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Research article Special Issues

Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription

  • Received: 10 January 2019 Accepted: 02 April 2019 Published: 03 July 2019
  • Transcription is a pivotal step in gene expression yet a complex biochemical process. It occurs often in a bursty manner, leading to cell-to-cell variability important for the cells surviving in complex environments. Quantitative experiments of such stochastic transcription call for model analysis in a systematic rather than case-by-case fashion. Here we analyze two general yet biologically-reasonable classes of stochastic transcription models: the first class called promoter models considers that promoter structure is general, and the second class called queuing models considers that waiting-time distributions are general. For the former, we show that there are conjugate relationships between models with different transcription exits. This property well reveals the mechanic principle of stochastic transcription in complex cases of transcription factor regulation. For the latter, we establish an integral equation for the mRNA moment-generating function, through which stationary mRNA moments of any orders can be analytically derived. Finally, we analyze parametric regions for robust unimodality and bimodality. The overall analysis not only lays a foundation for quantitative analysis of stochastic transcription but also reveals how the upstream promoter kinetics impact the downstream expression dynamics.

    Citation: Jiajun Zhang, Tianshou Zhou. Stationary moments, distribution conjugation and phenotypic regions in stochastic gene transcription[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 6134-6166. doi: 10.3934/mbe.2019307

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  • Transcription is a pivotal step in gene expression yet a complex biochemical process. It occurs often in a bursty manner, leading to cell-to-cell variability important for the cells surviving in complex environments. Quantitative experiments of such stochastic transcription call for model analysis in a systematic rather than case-by-case fashion. Here we analyze two general yet biologically-reasonable classes of stochastic transcription models: the first class called promoter models considers that promoter structure is general, and the second class called queuing models considers that waiting-time distributions are general. For the former, we show that there are conjugate relationships between models with different transcription exits. This property well reveals the mechanic principle of stochastic transcription in complex cases of transcription factor regulation. For the latter, we establish an integral equation for the mRNA moment-generating function, through which stationary mRNA moments of any orders can be analytically derived. Finally, we analyze parametric regions for robust unimodality and bimodality. The overall analysis not only lays a foundation for quantitative analysis of stochastic transcription but also reveals how the upstream promoter kinetics impact the downstream expression dynamics.


    Nonlinear oscillators have been widely used in various engineering and applied sciences, such as mathematics, physics, structural dynamics, mechanical engineering and other related fields of science [1,2,3,4,5,6]. Nonlinear differential equations (NDEs) can model many phenomena in various scientific aspects to present their effects and behaviors through mathematical principles. Perturbation methods are extremely beneficial when the nonlinear response is small [7,8,9,10]. In general, solving strongly nonlinear differential equations is very difficult. In [11], Mickens suggested an approximate expression for solving a truly nonlinear Duffing oscillator. Recently, various powerful analytical and numerical approximation techniques have been suggested for dealing with nonlinear oscillator differential equations. These include He's frequency-amplitude formulation [12], the harmonic balance method [13,14], the straightforward frequency prediction method [15], the modified harmonic balance method [16,17], the energy balance method [18,19,20], the homotopy perturbation method [21,22,23,24,25], the Hamiltonian approach [26,27], the weighted averaging method [28], the global residue harmonic balance method [29,30,31], the max-min approach [32,33], Newton's harmonic balance method [34], the variational iteration method [35], the parameter-expansion method [36], the Lindstedt-Poincaré method [37,38] and the global error minimization method [39,40,41,42].

    The global error minimization method (GEMM) is one of the most frequently used techniques for dealing with the solutions of nonlinear oscillators, as it provides more accurate results valid for both weakly and strongly nonlinear oscillators than other known methods [39,40,41,42]. In the previous example, the solution up to the first approximation is calculated.

    In this study, we extend and improve the global error minimization method up to a third order approximation to achieve the higher-order analytical solution of strongly nonlinear Duffing-harmonic oscillators. The present method is applied for two different problems, and the analytical results show that the modified global error minimization method (MGEMM) has better agreement with numerical solutions than the other analytical methods. Excellent agreement is observed between the approximate and exact solutions even for large amplitudes of the oscillation. Comparing the exact solutions with the approximate results has proved that the MGEMM is quite an accurate method in strongly nonlinear oscillator systems.

