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# Zeitschrift für Naturforschung A

### A Journal of Physical Sciences

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# Approximate Solutions for the Nonlinear Third-Order Ordinary Differential Equations

M.M. Fatih Karahan
Published Online: 2017-05-08 | DOI: https://doi.org/10.1515/zna-2016-0502

## Abstract

A new perturbation method, multiple scales Lindstedt–Poincare (MSLP) is applied to jerk equations with cubic nonlinearities. Three different jerk equations are investigated. Approximate analytical solutions and periods are obtained using MSLP method. Both approximate analytical solutions and periods are contrasted with numerical and exact results. For the case of strong nonlinearities, obtained results are in good agreement with numerical and exact ones.

## 1 Introduction

Perturbation methods are widely used to solve approximate analytical solutions of physical problems. From heat transfer to mechanical vibrations, from solid mechanics to fluid mechanics, many mathematical models corresponding to physical problems were solved successfully with these methods. In these methods, the solution is expressed as a series expansion of the small parameter. The simplest perturbation technique, called direct expansion, leads to problems in many models and causes blowing solutions as time tends to infinity. For this reason, many different perturbation techniques were developed to obtain realistic solutions [1], [2], [3].

One of the major limitations of perturbation methods is the need for a small parameter in the equations or artificial introduction of the small parameter to the equations. Solutions will then be valid for weak nonlinear systems. For strong nonlinear systems where the physical parameter is large, these solutions cease to be valid. An important study was presented by Cheung et al. [4] for strong nonlinear systems. In this article, a modified Lindstedt–Poincare method was developed. It is an effective method for dealing with strongly nonlinear systems. Recently, some research was conducted for developing new techniques for perturbation methods to be valid for large parameters also [5], [6], [7]. A generalised hyperbolic perturbation method [5] for heteroclinic solutions is proposed for strongly nonlinear self-excited oscillators. The method is applied to equations with quadratic–cubic and quintic–septic nonlinearities. Li and Tang [6] developed the generalised Padé–Lindstedt–Poincaré method for predicting homoclinic and heteroclinic bifurcations of strongly nonlinear autonomous oscillators. Very recently, Chen et al. [7] improved the perturbation method based on nonlinear time transformation to achieve purely analytical and explicit homoclinic solutions.

Multiple scales Lindstedt–Poincare (MSLP) method [8], [9], [10], [11], [12], [13], [14] is an algorithm for determining analytical approximations for the cases of both weak and strong nonlinearities. The method is an integration of the method of multiple scales and the Lindstedt–Poincare method with some transformations and expansion selections in [15], [16].

In this article, the MSLP method is applied to the nonlinear jerk equations for the first time. The nonlinear jerk equations involving third-order time derivative define some physical problems. The general form of the jerk function $\stackrel{⃛}{x}=J\left(x,\text{\hspace{0.17em}}\stackrel{˙}{x},\text{\hspace{0.17em}}\stackrel{¨}{x}\right)$ is given by Gottlieb [17]. Three different nonlinear jerk equations are investigated: Jerk functions containing displacement times velocity times acceleration and velocity $\left(x\stackrel{˙}{x}\stackrel{¨}{x}\right),$ velocity times acceleration-squared and velocity $\left(\stackrel{˙}{x}{\stackrel{¨}{x}}^{2}\right),$ velocity-cubed $\left({\stackrel{˙}{x}}^{3}\right),$ and velocity time displacement-squared and velocity $\left(\stackrel{˙}{x}{x}^{2}\right).$ Approximate analytical solutions are obtained using the MSLP method. Obtained solutions are contrasted with the numerical solutions for the cases of both weak and strong nonlinearities. Results of MSLP and numerical solutions are in good agreement with each other.

