# On New Modifications of Some Perturbation Procedures

Authors: A. I. Ismail
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.


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## Body

## 1. Introduction

In [1], the variation of parameters method is applied to solve the general nonhomogenous linear differential equation of the second order. The author solved the homogenous equation and constructed a particular solution depending on its solutions. He got the first and second differentiation of the particular solutions, substituted them into the considered equation, and performed some investigated steps, and he found the required general solution.We note that this is an exact solution for a linear differential equation and not an approximated one and so is not the small parameter technique or the large parameter one useful for this problem. Such techniques are useful only for approximated problems consisting of nonlinear differential equations containing a small parameter [2] or a large one [3, 4]. In the following sections, we modify the well-known previous method in terms of the large parameter for solving some systems for some perturbed problems [5, 6] containing quasi-linear systems. This modification gives us the chance to study the considered problems in new conditions and new domains of solutions.


## 2. The Method of Averaging

We consider the following examples for illustrating this method, and we will find the periodic solutions of each case:


### 2.1. Van Der Pol’s Technique

This technique is considered in previous works for finding the approximated periodic solutions as power series in terms of a small parameter defined for each problem. In our work, we aim to find new periodic solutions in terms of a large parameter instead of the small one of the equation(1)d2udt2+ω02u=ε−11−u2dudt+Kλcosλt,where ε is imposed to be large and λ=Oε−1 is the movement frequency which is assumed to be small to be different from the natural frequency ω0. By taking the solution to (1), we have assumed the following form:(2)ut=a1tcosλt+a2tsinλt,where a1t and a2t are taken to be slow functions that vary with time as well as dai/dt=0ε−1 and d2ai/dt2=0ε−2. By differentiating (2) twice, we obtain(3)u¨=−λ2a1cosλt−λ2a2sinλt−2a˙1λsinλt+2a˙2λcosλt+a¨1cosλt+a¨2sinλt,u˙≡dudt.Using equations (1)–(3), we obtain(4)2a˙1+a2λ−ω02λ−ε−1a11−ρ=0,2a˙2−a1λ−ω02λ−ε−1a21−ρ=ε−1K,(5)ρ=0.25a2=0.25a12+a22.Periodic solutions of (1) match solutions:(6)2σa20−a101−ρ0=0,(7)−2σa10−a201−ρ0=K,where σ=λ−εω0.By adding squares (6) and (7) and using (5), we obtain the solution of the frequency equation:(8)ρ04σ2+1−ρ02=0.25K2.


### 2.2. The Krylov–Bogoliubov Technique

This technique considered through the weak nonlinear equation of the second order in terms of the small parameter was defined in the problem [7–11]. Here, we consider applying the large parameter ε. Let us consider the following equation:(9)d2udt2+ω02u=ε−1fu,dudt.Atε⟶∞, the solution of (9) can be written as follows:(10)u=acosω0t+θ,where a and θ are constants. To determine the approximated solution of (9) for the large parameter ε, Krylov and Bogoliubov hypothesized that the solution is still given by (10) but with the time change a and θ, and it is subjected to the condition:(11)dudt=−aω0sinϕ,ϕ=ω0t+θ.By differentiation (10) w. r. t.t, we obtain(12)dudt=−aω0sinϕ+dadtcosϕ−adθdtsinϕ.So,(13)dadtcosϕ−adθdtsinϕ=0.By differentiation (11) w. r. t.t, we obtain(14)d2udt2=−aω02cosϕ−ω0dadtsinϕ−aω0dθdtcosϕ.Substituting this expression into (9) and using (10), we obtain(15)ω0dadtsinϕ+aω0dθdtcosϕ=−ε−1f,f=facosϕ,−aω0sinϕ.Solving (13) and (15) for da/dt and dθ/dt, we obtain that(16)ω0dadt=−ε−1sinϕf,aω0dθdt=−ε−1cosϕf.By integrating (16) into the period [t, t + T], where a and θ can be taken as constants on the right-hand side, we obtain(17)2ω0dadt=−ε−1f1a,2aω0dθdt=−ε−1g1a,where(18)πf1a=∫02πsinϕfdϕ,πg1a=∫02πcosϕfdϕ.We also note thatf1 and g1 are coefficients of the Fourier series expansion of the function f.As an example, let us consider the Duffing equation.In this example, we modify the Duffing equation in terms of the large parameter to be(19)d2udt2+ωo2u=−ε−1u3,where(20)fu,u˙=−u3.So,(21)f1a=0,g1a=−0.75a3.Thus, from (17), it is obtained(22)ωoθ=0.375ε−1a2t+ωoθo,a=const.Therefore, the first approximation is(23)u=acosωo1+0.375ε−1a2/ωo2t+oε−2.For the second example, consider the Van der Pol oscillator:(24)d2udt2+ωo2u=fu,u˙,where(25)fu,u˙=1−u2dudt.In this case,(26)f1=−ωoa1−0.25a2,g1=0.Hence and from (17), it follows that da/dt is obtained in terms of the large parameter, so(27)dadt=0.5ε−1a1−0.25a2,θ=θo=const.By integrating the first equation into (27), it becomes clear that(28)a2=41+4/ao−1eεt.


