵ2=ϵ2y~2θ~,0+oϵ2=ϵ2c2y1θ~,0+oϵ2=ϵc2ϵy1θ~,0+oϵ=ϵc2y0θ,0-yθ~,ϵ<y0θ,0-yθ~,ϵ.So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.Remark 9.
Letg be a real function such that gϵ=ϵmhϵ,m∈N∗, where hϵ can be expanded in powers of ϵashϵ=∑k≥1ϵkak, with ak real constants.
( 1 ) If m=1 or 2, according to the conditions of Theorem 8, we conclude that y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
( 2 ) If m>2 (in the case where m=3, the fourth term is given by Proposition 5), we can expect that y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.Remark 10.
We note here that, in the fractional case, the existence of a positive solution of (32) is studied in [18].Remark 11.
Although the Lindstedt-Poincaré method gives uniformly valid asymptotic expansions for periodic solutions of weakly nonlinear oscillations, i.e.,0<ϵ<C1, the technique does not work if the amplitude of the oscillation is a function of time (see [16, 17]).


## 3.1. General Formula

Practically, for many considerations we are forced to use a small number of terms in the perturbation expansion. We note here that the second and third terms are determined by (11) and (12) in [17]. In the following proposition, (13) which determines the fourth term is explicitly stated.Proposition 5.
The general formula of (13) is (25)y3¨+y3=G3θ=-2ω1y2¨-ω12+2ω2y1¨-2ω3+2ω1ω2y0¨+y2Fyy0,y0˙+y122Fyyy0,y0˙++y1ω1y0˙+y1˙Fyy˙y0,y0˙+ω2y0˙+ω1y1˙+y2˙Fy˙y0,y0˙+12ω1y0˙+y1˙2Fy˙y˙y0,y0˙.Proof.
First, (9) gives (26)ϵ3y3¨+y3+2ω1y2¨+ω12+2ω2y1¨+2ω3+2ω1ω2y0¨=ϵ32∂2Fy0,y0˙∂ϵ2⟹y3¨+y3=-2ω1y2¨-ω12+2ω2y1¨-2ω3+2ω1ω2y0¨+12∂2Fy0,y0˙∂ϵ2,such that (27)∂2Fy,ωy˙∂ϵ2=∂∂ϵ∂F∂y∂y∂ϵ+∂F∂y˙∂ωy˙∂ϵ=∂y∂ϵ∂∂ϵ∂F∂y+∂2y∂ϵ2∂F∂y+∂ωy˙∂ϵ∂∂ϵ∂F∂y˙+∂2ωy˙∂ϵ2∂F∂y˙=∂y∂ϵ∂2F∂y2∂y∂ϵ+∂2F∂y∂y˙∂ωy˙∂ϵ+∂2y∂ϵ2Fy+∂2ωy˙∂ϵ2Fy˙+∂2ωy˙∂ϵ∂2F∂y∂y˙∂y∂ϵ+∂2F∂y2∂ωy˙∂ϵ=∂y∂ϵ2Fyy+2∂y∂ϵ∂ωy˙∂ϵFyy˙+∂2y∂ϵFy+∂2ωy˙∂ϵ2Fy˙+∂ωy˙∂ϵ2Fy˙y˙, with (28)Fyy≡∂2F∂y2,Fyy˙≡∂2F∂y∂y˙,Fy˙y˙≡∂2F∂y˙∂y˙. On the other hand, in the third order we have (29)y=y0+ϵy1+ϵ2y2+ϵ3y3⟹∂y∂ϵ=y1+2ϵy2+3ϵ2y3⟹∂2y∂ϵ2=2y2+6ϵy3, and (30)∂ωy˙∂ϵ=∂ω∂ϵy˙+∂y˙∂ϵω=ω1+2ϵω2+3ϵ2ω3y0˙+ϵy1˙+ϵ2y2˙+ϵ3y3˙+y1˙+2ϵy2˙+3ϵ2y3˙1+ϵω1+ϵ2ω2+ϵ3ω3, and also (31)∂2ωy˙∂ϵ2=2ω2+6ϵω3y0˙+ϵy1˙+ϵ2y2˙+ϵ3y3˙+2y1˙+2ϵy2˙+3ϵ2y3˙ω1+2ϵω2+3ϵ2ω3+2y2˙+6ϵy3˙1+ϵω1+ϵ2ω2+ϵ3ω3⟹12∂2Fy0,y0˙∂ϵ2=y122Fyyy0,y0˙+ω1y0˙+y1˙y1Fyy˙y0,y0˙+y2Fyy0,y0˙+ω2y0˙+ω1y1˙+y2˙Fy˙y0,y0˙+ω1y0˙+y1˙2Fy˙y˙y0,y0˙,and when we substitute (31) into (26), we get (25).