Finally, for the case of third-order differential equations, some recent works are mentioned [17], [18], [19], [20], [21], [22], [23], [24], [25]. Gottlieb [17] applied lowest order harmonic balance method to nonlinear jerk equations with cubic nonlinearities. Periods obtained by Gottlieb are contrasted with the exact periods of the jerk functions. They are not accurate enough for the large range of velocity amplitude. Wu et al. [18] employed improved harmonic balance approach to jerk function containing velocity-cubed and velocity times displacement-squared. Their method gives highly accurate results in terms of large velocity amplitudes. Hu [19], [20] applied the parameter perturbation and Micken’s iteration procedure to abovementioned jerk equation. For the case of large velocity amplitudes, his periods are in good agreement with the exact periods. Residue harmonic balance [21] and He’s homotopy perturbation methods [22] are applied to seek approximate solutions of three cases of nonlinear jerk equations. Their results are very accurate for a large range of the initial velocity amplitude. Ramos [23], [24] implemented some approximation methods to nonlinear jerk equations involving third-order time derivative. Feng and Chen [25] applied homotopy analysis method to nonlinear jerk equation.

## 2 General Nonlinear Jerk Functions

Following Gottlieb [17], the general jerk equation has the form

$x⃛=−γx˙−αx˙3−βx2x˙+δxx˙x¨−εx˙x¨2$(1)

where the overdot denotes differentiation with respect to time t and the parameters γ, α, β, δ, and ε are constants. The initial conditions are

$x(0)=0, x˙(0)=a0, x¨(0)=0$(2)

In this section, the MSLP method is applied to solve three different nonlinear jerk equations.

## 2.1 Jerk Function Containing Displacement Times Velocity Times Acceleration and Velocity

For this special case, the general nonlinear jerk function is in the following form

$x⃛+γx˙−δxx˙x¨=0$(3)

with the initial conditions

$x(0)=0, x˙(0)=a0, x¨(0)=0$(4)

Cubic nonlinear term is reordered

$δ=pδ^$(5)

where p is the perturbation parameter.

For details of the MSLP method, see Pakdemirli et al. [8]. First, the time transformation is applied to (3)

$ω3x‴+γωx′−pδ^ω3xx′x″=0$(6)

with the initial conditions

$x(0)=0, x′(0)=a0ω, x″(0)=0$(7)

where prime is derivative with respect to the new time variable τ=ωt. Fast and slow timescales are

$T0=τ, T1= pτ, T2=p2τ$(8)

Using

$ddτ=D0+pD1+p2D2+p3D3+…$(9)

$d2dτ2=D02+2pD0D1+p2(D12+2D0D2)+…$(10)

$d3dτ3=D03+3pD02D1+p2(3D0D12+3D02D2)+…$(11)

and substituting the expansions

$x=x0(T0, T1, T2)+px1(T0, T1, T2)+p2x2(T0, T1, T2)+…$(12)

$γ=ω2−pω1−p2ω2$(13)

into (6) and initial conditions (7) yield after separation

$O(1):ω3D03x0+ω3D0x0=0 x0(0)=0, D0x0(0)=a0ω, D02x0(0)=0$(14)

$O(p):ω3D03x1+ω3D0x1=−3ω3D02D1x0−ω3D1x0+ωω1D0x0 +δ^ω3x0(D0x0)(D02x0) x1(0)=0, D0x1(0)+D1x0(0)=0, D02x1(0)+2D0D1x0(0)=0$(15)

$O(p2):ω3D03x2+ω3D0x2=−3ω3D02D1x1−3ω3D0D12x0 −3ω3D02D2x0−ω3D1x1−ω3D2x0+ωω1(D0x1+D1x0) +ωω2D0x0+δ^ω3[x0(D0x0)(D02x1+2D0D1x0) +x0(D0x1+D1x0)(D02x0)+x1(D0x0)(D02x0)]$(16)

The solution at the first order is

$x0=AeiT0+cc$(17)

where

$A=12aeiλ$(18)

is the polar representation of complex amplitudes. In terms of the real amplitudes and phases, the first-order solution is

$x0=acos(T0+λ)$(19)

Applying the initial conditions yield

$a(0)=−a0ω, λ(0)=π2$(20)

Equation (17) is substituted into (15) and secular terms are eliminated

$2D1A+iω1Aω2−iδ^A2A¯=0$(21)

In the MSLP as outlined in [8], first D1A=0 is selected and if the frequency correction is real, this choice is admissible. If ω1 turns out to be complex, then D1A≠0 which implies ω1=0 and secularities are eliminated by choosing D1A. For (21), selection of