### 2.3. The Generalized Method of Averaging

This method was applied for obtaining the periodic solutions of the differential equations applying the small parameter technique which was defined for each case of the motion [12–14]. In our work, we use the large parameter method for obtaining new solutions dependent on the large parameter ε. Let us consider (10) and (11) as the conversion from u and du/dt to a and ϕ in terms of the large parameter, so(29)ωodadt=−ε−1sinϕf,aωodϕdt=aωo2−ε−1cosϕf,where ϕ is called the rapid rotation phase. Let us take the conversions from a,ϕ to a¯,ϕ¯ such that(30)a=a¯+ε−1a1a¯,ϕ¯+ε−2a2a¯,ϕ¯+…,ϕ=ϕ¯+ε−1ϕ1a¯,ϕ¯+ε−2ϕ2a¯,ϕ¯+….Using (30), we find that system (29) takes the following form:(31)da¯dt=ε−1A1a¯+ε−2A2a¯+…,dϕ¯dt=ωo+ε−1B1a¯+ε−2B2a¯+…,where Ai and Bi do not depend on ϕ¯, for each i=1,2.Using (29)–(31), we obtain(32)ωo∂an∂ϕ¯+An=Fna¯,ϕ¯,ωo∂ϕn∂ϕ¯+Bn=Gna¯,ϕ¯.Generally,Fn and Gn contain short periodic terms (referred to as superscript s) and long periodic terms (referred to by superscriptℓ).We chooseAn and Bn such that it is equal to the long periodic terms, so(33)An=Fnℓ,Bn=Gnℓ,then(34)ωo∂an∂ϕ¯=Fns,ωo∂ϕn∂ϕ¯=Gns,which can be solved, respectively, in an and ϕn.As an example, consider the Van der Pol oscillator (24) in which(35)fu,u˙=1−u2u˙,ωo=1.In this case, equation (29) in terms of the large parameter becomes:(36)dadt=0.125ε−1a4−a2−4acos2ϕ+a3cos4ϕ,dϕdt=1+0.125ε−122−a2sin2ϕ−a2sin4ϕ.By substituting (30) and (31) into (36) and equating the coefficients of similar powers of ε we obtain the following.Orderε−1:(37)∂a1∂ϕ¯+A1=0.125a¯4−a¯2−0.5a¯cos2ϕ¯+0.125a¯3cos4ϕ¯,∂ϕ1∂ϕ¯+B1=0.252−a¯2sin2ϕ¯−0.125a¯2sin4ϕ¯.Orderε−2:(38)∂a2∂ϕ¯+A2=−∂a1∂a¯A1−∂a1∂a¯B1+0.125a14−3a¯2−4cos2ϕ¯+3a¯2cos4ϕ¯+0.5a¯ϕ2sin2ϕ¯−a¯2sin4ϕ¯,∂ϕ2∂ϕ¯+B2=−∂ϕ1∂a¯A1−∂ϕ1∂ϕ¯B1−0.25a¯a12sin2ϕ¯+sin4ϕ¯+0.5ϕ12−a¯2cos2ϕ¯−a¯2cos4ϕ¯.Equally,A1 and B1 by the long periodic terms on the right-hand side in (37) is obtained as(39)A1=0.125a¯4−a¯2,B1=0.Hence, system (37) becomes(40)∂a1∂ϕ¯=−0.5a¯cos2ϕ¯+0.125a¯3cos4ϕ¯,∂ϕ1∂ϕ¯=0.252−a¯2sin2ϕ¯−0.125a¯2sin4ϕ¯.So, we obtain the following solutions(41)a1=−0.25a¯sin2ϕ¯+0.031a¯3sin4ϕ¯,ϕ1=−0.1252−a¯2cos2ϕ¯+0.031a¯2cos4ϕ¯.By using (39) and (41), the relation (38) becomes:(42)∂a2∂ϕ¯+A2=short−period terms,∂ϕ2∂ϕ¯+B2=−0.125+0.188a¯2−0.043a¯4+short−period terms.Equally,A2 and B2 by the long periodic terms on the right-hand side in (42) is obtained as(43)A2=0,B2=−0.125+0.188a¯2−0.043a¯4.Therefore, the solution up to the second approximation1/ε2 becomes(44)u=acosϕ,where(45)a=a¯−0.125ε−1a¯sin2ϕ¯−0.125a¯2sin4ϕ¯+oε−2,ϕ=ϕ¯−0.125ε−12−a¯2cos2ϕ¯−0.25a¯2cos4ϕ¯+oε−2,da¯dt=0.125ε−1a¯4−a¯2+oε−2,dϕ¯dt=1−0.125ε−21−1.5a¯2+0.344a¯4+oε−3.