## 3.2. Main Result

In this essential part of our work, we deal with some nonclassical equations, more general than (1), and also different from the equation studied in [2]. We consider equations in the following form:(32)d2y~dt2t,ϵ+y~t,ϵ=gϵFy~t,ϵ,dy~dtt,ϵ,0<ϵ<<1,with y~0,ϵ=A,dy~/dt0,ϵ=0, where ϵ is a small positive parameter and F is supposed to be an analytical function of y~t,ϵ and dy~/dtt,ϵ.To compute an uniformly approximate periodic solution, a new variableθ~=ω~t is introduced, and both y~ and ω~ are expanded in powers of ϵ as follows:(33)y~θ~,ϵ=y~0θ~,0+ϵy~1θ~,0+…+ϵny~nθ~,0+…,with(34)ω~ϵ=1+ϵω~1+…+ϵnω~n+….We note that, in this step, ω~j are unknowns, and we obtain them by elimination of the secular terms.To use the uniformly approximate periodic solution (33), we give firstly the general formula of y~0θ~,0,y~1θ~,0, and y~2θ~,0 in the following proposition.Proposition 6.
The termsy~0θ~,0,y~1θ~,0andy~2θ~,0 are, respectively, solutions of (35), (36), and (37) such that(35)y~¨0+y~0=0,(36)y~¨1+y~1=G~1θ~,(37)y~¨2+y~2=G~2θ~,with G~1θ~=-2ω~1y~¨0+c1Fy~0θ~,0,y~˙0θ~,0,(38)G~2θ~=-2ω~1y~¨0+c1Fy~0θ~,0,y~˙0θ~,0-2ω~1y~¨1-ω~12+2ω~2y~¨0++c1Fy~y~0θ~,0,y~˙0θ~,0y~1+Fy~˙y~0θ~,0,y~˙0θ~,0ω1~y~˙0+y~˙1+c2Fy~0θ~,0,y~˙0θ~,0.Proof.
Equation (32) will be written as (39)d2y~θ~,ϵdt2+y~θ~=Σk≥1ϵkckFy~θ~,ϵ,dy~θ~,ϵdt=ϵΣk≥0ϵkck+1Fy~θ~,ϵ,dy~θ~,ϵdt=ϵKy~θ~,ϵ,dy~θ~,ϵdt, where K is an analytical function of y and dy/dt.
Substituting (33) into (32), we have (40)1+ϵω~1+ϵ2ω~2+ϵ3ω~3+…2y~¨0+ϵy~¨1+ϵ2y~¨2+ϵ3y~¨3+…+y~0+ϵy~1+ϵ2y~2+ϵ3y~3+…=ϵKy~0,y~˙0+ϵ2∂Ky~0,y~˙0∂ϵ+ϵ32∂2Ky~0,y~˙0∂ϵ2+…,=ϵc1Fy~0,y~˙0+ϵ2c1∂Fy~0,y~˙0∂ϵ+c2Fy~0,y~˙0+…,=ϵc1Fy~0,y~˙0+ϵ2c1Fy~y~0,y~˙0y~1+Fy~˙y~0,y~˙0ω1~y~˙0+y~˙1+c2Fy~0,y~˙0+⋯, then we put all different powers of ϵ to zero, and we obtain (35), (36), (37), and so on.The aim of this study is to construct a new approach to (2), which gives a closer approximate solution of (2) more than an approximate solution of (1). The relations between an approximate solution of (32) and that of (1) are determined by the following lemma.Lemma 7.
If the functiongϵ is expanded in powers of ϵ with g0=0i.egϵ=∑k≥1ϵkck, where ck are real constants, we have
( 1 )  y ~ 0 θ ~ , 0 = y 0 θ ~ , 0.
( 2 )  y ~ 1 θ ~ , 0 = c 1 y 1 θ ~ , 0.
( 3 )  y ~ 2 θ ~ , 0 = c 1 2 y 2 θ ~ , 0 + c 2 y 1 θ ~ , 0.Proof.
( 1 ) Equation (35) gives y~0θ~,0=Acos⁡θ~=Acos⁡ω~0t=Acos⁡t=Acos⁡ω0t=Acos⁡θ=y0θ,0.
( 2 ) When we apply the periodicity condition (22) to (36), we have (41)∫02πcos⁡θ~G~1θ~dθ~=0,∫02πsin⁡θ~G~1θdθ~=0.⇓∫02πcos⁡θ~-2ω~1y~¨0+c1Fy~0,y~˙0dθ~=0,∫02πsin⁡θ~-2ω~1y~¨0+c1Fy~0,y~˙0dθ~=0.⇓ω~1=-c12πA∫02πcos⁡θ~Fy~0,y~˙0dθ~=c1ω1,∫02πsin⁡θ~Fy~0,y~˙0dθ~=0. On the other hand, according to (17) the solution y~1θ~,0 of (36) is given by (42)y~1θ~,0=∫0θ~sin⁡θ~-τ~G~1τ~dτ~=c1∫0θ~sin⁡θ~-τ~G1τ~dτ~=c1y1θ~,0.(3) When we apply the periodicity condition (22) to (12), we have(43)∫02πcos⁡θ~G~2θ~dθ~=0,∫02πsin⁡θ~G~2θdθ~=0.⇓∫02πcos⁡θ~c12G2θ~+c2Fy~0,y~˙0dθ~=0,∫02πsin⁡θ~c12G2θ~+c2Fy~0,y~˙0dθ~=0.⇓ω~2=c12ω2+c2ω1,∫02πsin⁡θ~Fy~0,y~˙0dθ~=0.