$D1A=0⇒A=A(T2)$(22)

produces

$ω1=δ^AA¯ω2=14δ^a2ω2$(23)

which is suitable because ω1 is real. The solution at order (p) is

$x1=BeiT0+δ^24A3e3iT0+cc=bcos(T0+θ)+δ^96a3cos(3T0+3λ)$(24)

where

$B=12beiθ$(25)

The initial conditions at this order imply

$b(0)=−δ^32ω3a03, θ(0)=λ(0)=π2$(26)

At the last order, (17) and (24) are inserted into (16) and secular terms are eliminated

$2D2A+i(ω2Aω2−δ^A2B¯−δ^AA¯B+7δ^224A3A¯2)=0$(27)

According to the method, first D2A=0 should be tried. This choice leads to a real ω2 and therefore admissible. After algebraic manipulations, (27) yields

$a=−a0ω, b=−δ^32ω3a03, θ=λ=π2, ω2=ω2(−7δ^2a4384+δ^ab2)$(28)

The frequency is

$ω2=γ+pδ^4a02−p2δ^2384ω2a04$(29)

Solving for the frequency yields

$ω=12(γ+δ4a02)+12(γ+δ4a02)2−δ296a04$(30)

The final solution in terms of this frequency is

$x(t)=−a0ωcos(ωt+π2)−δa0396ω3[3cos(ωt+π2) +cos(3ωt+3π2)]+O(p2)$(31)

## 2.2 Jerk Function Containing Velocity Times Acceleration-Squared and Velocity

For this special case, the general nonlinear jerk function is in the following form

$x⃛+γx˙+εx˙x¨2=0$(32)

with the initial conditions

$x(0)=0, x˙(0)=a0, x¨(0)=0$(33)

Cubic nonlinear term is reordered

$ε=pε^$(34)

where p is the perturbation parameter.

First, the time transformation is applied to (32)

$ω3x‴+γωx′+pε^ω5x′x″2=0$(35)

with the initial conditions

$x(0)=0, x′(0)=a0ω, x″(0)=0$(36)

Selecting the fast and slow time variables and expansions as in Section 2.1, the equations at each order are

$O(1):ω3D03x0+ω3D0x0=0 x0(0)=0, D0x0(0)=a0ω, D02x0(0)=0$(37)

$O(p):ω3D03x1+ω3D0x1=−3ω3D02D1x0−ω3D1x0+ωω1D0x0−ε^ω5(D0x0)(D02x0)2x1(0)=0,D0x1(0)+D1x0(0)=0,D02x1(0)+2D0D1x0(0)=0$(38)

$O(p2):ω3D03x2+ω3D0x2=−3ω3(D02D1x1+D0D12x0+D02D2x0) −ω3(D1x1+D2x0)+ωω1(D0x1+D1x0)+ωω2D0x0 −ε^ω5[2(D0x0)(D02x0)(D02x1+2D0D1x0) +(D02x0)2(D0x1+D1x0)]$(39)

The solution at the first order is

$x0=AeiT0+cc=acos(T0+λ)$(40)

Applying the initial conditions yield

$a(0)=−a0ω, λ(0)=π2,$(41)

The first-order solution is substituted into (38) and secular terms are eliminated

$2D1A+iω1Aω2−iε^ω2A2A¯=0$(42)

For D1A=0, ω1 is a real number, so this choice is appropriate

$D1A=0⇒A=A(T2)$(43)

$ω1=ε^AA¯ω4=14ε^a2ω4$(44)

The solution at order (p) is

$x1=BeiT0+ε^ω224A3e3iT0+cc=bcos(T0+θ) +ε^ω296a3cos(3T0+3λ)$(45)

The initial conditions at this order imply

$b(0)=−ε^32ωa03, θ(0)=λ(0)=π2$(46)

At the last order, (40) and (45) are inserted into (39) and secular terms are eliminated

$2D2A+i(ω2Aω2−ε^ω2A2B¯−ε^ω2AA¯B+5ε^2ω48A3A¯2)=0$(47)