## 2.1. Van Der Pol’s Technique

This technique is considered in previous works for finding the approximated periodic solutions as power series in terms of a small parameter defined for each problem. In our work, we aim to find new periodic solutions in terms of a large parameter instead of the small one of the equation(1)d2udt2+ω02u=ε−11−u2dudt+Kλcosλt,where ε is imposed to be large and λ=Oε−1 is the movement frequency which is assumed to be small to be different from the natural frequency ω0. By taking the solution to (1), we have assumed the following form:(2)ut=a1tcosλt+a2tsinλt,where a1t and a2t are taken to be slow functions that vary with time as well as dai/dt=0ε−1 and d2ai/dt2=0ε−2. By differentiating (2) twice, we obtain(3)u¨=−λ2a1cosλt−λ2a2sinλt−2a˙1λsinλt+2a˙2λcosλt+a¨1cosλt+a¨2sinλt,u˙≡dudt.Using equations (1)–(3), we obtain(4)2a˙1+a2λ−ω02λ−ε−1a11−ρ=0,2a˙2−a1λ−ω02λ−ε−1a21−ρ=ε−1K,(5)ρ=0.25a2=0.25a12+a22.Periodic solutions of (1) match solutions:(6)2σa20−a101−ρ0=0,(7)−2σa10−a201−ρ0=K,where σ=λ−εω0.By adding squares (6) and (7) and using (5), we obtain the solution of the frequency equation:(8)ρ04σ2+1−ρ02=0.25K2.


## 2.2. The Krylov–Bogoliubov Technique

This technique considered through the weak nonlinear equation of the second order in terms of the small parameter was defined in the problem [7–11]. Here, we consider applying the large parameter ε. Let us consider the following equation:(9)d2udt2+ω02u=ε−1fu,dudt.Atε⟶∞, the solution of (9) can be written as follows:(10)u=acosω0t+θ,where a and θ are constants. To determine the approximated solution of (9) for the large parameter ε, Krylov and Bogoliubov hypothesized that the solution is still given by (10) but with the time change a and θ, and it is subjected to the condition:(11)dudt=−aω0sinϕ,ϕ=ω0t+θ.By differentiation (10) w. r. t.t, we obtain(12)dudt=−aω0sinϕ+dadtcosϕ−adθdtsinϕ.So,(13)dadtcosϕ−adθdtsinϕ=0.By differentiation (11) w. r. t.t, we obtain(14)d2udt2=−aω02cosϕ−ω0dadtsinϕ−aω0dθdtcosϕ.Substituting this expression into (9) and using (10), we obtain(15)ω0dadtsinϕ+aω0dθdtcosϕ=−ε−1f,f=facosϕ,−aω0sinϕ.Solving (13) and (15) for da/dt and dθ/dt, we obtain that(16)ω0dadt=−ε−1sinϕf,aω0dθdt=−ε−1cosϕf.By integrating (16) into the period [t, t + T], where a and θ can be taken as constants on the right-hand side, we obtain(17)2ω0dadt=−ε−1f1a,2aω0dθdt=−ε−1g1a,where(18)πf1a=∫02πsinϕfdϕ,πg1a=∫02πcosϕfdϕ.We also note thatf1 and g1 are coefficients of the Fourier series expansion of the function f.As an example, let us consider the Duffing equation.In this example, we modify the Duffing equation in terms of the large parameter to be(19)d2udt2+ωo2u=−ε−1u3,where(20)fu,u˙=−u3.So,(21)f1a=0,g1a=−0.75a3.Thus, from (17), it is obtained(22)ωoθ=0.375ε−1a2t+ωoθo,a=const.Therefore, the first approximation is(23)u=acosωo1+0.375ε−1a2/ωo2t+oε−2.For the second example, consider the Van der Pol oscillator:(24)d2udt2+ωo2u=fu,u˙,where(25)fu,u˙=1−u2dudt.In this case,(26)f1=−ωoa1−0.25a2,g1=0.Hence and from (17), it follows that da/dt is obtained in terms of the large parameter, so(27)dadt=0.5ε−1a1−0.25a2,θ=θo=const.By integrating the first equation into (27), it becomes clear that(28)a2=41+4/ao−1eεt.


