Yusry O El-Dib

Department of Mathematics,Faculty of Education,Ain Shams University,Roxy,Cairo,Egypt

Abstract The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest method.An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach.The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order.A detailed solving process is given for the obtained oscillator with the traditional system.

Keywords:Van der Pol oscillator,Duffing jerk oscillator,fractional oscillator,non-perturbative approach,numerical simulation

1.Introduction

In the preceding few decades,non-integer order differential equations became an area of interest for researchers owing to their accuracy and applicability in various fields of physics,science,and technology.Many physical,dynamical,biological,and chemical phenomena are represented in a highly effective style by using differential equations having a noninteger rather than integer-order.Being a more accurate approach is the major reason for attracting the attention of researchers.Differential equations having fractional order are suitable for mathematicians,engineers,and physicists.The fractional-order differential equations have a large number of applications in several fields of science and technology,for example,porous media,rheology,optics,electromagnetism,electrochemistry,bio-science,bioengineering,medicine,geology,probability,statistics,etc.The fractional-order differential equations are also applicable in control theory,control of power electrons,tomography,polymer physics,polymer science,and neural networks.Furthermore,it has various applications in the modeling of other phenomena,such as the absorption of drugs in the bloodstream,seepage flow,traffic modeling of fluid dynamics,porous media,image processing,mathematical biology,genetic properties,and nonlinear oscillations resulting from earthquakes.These equations are also used for the calculations of genetically and chemically acquired properties of different materials and phenomena,(see for details[1–7]).

Several physical phenomena are modeled using systems of nonlinear fractional differential equations,which are more accurate for practical applications.Several real phenomena emerging in engineering and science fields can be successfully modeled by developing models which are more accurate than the integer ones[8–10].The essential quality of fractional-order differential equations is that it yields accurate and stable results.Some of these equations are time-fractional heat equations,time-fractional heat-like equations,time-fractional wave equations,time-fractional telegraphic equations,fractional-order oscillators such as the Van der Pol equation,and so on.These physical equations are performed as linear or nonlinear formulations;and since they have several employments in the field of applied engineering and science,then the analysis of solving such equations is very important.As these equations are generally strenuous to solve,many alternative and powerful methods have been expanded over the last few years.The main advantage of fractional order differential equations is that they produce accurate and stable results.Therefore,these equations represent a significant class of differential equations[11–18].

Fractional calculus(FC)represents a field of mathematics that discusses the non-integer order differentiation and integration and their implications on different physical systems.Generally,the physical situation might predicate its present state and its historical status,which can be confidently modeled by using the technology of FC tools[19,20].Accordingly,many analytic methods are derived to deduce exact,explicit,and numerical techniques for nonlinear fractional partial differential equations[21–25].A lot of fractional-order definitions;while Grunwald–Letnikov,Riemann–Liouville,and Caputo definitions are the majority used in FC research[26].

Schot[27]presented the definition of a jerk as referring to the rate of variation of acceleration.A jerk equation has vast applications in physics and daily life.It has been set to have numerous applications in various areas of science,such as laser physics electrical circuits,acoustics,dynamical processes,and mechanics[28–32].Jerk is also organized to be governed the flow of a thin-film viscous fluid with a free surface where the surface tension effects play a role typically leading to a third-order equation governing the form of the free surface of the fluid.Jerk also plays a mighty role in the physiological balancing of the human body.

Another employment of jerk is in the accelerated charged particle which sends off radiation,which is correlated to the jerk[33,34].In most cases,it is totally hard to find the exact solutions to nonlinear problems,where estimated solutions to the fractional nonlinear problems are considered.

Over the last two decades,there was significant progress in the area of fractional differential equations.Many efforts have been done to the existence of solutions to fractional differential equations[35].Since the fractional nonlinear operators have a vital role in differential equations,the investigation to find a simple approach alternative to the mathematical hard work is urgent use.One of the several significant operators used to simplify the procedure of nonlinear differential equations is the equivalent linearized method which leads to obtaining the solution easily.A considerable number of valuable research articles can be obtained in the literature regarding this topic,(see for detail[36–38]).