After algebraic manipulations, (47) yields

$a=−a0ω, b=−ε^a0332ω, θ=λ=π2 ω2=ε^ω4ab2−5ε^2ω6a4128$(48)

The frequency is

$ω2=γ+pε^ω24a02−p23ε^2ω2128a04$(49)

Solving for the frequency yields

$ω=128γ128−32εa02+3ε2a04$(50)

The final solution in terms of this frequency is

$x(t)=−a0ωcos(ωt+π2)−εa0396ω[3cos(ωt+π2) +cos(3ωt+3π2)]+O(p2)$(51)

## 2.3 Jerk Function Containing Velocity-Cubed, Velocity Times Displacement-Squared and Velocity

For this special case, the general nonlinear jerk function is in the following form

$x⃛+γx˙+αx˙3+βx2x˙=0$(52)

with the initial conditions

$x(0)=0, x˙(0)=a0, x¨(0)=0$(53)

Cubic nonlinear terms are reordered

$α=pα^, β=pβ^$(54)

where p is the perturbation parameter.

First, the time transformation is applied to (52)

$ω3x‴+γωx′+pα^ω3x′3+pβ^ωx2x′=0$(55)

with initial conditions

$x(0)=0, x′(0)=a0ω, x″(0)=0$(56)

Selecting the fast and slow time variables and expansions as in the previous sections, the equations at each order are

$O(1):ω3D03x0+ω3D0x0=0 x0(0)=0, D0x0(0)=a0ω, D02x0(0)=0$(57)

$O(p):ω3D03x1+ω3D0x1=−3ω3D02D1x0−ω3D1x0+ω1ωD0x0−α^ω3(D0x0)3−β^ωx02D0x0x1(0)=0,(D0x1+D1x0)(0)=0(D02x1+2D0D1x0)(0)=0$(58)

$O(p):ω3D03x1+ω3D0x1=−3ω3D02D1x0−ω3D1x0+ω1ωD0x0−α^ω3(D0x0)3−β^ωx02D0x0x1(0)=0,(D0x1+D1x0)(0)=0(D02x1+2D0D1x0)(0)=0$(59)

The solution at the first order is

$x0=AeiT0+cc=acos(T0+λ)$(60)

Applying the initial conditions yield

$a(0)=−a0ω, λ(0)=π2$(61)

The first-order solution is substituted into (58) and secular terms are eliminated

$2D1A+iAω1ω2−iA2A¯(3α^ω2+β^ω2)=0$(62)

For D1A=0, ω1 is a real number, so this choice is appropriate

$D1A=0⇒A=A(T2)$(63)

$ω1=AA¯(3α^ω2+β^)$(64)

The solution at order p is

$x1=BeiT0+(β^−α^ω224ω2)A3e3iT0+cc$(65)

or

$x1=bcos(T0+θ)+a396ω2(β^−α^ω2)[cos(3T0+3λ)]$(66)

The initial conditions at this order imply

$b(0)=−a0332ω5(β^−α^ω2), θ(0)=π2$(67)

At the last order, (60) and (66) are inserted into (59) and secular terms are eliminated

$2D2A+ω2Aω2−iAA¯B(3α^ω2+β^ω2)−iA2B¯(3α^ω2+β^ω2) +iA3A¯2((9α^ω2−β^)(β^−α^ω2)24ω4)=0$(68)

According to the method, first D2A=0 should be tried. This choice leads to a real ω2 and therefore admissible. After algebraic manipulations, (68) yields

$a=−a0ω, b=−a03(β^−α^ω2)32ω5, θ=λ=π2$(69)

$ω2=12ab(3α^ω2+β^)+116a4((β^−9α^ω2)(β^−α^ω2)24ω2)$(70)

The approximate expansion is

$ω2=γ+pa024ω2(3α^ω2+β^)+p2a04384ω6(−9α^2ω4+2α^β^ω2+7β^2)$(71)

From which the frequency is solved

$ω=D4+κ2+12D22−4E3−43(E2−3DF−12G)+Γ2323Γ+D3−4DE+8F4κ$(72)

where

$D=γ+3αa024$(73)