## 2.3. The Generalized Method of Averaging

This method was applied for obtaining the periodic solutions of the differential equations applying the small parameter technique which was defined for each case of the motion [12–14]. In our work, we use the large parameter method for obtaining new solutions dependent on the large parameter ε. Let us consider (10) and (11) as the conversion from u and du/dt to a and ϕ in terms of the large parameter, so(29)ωodadt=−ε−1sinϕf,aωodϕdt=aωo2−ε−1cosϕf,where ϕ is called the rapid rotation phase. Let us take the conversions from a,ϕ to a¯,ϕ¯ such that(30)a=a¯+ε−1a1a¯,ϕ¯+ε−2a2a¯,ϕ¯+…,ϕ=ϕ¯+ε−1ϕ1a¯,ϕ¯+ε−2ϕ2a¯,ϕ¯+….Using (30), we find that system (29) takes the following form:(31)da¯dt=ε−1A1a¯+ε−2A2a¯+…,dϕ¯dt=ωo+ε−1B1a¯+ε−2B2a¯+…,where Ai and Bi do not depend on ϕ¯, for each i=1,2.Using (29)–(31), we obtain(32)ωo∂an∂ϕ¯+An=Fna¯,ϕ¯,ωo∂ϕn∂ϕ¯+Bn=Gna¯,ϕ¯.Generally,Fn and Gn contain short periodic terms (referred to as superscript s) and long periodic terms (referred to by superscriptℓ).We chooseAn and Bn such that it is equal to the long periodic terms, so(33)An=Fnℓ,Bn=Gnℓ,then(34)ωo∂an∂ϕ¯=Fns,ωo∂ϕn∂ϕ¯=Gns,which can be solved, respectively, in an and ϕn.As an example, consider the Van der Pol oscillator (24) in which(35)fu,u˙=1−u2u˙,ωo=1.In this case, equation (29) in terms of the large parameter becomes:(36)dadt=0.125ε−1a4−a2−4acos2ϕ+a3cos4ϕ,dϕdt=1+0.125ε−122−a2sin2ϕ−a2sin4ϕ.By substituting (30) and (31) into (36) and equating the coefficients of similar powers of ε we obtain the following.Orderε−1:(37)∂a1∂ϕ¯+A1=0.125a¯4−a¯2−0.5a¯cos2ϕ¯+0.125a¯3cos4ϕ¯,∂ϕ1∂ϕ¯+B1=0.252−a¯2sin2ϕ¯−0.125a¯2sin4ϕ¯.Orderε−2:(38)∂a2∂ϕ¯+A2=−∂a1∂a¯A1−∂a1∂a¯B1+0.125a14−3a¯2−4cos2ϕ¯+3a¯2cos4ϕ¯+0.5a¯ϕ2sin2ϕ¯−a¯2sin4ϕ¯,∂ϕ2∂ϕ¯+B2=−∂ϕ1∂a¯A1−∂ϕ1∂ϕ¯B1−0.25a¯a12sin2ϕ¯+sin4ϕ¯+0.5ϕ12−a¯2cos2ϕ¯−a¯2cos4ϕ¯.Equally,A1 and B1 by the long periodic terms on the right-hand side in (37) is obtained as(39)A1=0.125a¯4−a¯2,B1=0.Hence, system (37) becomes(40)∂a1∂ϕ¯=−0.5a¯cos2ϕ¯+0.125a¯3cos4ϕ¯,∂ϕ1∂ϕ¯=0.252−a¯2sin2ϕ¯−0.125a¯2sin4ϕ¯.So, we obtain the following solutions(41)a1=−0.25a¯sin2ϕ¯+0.031a¯3sin4ϕ¯,ϕ1=−0.1252−a¯2cos2ϕ¯+0.031a¯2cos4ϕ¯.By using (39) and (41), the relation (38) becomes:(42)∂a2∂ϕ¯+A2=short−period terms,∂ϕ2∂ϕ¯+B2=−0.125+0.188a¯2−0.043a¯4+short−period terms.Equally,A2 and B2 by the long periodic terms on the right-hand side in (42) is obtained as(43)A2=0,B2=−0.125+0.188a¯2−0.043a¯4.Therefore, the solution up to the second approximation1/ε2 becomes(44)u=acosϕ,where(45)a=a¯−0.125ε−1a¯sin2ϕ¯−0.125a¯2sin4ϕ¯+oε−2,ϕ=ϕ¯−0.125ε−12−a¯2cos2ϕ¯−0.25a¯2cos4ϕ¯+oε−2,da¯dt=0.125ε−1a¯4−a¯2+oε−2,dϕ¯dt=1−0.125ε−21−1.5a¯2+0.344a¯4+oε−3.