The fractional Van der Pol–Duffing jerk vibration is investigated to give a more perfect description and shed more light on the dynamics of the suggested oscillator.Although the suggested oscillator is simple,it improves complex and noticeable phenomena such as symmetry-breaking bifurcation,bistability,reverses period-doubling,symmetry-restoring crisis,and coexisting attractors[39].In the present proposal,the linearized equivalent method has been derived or developed to be consistent with solving the fractional jerk oscillator.Afterward,the proposed method is applied to obtain an analytical solution to different cases of the fractional Van der Pol–Duffing jerk oscillator.The technique consists of obtaining the equivalent linearized form with an integer order employing the principle of minimum mean-square error.Then the solution will be available.

2.The methodology

The present section describes the purpose of the fractional jerk oscillator and explains how to find the solution most simply.The simple effective approach is adopted herein utilizing the equivalent linearization technique to solve the fractional Duffing jerk oscillation of the form

3.The case of the Duffing jerk oscillator with the fractional damping forces

3.1.Solution utilizing the modified HPM coupling with He’s frequency formula

4.The case of the Van der Pol–Duffing jerk oscillator in its fractional-order form

It is noted that asε→0 into(54)and(55)the results reduce to those obtained before in(31)and(32).

Thirdly,the case of the full fractional orders form is discussed in the next section:

5.Van der Pol–Duffing jerk in full fractional order

In this section,an important cause of the nonlinear jerk oscillation with the full fractional order is considered in the form

Because of the approximate relations(21)–(23)and by utilizing He’s frequency formula,the above equation can be sought in the equivalent linearized form as

The solution of the linear jerk oscillator(57)is still given by the solution(28)except that the frequency Ω and the parameterφhave been replaced by

The above equation(60)is the fractional frequency-amplitude formula of the full fractional Duffing jerk oscillator.

6.The equivalent linearized approach to reducing the rank of the jerk oscillator

In this section,the effort in converting the third-order oscillator into an equivalent linear second-order oscillator is interesting.This aim is done for the first time,and the linear equivalence method can be used to reduce the rank of the oscillation.The new approach will be applied to the oscillator given in(4).To illustrate the procedure,equation(4)can be arranged in the form As mentioned before the total frequency is selected to be Ω.The application of the equivalent linearized technique transforms the above equation to

whereω3is the He’s frequency formula as given in(27),while the coefficientceqis estimated as

At this stage,equation(63)can be solved as a second-order linear equation that has solution arises in the form

where the frequency Ω is given by

This can be read as

It is noted that this approach can be used when the coefficientηhas a larger value than the other parameters of the oscillator(56).

7.Numerical simulation

To have a clear and proper understanding of the properties of the present approach,an illustrative numerical interpolation is done.The comparison of the obtained results in this paper with the numerical solution obtained by the Mathematica software is investigated.The comparison of the quasi-exact solution performed in(28)and the homotopy perturbation solution given by(46)with the numerical solution of the traditional integer-order is studied herein for the case of(i.e.equation(4)),to illustrate how the equivalent method of the fractional-order is close to the numerically exact solution.Two systems of the numerical values of the given coefficients are considered.One of them produces a growing behavior in the time-history profile,while the other deals with the damping behavior in the time-history description.The comparison of the behavior of the influence of0＜α＜1 is between the quasi-exact solution and the solution obtained by HPM.

In figure 1,the numerical solution of equation(4),the analytical solution(46)with HPM,and the quasi-exact solution(28)are plotted together,for the case ofε= 0,in one graph for the system ofη=μ=Q =ω0=A =α=1.The numerical solution is plotted by the solid red line;the solution of HPM is drawn by the green dashed line;the quasiexact solution is displayed by the blue dotted line.This graph shows that there is an excellent agreement among the three different solutions for a system producing a growing behavior.When the system is selected to produce a damping behavior,the same conclusion is found as found in figure 2.The system used to produce the graph in figure 2 isη=μ= 1.3,ε= 0,Q =ω0= A =α=1.The variation in the fractional-orderαof the solution(28)has been collected together in figure 3,for the same system given in figure 2.It is observed from the investigation of this graph that the damping behavior for the caseα= 1 still occurs with an increase in its amplitude due to a small decrease in the values ofα.Continue in decreasing ofαleads to a dramatic change in the damping behavior for the oscillation.In which the growing behavior in the oscillation amplitude is observed.