$E=9α2a04384−βa024$(74)

$F=αβa04192$(75)

$G=7β2a04384$(76)

$Γ=(2E3−9DEF+27F2−27D2G+72EG+−4(E2−3DF−12G)3+(2E3−9DEF+27F2−27D2G+72EG)2)13$(77)

$κ=D24−2E3+23(E2−3DF−12G)3Γ+Γ323$(78)

The final solution in terms of this frequency is

$x(t)=−a0ωcos(ωt+π2)−a0396ω5(β−αω2)[3cos(ωt+π2) +cos(3ωt+3π2)]+O(p2)$(79)

## 3 Comparison of the Results

In this section, approximate analytical solutions of the MSLP method are compared with numerical solutions obtained by directly integrating the jerk functions using computational software ODE Workbench.

To verify the results, time histories and phase curves of the MSLP method are contrasted with the numerical solutions. Another task can be to compare exact and approximate periods.

## 3.1 Jerk Function Containing Displacement Times Velocity Times Acceleration and Velocity

The time histories and the phase curves of the present method are contrasted with the numerical simulations in Figures 16 . In all simulations, γ=1 and a0=1 are selected. In Figures 1 and 2, results are compared for δ=1. Figures demonstrate that MSLP solutions provide excellent approximations to the numerical solutions. Both solutions are indistinguishable. In Figures 36, the effect of cubic nonlinearity is amplified by increasing δ. In Figures, separation starts, but MSLP is in good agreement with the numerical solutions for strong nonlinearities.

Figure 1:

Comparison of the time histories of the MSLP method and numerical simulation for (3),

denotes numerical solution,
denotes MSLP method, δ=1, γ=1, a0=1.

Figure 2:

Comparison of the phase curves of the MSLP method and numerical simulation for (3),

denotes numerical solution, • • • • denotes MSLP method, δ=1, γ=1, a0=1.

Figure 3:

Comparison of the time histories of the MSLP method and numerical simulation for (3),

denotes numerical solution,
denotes MSLP method, δ=5, γ=1, a0=1.

Figure 4:

Comparison of the phase curves of the MSLP method and numerical simulation for (3),

denotes numerical solution, • • • • denotes MSLP method, δ=5, γ=1, a0=1.

Figure 5:

Comparison of the time histories of the MSLP method and numerical simulation for (3),

denotes numerical solution,
denotes MSLP method, δ=10, γ=1, a0=1.

Figure 6:

Comparison of the phase curves of the MSLP method and numerical simulation for (3),

denotes numerical solution, • • • • denotes MSLP method, δ=10, γ=1, a0=1.

Table 1 compares the second-order approximate periods of MSLP method (T2) and the previous approximate results with the exact period (Te). The relative errors are defined as (TTe/Te)×100. As can be seen from Table 1, the periods of the present (MSLP) method and the previous approximate results are in excellent agreement with exact periods for the wide range of initial velocity amplitude (a0).

Table 1:

Comparison of the periods of the MSLP method and the previous approximate results with exact ones (δ=1, γ=1).

## 3.2 Jerk Function Containing Velocity Times Acceleration-Squared and Velocity

Time histories and phase curves of the obtained solutions are contrasted with the numerical solutions. In all figures, γ=1 and a0=1 are selected. In Figures 7 and 8 , the numerical and the MSLP solutions are compatible for ε=1. In Figures 9 and 10 , ε=4 is selected. For this choice, separations are observed, MSLP being in a good agreement with the numerical solutions.

Figure 7:

Comparison of the time histories of the MSLP method and numerical simulation for (32),

denotes numerical solution,
denotes MSLP method, ε=1, γ=1, a0=1.

Figure 8:

Comparison of the phase curves of the MSLP method and numerical simulation for (32),

denotes numerical solution, • • • • denotes MSLP method, ε=1, γ=1, a0=1.

Figure 9:

Comparison of the time histories of the MSLP method and numerical simulation for (32),

denotes numerical solution,
denotes MSLP method, ε=4, γ=1, a0=1.