## 3. Struble’s Technique

Struble developed a technique for treating weak nonlinear oscillatory systems such as those governed by a second-order differential equation in terms of a small parameter [15]. We treat this equation in terms of the large parameter as follows:(46)u¨+ω02u=ε−1fu,u˙,t.Let us consider the solution of (46) in terms of the large parameter ε, which takes the form(47)u=acosω0t−θ+∑n=1Nε−nunt+oε−N−1,where a and θ are functions that are slowly changing with time.As an example, we take Duffing’s equation (19) and use (47) to give(48)2aω0dθdt+d2adt2−adθdt2cosω0t−θ+−2ω0dadt+ad2θdt2+2dadtdθdtsinω0t−θ+ε−1d2u1dt2+ω02u1+ε−2d2u2dt2+ω02u2+…=−ε−1a3cos3ω0t−θ−3ε−2u1a2cos2ω0t−θ+….Considering the terms of orderOε−1 and equality of coefficients cosω0t−θ and sinω0t−θ on both sides, we get the so-called variation equations:(49)2aω0dθdt+d2adt2−adθdt2=−0.75ε−1a3,−2aω0dadt+ad2θdt2+2dadtdθdt=0.While the perturbed equations remain, so(50)d2u1dt2+ω02u1=−0.25a3cos3ω0t−θ.For the first order of1/ε, equation (49) reduces to(51)dadt=0,dθdt=−0.375ε−1a2ω0.Hence,(52)a=a0,θ=−0.375ε−1a02tω0+θ0,where ao and θo are constants. If we consider θ and a are constants, the solution of (50) for the first order becomes(53)u1=0.031ω0−2a3cos3ω0t−θ.Hence, the solution for the first-orderO1/ε becomes(54)u=acosω0t−θ+0.031ω0−2ε−1a3cos3ω0t−θ,where a and θ are given from equation (52).Satisfying the solution for second order, we use (53) for calculating the term:(55)−3ε−2u1a2cos2ω0−θ=−0.023ω0−2ε−2a5cosω0t−θ+2cos3ω0t−θ+cos5ω0t−θ.We note that the expressiond2u1/dt2+ω02u1 contains a term of order O1/ε in the following form:(56)0.563ω0−2a3dθdtcos3ω0t−θ.Now, we consider the orderO1/ε2 leads us to the variation equations:(57)2aω0dθdt+d2adt2−adθdt2=−0.75ε−1a3−0.023ω0−2ε−2a5−2aω0dadt+ad2θdt2+2dadtdθdt=0,and the perturbed equation is(58)d2u2dt2+ω02u2=−0.023ω0−2a52cos3ω0t−θ+cos5ω0t−θ−0.563ω0−1ε−1a3dθdtcos3ω0t−θ.The solution of equation (57) can be obtained from the recurrence principle of (52) to obtain(59)a=a0,θ=−0.375ω0−1ε−1a02t+0.059ω0−3ε−2a04t+θ0+oε−3,where ao and θo are constants. Substituting by dθ/dt from (52) into (58) and solving the resulted equation, we obtain(60)u2=−0.001ω0−4a521cos3ω0t−θ−cos5ω0t−θ.Thus, we obtain the solution up to the second approximation in the form(61)u=acosωt−θ0+0.031ε−1a3ω0−21−0.656ε−1ω0−2a2cos3ωt−θ0+0.001ε−2a5ω0−4cos5ωt−θ0+oε−3,where(62)ω=ω01+0.375ε−1a2ω0−2−0.059ε−2a4ω0−4+oε−3.The periodic timeT=2π/ω is given in the form(63)T=2πω0−1+αε−1,where the correction of the period is obtained in the form(64)αε−1=2πω0−1−0.375ε−1a2ω0−2+0.199ε−2a4ω0−4+oε−3.