It is worth noting that the forgoing graphs are plotted for the solution(28)in the case ofε= 0.In figures 4–6,the calculations should deal with the case ofε＞0.Figure 4 is plotted using the same system given in the graph of figure 1 except thatε= 1.The comparison between the two graphs of figures 1 and 4 shows that replacingε= 0 byε= 1 leads to an increase in the displacement scale of the growth behavior.Figure 5 is plotted for the same system given in figure 2 but withε= 1.The comparison between these figures shows a dramatic change occurs.The time-history curve starts as damping behavior and then changes to behave as a growing behavior.The examination of the variation inεhas been demonstrated in figure 6.In this graph,the solution(28)is plotted for the same system considered in figure 2 with different valuesεto illustrate the influence ofε.It is worthwhile to note that the system is given in figure 6 starts with damping behavior atε= 0.It is observed that the amplitude of the damping influence has increased with the increase inε.Atε= 0.5,the oscillation becomes with constant amplitude and there is no damping oscillation as shown in figure 7.Whenεbecomesε= 0.5,the oscillation behaves with the same amplitude as a non-damped behavior.After this state ofε= 0.5,the growing behavior in the amplitude of the oscillation will be increased.This behavior should be increased more rapidly for more increase inε.This means that there are three cases observed for the variation ofε,namely,the damping behavior in the interval ofno damped at,and growing behavior in the interval of

Figure 1.Comparison of the quasi-exact solution(28)(Blue-dashing curve)with the numerical solution of equation(4),for the system ofη = μ =Q = ω 0 =A = α=1.

Figure 2.Comparison of the quasi-exact solution(28)with the numerical solution for the system ofη = 1.2,μ = 1.3,ε=0,Q= ω 0 =A = α=1.

Figure 3.The variation of the parameter α in the profile of the timehistory for the same system is considered in figure 2.

Figure 4.The comparison of two solutions as given in figure 1 except thatε = 1.

Figure 5.The comparison of the two solutions as given in figure 2 except thatε = 1.

Figure 6.Influence of the variationε on the time-history profile for the same system of figure 2.

Figure 7.Influence ofε = 0.5 on the time-history profile for the same system of figure 2.

Figure 8.Comparison of the numerical solution of equation(56)versus its quasi-exact solution(28)for the same system of figure 2 with the variation in the parameter α withε = 0.5.

Figure 9.Comparison of the numerical solution of equation(4)versus its reduced equation(63)for the system ofη = 10,μ =ω0= Q = A=1.

Figure 8 is plotted to examine the solution of equation(56)with the variation in the parameterα.The comparison with its numerical solution is done for the case ofα= 1 withε= 0.5.The remaining parameters are as given in figure 2.This graph shows that the decrease inαhas a contribution to the damping influence of the jerk oscillator.

Figure 9 is graphed to clarify the new feature that reduces the rank of the jerk oscillator from the third rank to the second rank.The numerical solution of the original third-order equation(4)was compared with the new analytical solution(65)of the second-order equation(63),taking into account thatThis graph shows that the solution of the reduced equation(63)becomes an excellent equivalent to the original equation(4).

8.Conclusion

A qualitative study of the efficiency of the simplest analysis of the differential equations based on fractional-order has been considered in the present proposal.The solution of the fractional nonlinear oscillator is obtained by converting it to the linear ordinary differential equation of integer order.This aim is accomplished using the linearized equivalent approach.In the reduced equations,the solution is obtained easily.It has been utilized for the time-fractional Van der Pol–Duffing jerk oscillator.The validation of the proposed technique is emulated with the numerical simulation of the oscillator with integer orders.The obtained solutions might be of huge importance in several areas of applied mathematics in elucidating some physical phenomena.Furthermore,the present approach can be used to reduce the rank of the jerk oscillator under certain conditions.The simplicity of the present approach provides extra advantages for fractional-order solutions.

Competing interests

The author declares that there are no competing interests regarding the publication of the present paper.

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