Figure 10:

Comparison of the phase curves of the MSLP method and numerical simulation for (32),

denotes numerical solution, • • • • denotes MSLP method, ε=4, γ=1, a0=1.

From Table 2, all approximate periods are in excellent agreement with the exact one for small initial velocity amplitude a0. With increasing a0, present approximate (MSLP) period is more accurate than previous approximate results.

Table 2:

Comparison of the periods of the MSLP method and the previous approximate results with exact ones (ε=1, γ=1).

## 3.3 Jerk Function Containing Velocity-Cubed, Velocity Times Displacement-Squared and Velocity

Time histories and phase curves of the MSLP and the numerical solutions are investigated here for the cases of both weak and strong nonlinearities. In all Figures, γ=1 and a0=1 are selected. For the case of weak nonlinearity, α=1 and β=1 are selected in Figures 11 and 12 . As expected, there is no separation between the approximate and numerical solutions. In Figures 1316 , the effect of cubic nonlinearity is amplified by increasing α and β. With increasing α in Figures 13 and 14 (α=100, β=1), the MSLP and the numerical solutions have an excellent agreement. In Figures 15 and 16, α=1 and β=100 are selected. Separations are observed, both solutions are compatible for increasing β. These figures show that MSLP solutions provide most excellent approximations to the numerical solutions for the case of weak and strong nonlinear systems.

Figure 11:

Comparison of the time histories of the MSLP method and numerical simulation for (52),

denotes numerical solution,
denotes MSLP method, α=1, β=1, γ=1, a0=1.

Figure 12:

Comparison of the phase curves of the MSLP method and numerical simulation for (52),

denotes numerical solution, • • • • denotes MSLP method, α=1, β=1, γ=1, a0=1.

Figure 13:

Comparison of the time histories of the MSLP method and numerical simulation for (52),

denotes numerical solution,
denotes MSLP method, α=100, β=1, γ=1, a0=1.

Figure 14:

Comparison of the phase curves of the MSLP method and numerical simulation for (52),

denotes numerical solution, • • • • denotes MSLP method, α=100, β=1, γ=1, a0=1.

Figure 15:

Comparison of the time histories of the MSLP method and numerical simulation for (52),

denotes numerical solution,
denotes MSLP method, α=1, β=100, γ=1, a0=1.

Figure 16:

Comparison of the phase curves of the MSLP method and numerical simulation for (52),

denotes numerical solution, • • • • denotes MSLP method, α=1, β=100, γ=1, a0=1.

In Table 3, second-order approximate periods of MSLP (T2) are contrasted with the exact and the previous approximate results obtained by Wu et al. [18], Hu [19], Hu et al. [20], Leung and Guo [21], Ma et al. [22], Ramos [24], and Feng and Chen [25]. Five different approximate periods are obtained by Ramos [24]. His best result is only given here. As can be seen from Table 3, the periods of the present (MSLP) method and the previous approximate results are in excellent agreement with the exact periods for both small and large initial velocity amplitudes.

Table 3:

Comparison of the periods of the MSLP method and the previous approximate results with exact ones (α=1, β=1, γ=0).

## 4 Concluding Remarks

The new perturbation method (MSLP) combining the multiple scales and the Lindstedt–Poincare method developed in [8] is applied to jerk functions for the first time. The method is applied to three different nonlinear jerk equations: (1) Jerk function containing displacement times velocity times acceleration and velocity $\left(x\stackrel{˙}{x}\stackrel{¨}{x}\right),$ (2) Jerk function containing velocity times acceleration-squared and velocity $\left(\stackrel{˙}{x}{\stackrel{¨}{x}}^{2}\right),$ and (3) Jerk function containing velocity-cubed $\left({\stackrel{˙}{x}}^{3}\right)$ and velocity times displacement-squared and velocity $\left(\stackrel{˙}{x}{x}^{2}\right).$ In all cases, MSLP method produced approximate solutions and periods with good agreement with the numerical solutions and the exact periods for strongly nonlinear systems.

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Accepted: 2017-04-06

Published Online: 2017-05-08

Published in Print: 2017-05-24

Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 6, Pages 547–557, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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©2017 Walter de Gruyter GmbH, Berlin/Boston.