## 4. The Krylov–Bogoliubov–Mitropolski Technique

This technique is devoted to finding approximated periodic solutions using the small parameter as in [7–11]. In our work, we modify the approximated expansions using the large parameter to become(65)u=acosψ+∑n=1Nε−nuna,ψ+oε−N−1,where each un is a periodic function in ψ, and its period 2π, a, and ψ are imposed to change with time as follows:(66)dadt=∑n=1Nε−nAna+oε−N−1,(67)dψdt=ωo+∑n=1Nε−nψna+oε−N−1,where the functions un,An,andψn are arbitrary so that equations (65) to (67) satisfy the differential equation (9). The following derivatives series are obtained:(68)ddt=dadt∂∂a+dψdt∂∂ψ,d2dt2=dadt2∂2∂a2+d2adt2∂∂a+2dadtdψdt∂2∂a∂ψ+dψdt2∂2∂ψ2+d2ψdt2∂∂ψ;(69)d2adt2=dadt∑n=1Nε−ndAnda=ε−2A1dA1da+oε−3,d2ψdt2=dadt∑n=1Nε−ndψnda=ε−2A1dψ1da+oε−3.We will visualize this method in the following example.


### 4.1. Duffing Equation

We consider a nonlinear oscillator (19), using equations (65) to (69), and equate the coefficients of similar powers to 1/ε, we obtain(70)ωo∂2u1∂ψ2+u1=2ψ1acosψ+A1sinψ−a3ωo−1cos3ψ,(71)ωo2∂2u2∂ψ2+u2=a2ωoψ2+ψ12−A1dA1dacosψ+2ωoA2+A1ψ1+aA1dψ1dasinψ−3u1a2cos2ψ−2ωoψ1∂2u1∂ψ2+A1∂2u1∂a∂ψ.Letu1 be periodic, and the singular terms in the right-hand side of (70) must be deleted. Since(72)cos3ψ=0.753cosψ+cos3ψ,then equation (70) gives(73)A1=0,ψ1=0.375a2ωo−1.The solution of equation (70) becomes(74)u1=0.031a3ωo−2cos3ψ.Substituting (74) into (71) gives(75)ωo2∂2u2∂ψ2+u2=2ωoψ2+0.117a4ωo−2acosψ+2ωoA2sinψ+0.008a5ωo−221cos3ψ−3cos5ψ.Canceling the singular terms yields(76)A2=0,ψ2=−0.059a4ωo−3.The solution of (75) becomes(77)u2=−0.001a5ωo−421cos3ψ−cos5ψ.Therefore, the second approximated solutionu is given in the following form:(78)u=acosψ+0.031ε−1a3ω0−2cos3ψ−0.001ε−2a5ω0−421cos3ψ−cos5ψ+oε−3,where(79)dadt=0ora=a0=const.,dψdt=ω0+0.375ε−1a2ω0−1−0.059ε−2a4ω0−3+oε−3,ψ=ω01+0.375ε−1a2ω0−2−0.059ε−2a4ω0−4t+ψ0+oε−3,where ψo is constant.