On the other hand, according to (17) the solution y~2θ~,0 of (11) is given by (44)y~2θ~,0=∫0θ~sin⁡θ~-τ~G~2τ~dτ~=c12∫0θ~sin⁡θ~-τ~G2τ~dτ~+c2∫0θ~sin⁡θ~-τ~-2ω~1y~¨0+c1Fy~0,y~˙0dτ~⟹y~2θ~,0=c12y2θ~,0+c2y1θ~,0.Theorem 8.
If the functiongϵ is expanded in powers of ϵ with g0=0i.egϵ=∑k≥1ϵkck, where ck are real constants, so one has the following:
(1) Ifc1≠0,withc1<1, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
(2) Ifc1=0 and c2<1,withc2≠0, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
Moreover, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1) where c1=0,c2≠0.Proof.
According to the results given by Lemma7, we have the following:
( 1 ) If c1<1 and c1≠0 we have (45)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵy~1θ~,0-oϵ=ϵy~1θ~,0+oϵ=c1ϵy1θ~,0+oϵ=c1ϵy1θ~,0+oϵ<ϵy1θ~,0+oϵ=y0θ,0-yθ~,ϵ.So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
( 2 ) If c1=0andc2<1withc2≠0, we have(46)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵ2y~2θ~,0-oϵ2=ϵ2y~2θ~,0+oϵ2=ϵ2c2y1θ~,0+oϵ2=c2ϵ2y1θ~,0+oϵ2=c2ϵ2y0θ,0-yθ~,ϵ+ϵ2y2θ~,0<ϵ2y0θ,0-yθ~,ϵ+ϵ2y2θ~,0<ϵ2y0θ,0-yθ~,ϵ+ϵ4y2θ~,0.Since y0θ,0-yθ~,ϵ=o(ϵ), for ϵ small enough, there exists a positive real constant C such that y0θ,0-yθ~,ϵ≤Cϵ.
Leth be a function defined by hϵ:=ϵ2-1C+ϵ3y2θ~,0. So, h is continuous with h0=-C<0, then ∃ϵ0>0suchthat∀ϵ∈0,ϵ0, we have hϵ<0.
Therefore, for allϵ∈0,ϵ0 we get(47)ϵ2-1y0θ,0-yθ~,ϵ+ϵ4y2θ~,0≤ϵϵ2-1C+ϵ3y2θ~,0=ϵhϵ<0which implies from what precedes that y0θ,0-y~θ~,ϵ<y0θ,0-yθ~,ϵ. So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
Moreover, ifc1=0andc2≠0, then ∃ϵ0<1/c2>0 such that, ∀ϵ∈0,ϵ0,(48)y0θ,0-y~θ~,ϵ=y0θ,0-y0θ~,0-ϵ2y~2θ~,0-oϵ2=ϵ2y~2θ~,0+oϵ2=ϵ2c2y1θ~,0+oϵ2=ϵc2ϵy1θ~,0+oϵ=ϵc2y0θ,0-yθ~,ϵ<y0θ,0-yθ~,ϵ.So y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.Remark 9.
Letg be a real function such that gϵ=ϵmhϵ,m∈N∗, where hϵ can be expanded in powers of ϵashϵ=∑k≥1ϵkak, with ak real constants.
( 1 ) If m=1 or 2, according to the conditions of Theorem 8, we conclude that y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.
( 2 ) If m>2 (in the case where m=3, the fourth term is given by Proposition 5), we can expect that y~θ~,ϵ is an approximation to y0θ,0 closer than yθ~,ϵ.Remark 10.
We note here that, in the fractional case, the existence of a positive solution of (32) is studied in [18].Remark 11.
Although the Lindstedt-Poincaré method gives uniformly valid asymptotic expansions for periodic solutions of weakly nonlinear oscillations, i.e.,0<ϵ<C1, the technique does not work if the amplitude of the oscillation is a function of time (see [16, 17]).



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