## 4.1. Duffing Equation

We consider a nonlinear oscillator (19), using equations (65) to (69), and equate the coefficients of similar powers to 1/ε, we obtain(70)ωo∂2u1∂ψ2+u1=2ψ1acosψ+A1sinψ−a3ωo−1cos3ψ,(71)ωo2∂2u2∂ψ2+u2=a2ωoψ2+ψ12−A1dA1dacosψ+2ωoA2+A1ψ1+aA1dψ1dasinψ−3u1a2cos2ψ−2ωoψ1∂2u1∂ψ2+A1∂2u1∂a∂ψ.Letu1 be periodic, and the singular terms in the right-hand side of (70) must be deleted. Since(72)cos3ψ=0.753cosψ+cos3ψ,then equation (70) gives(73)A1=0,ψ1=0.375a2ωo−1.The solution of equation (70) becomes(74)u1=0.031a3ωo−2cos3ψ.Substituting (74) into (71) gives(75)ωo2∂2u2∂ψ2+u2=2ωoψ2+0.117a4ωo−2acosψ+2ωoA2sinψ+0.008a5ωo−221cos3ψ−3cos5ψ.Canceling the singular terms yields(76)A2=0,ψ2=−0.059a4ωo−3.The solution of (75) becomes(77)u2=−0.001a5ωo−421cos3ψ−cos5ψ.Therefore, the second approximated solutionu is given in the following form:(78)u=acosψ+0.031ε−1a3ω0−2cos3ψ−0.001ε−2a5ω0−421cos3ψ−cos5ψ+oε−3,where(79)dadt=0ora=a0=const.,dψdt=ω0+0.375ε−1a2ω0−1−0.059ε−2a4ω0−3+oε−3,ψ=ω01+0.375ε−1a2ω0−2−0.059ε−2a4ω0−4t+ψ0+oε−3,where ψo is constant.


## 5. Multiple Scale Method

This technique is considered for solving the differential equations applying the small parameter [1, 16]. In our work, we modify this method to be suitable for the large parameter technique.From equation (46), we put ωo=1, and we obtain(80)u¨+u=ε−1fu,u˙,t.To have accurate and regular expansion over all periods for the solution of this equation, we should be finding an approximated representation of the functionut,ε−1 as shown in the form(81)ut,ε−1=∑n=0Nε−nun∗t0,…,tN+oε−N−1,where(82)tm=ε−mt,m=0,1,2,…,N.Therefore, the total differential of (80) gives(83)ddt=∑n=oNε−n∂∂tn.Substituting expansions (81) and (83) into equation (80), then equalizing the coefficients of equal powers for ε−1on both sides, and by solving the resulting equations, the solutions are obtained. Such solutions contain arbitrary functions in the variables t1,…,tN and will be determined with a condition uo∗ and un∗/un−1∗ which are bounded to each tn.Now, we will apply the modified method to the following example.


### 5.1. Duffing Equation

Consider the Duffing equation in terms of the large parameter which takes the form(84)d2xdt2+ω02x+ε−1x3=0.Taking(85)x=∑n=o2ε−nxnto,t1,t2+oε−3,substituting (85) in (84), and equating the like powers of ε−1 to zero, we obtain(86)D02x0+ω02x0=0,(87)D02x1+ω02x1+2D0D1x0+x03=0,(88)D02x2+ω02x2+2D0D1x1+2D0D2+D12x0+3x02x1=0,where Dn=∂/∂tn.The solution of equation (86) is(89)x0=At1,t2eiω0t+A¯t1,t2e−iω0t,where A¯ is the complex conjugate of A. Then, equation (87) becomes as follows:(90)D02x1+ω02x1=2iω0D1A¯−3AA¯2e−iω0t−2iω0D1A+3A2A¯eiω0t−A3e3iω0t−A¯3e−3iω0t.Sincex1/x0 is finite for each t, the singular terms must be neglected. So,(91)2iω0D1A+3A2A¯=0,2iω0D1A¯−3AA¯2=0,thus(92)A=A1t2e3iA1A¯1t1/2ω0,x1=Beiω0t+B¯e−iω0t+0.125ω0−2A3e3iω0t+A¯3e−3iω0t.Usingx0andx1 into (88) yields(93)D02x2+ω02x2=−0.375ω0−2A5e5iω0t+A¯5e−5iω0t+3A20.875ω0−2A2A¯−Be3iω0t+3A¯20.875ω0−2AA¯2−B¯e−3iω0t+Qeiω0t+Q¯e−iω0t,where(94)Q=−2iω0D1B+D2A−3AAB¯+2BA¯+1.875ω0−2A3A¯2.By eliminating the singular terms, thereforex2/x1 will be finite for all t; then,(95)B=0,2iω0D2A1=1.875ω0−2A13A¯12.Therefore,(96)A1=A2e−0.938iA22A¯22ω0−3t2.A solutionx2 regardless of the homogeneous solution is given:(97)x2=0.016ω0−4A5e5iω0t+A¯5e−5iω0t−21AA¯A3e3iω0t+A¯3e−3iω0t.Finally, we get the solutionx in terms of the large parameter up to the second approximation of 1/ε in the form(98)x=acosωt+ϕ+0.031ε−1a3ω0−21−0.656ε−1a2ω0−2cos3ωt+ϕ+0.001ε−2a5ω0−4cos5ωt+ϕ,where A=0.5aeiϕ and ω=ω0+0.375ε−1a2ω0−1−0.059ε−2a4ω0−3. The obtained solution has the same period (63).


## 5.1. Duffing Equation

Consider the Duffing equation in terms of the large parameter which takes the form(84)d2xdt2+ω02x+ε−1x3=0.Taking(85)x=∑n=o2ε−nxnto,t1,t2+oε−3,substituting (85) in (84), and equating the like powers of ε−1 to zero, we obtain(86)D02x0+ω02x0=0,(87)D02x1+ω02x1+2D0D1x0+x03=0,(88)D02x2+ω02x2+2D0D1x1+2D0D2+D12x0+3x02x1=0,where Dn=∂/∂tn.The solution of equation (86) is(89)x0=At1,t2eiω0t+A¯t1,t2e−iω0t,where A¯ is the complex conjugate of A. Then, equation (87) becomes as follows:(90)D02x1+ω02x1=2iω0D1A¯−3AA¯2e−iω0t−2iω0D1A+3A2A¯eiω0t−A3e3iω0t−A¯3e−3iω0t.Sincex1/x0 is finite for each t, the singular terms must be neglected. So,(91)2iω0D1A+3A2A¯=0,2iω0D1A¯−3AA¯2=0,thus(92)A=A1t2e3iA1A¯1t1/2ω0,x1=Beiω0t+B¯e−iω0t+0.125ω0−2A3e3iω0t+A¯3e−3iω0t.Usingx0andx1 into (88) yields(93)D02x2+ω02x2=−0.375ω0−2A5e5iω0t+A¯5e−5iω0t+3A20.875ω0−2A2A¯−Be3iω0t+3A¯20.875ω0−2AA¯2−B¯e−3iω0t+Qeiω0t+Q¯e−iω0t,where(94)Q=−2iω0D1B+D2A−3AAB¯+2BA¯+1.875ω0−2A3A¯2.By eliminating the singular terms, thereforex2/x1 will be finite for all t; then,(95)B=0,2iω0D2A1=1.875ω0−2A13A¯12.Therefore,(96)A1=A2e−0.938iA22A¯22ω0−3t2.A solutionx2 regardless of the homogeneous solution is given:(97)x2=0.016ω0−4A5e5iω0t+A¯5e−5iω0t−21AA¯A3e3iω0t+A¯3e−3iω0t.Finally, we get the solutionx in terms of the large parameter up to the second approximation of 1/ε in the form(98)x=acosωt+ϕ+0.031ε−1a3ω0−21−0.656ε−1a2ω0−2cos3ωt+ϕ+0.001ε−2a5ω0−4cos5ωt+ϕ,where A=0.5aeiϕ and ω=ω0+0.375ε−1a2ω0−1−0.059ε−2a4ω0−3. The obtained solution has the same period (63).


## 6. Conclusion

In this paper, we modified some of the perturbation methods that are considered ones of the innovations of mathematicians and physicists for finding approximated solutions to important problems. We cannot find a complete solution to these problems. Nayfeh, Poincare, and Krylov are among the prominent scientists in this field as they dealt with many problems in mathematics and physics which are applied in many branches, such as classical mechanics, flexible body mechanics, and many other branches. All these methods depend on a small parameter that was defined for each problem.The study of such methods shows that the large parameter method is the easiest, most accurate, and one of the methods most used to solve many important problems efficiently. The large parameter method has an advantage over the other methods because it solves the problem in a new domain when it fails all other methods for solving the problem in such a domain. One of the most important application is when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment [4], the rotational motion of a heavy solid in a uniform gravity field [3], or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems.



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*Source: 6681932-2021-02-09.xml*