    To describe the proposed modified global error minimization method (MGEMM), we consider general second order nonlinear oscillator differential equations as follows:

    ¨u+F(˙u,u,t)=0,u(0)=A,˙u(0)=0. (1)

    We introducing E(u)as a new function, defined as in [40,43], in the following form:

    E(u)=T0(¨u+F(˙u,u,t))2dt,T=2πω1. (2)

    By assuming that F(u) is an odd function, a general n-th order trial function of Eq (1) can be expressed as a sum of trigonometric functions as follows:

    u(t)=n=0(a2n+1)cos((2n+1)ωt)), (3)

    where a(2n+1) are unknown constant values which satisfy the relation

    A=n=0a(2n+1). (4)

    The following conditions were used to obtain the unknown parameters (i.e., a(2n+1) and ω):

    E(u)ω=0,E(u)a(2n+1)=0,n1. (5)

    By solving the n Eq (5) with the aid of Eq (4), the constants a1,a3,a5 and the frequency of vibration ω are obtained.

    In this section, two practical examples of nonlinear Duffing-harmonic oscillators are illustrated to show the effectiveness, accuracy and applicability of the proposed approach.

    In this application, we consider the following nonlinear Duffing-harmonic oscillator:

    ¨u+k1u+k3u1+k2u2=0,u(0)=A,˙u(0)=0, (6)

    where dots denote differentiation with respect to t. Now, we shall study some different relevant cases considering Eq (6).

    First, we consider k1=1, k2=1 and k3=1 in Eq (6). Then, we have a nonlinear oscillator system having an irrational elastic item [12,16].

    ¨u+u+u1+u2=0,u(0)=A,˙u(0)=0. (7)

    According to the basic idea of the global error minimization method, the minimization problem of Eq (7) is

    E(u)=T0(¨u+u+u1+u2)2dt,T=2π/ω. (8)

    The first-order approximate solution for Eq (7) can be represented as a trial function in the form

    u1(t)=a1cos(ωt). (9)

    Substituting Eq (9) into Eq (8) and choosing a1=A, it follows that

    E(u1)=4A2πω+3A4πω+5A6π8ω4A2πω92A4πω54A6πω+A2πω3+32A4πω3+58A6πω3=0. (10)

    Applying E(u1)/ω=0, the frequency of the nonlinear oscillator is obtained as follows:

    ω=ω1=16+18A2+5A4+2256+576A2+487A4+180A6+25A83(8+12A2+5A4). (11)

    In order to illustrate the capacity of the global error minimization method, the second-order approximation is applied to the Duffing-harmonic oscillator by using the following new trial solution:

    u2(t)=a1cos(ωt)+a3cos(3ωt), (12)

    where A=a1+a3. By substituting Eq (12) into Eq (8), we have

    E(u2)=π8ω(5(ω21)2a61+5(11ω414ω2+3)a51a3+2a31a3(8(29ω2+5ω4)+3(550ω2+109ω4)a23)+3a41(4(23ω2+ω4)+(15110ω2+159ω4)a23)+a23(8(29ω2)2+12(227ω2+81ω4)a23+5(19ω2)2a43)+a21(8(2+ω2)2+16(645ω2+59ω4)a23+3(15190ω2+559ω4)a43))=0.) (13)

    Setting E(u2)/ω=0 and E(u2)/a3=0 leads to

    π8ω(20ω(1+ω2)a61+5(28ω+44ω3)a51a3+2a31a3(8(18ω+20ω3)+3(100ω+436ω3)a23+3a41(4(6ω+4ω3)+(220ω+636ω3)a23)+a23(288ω(29ω2)+12(54ω+324ω3)a23180ω(19ω2)a43)+a21(32ω(2+ω2)+16(90ω+236ω3)a23+3(380ω+2236ω3)a43))18ω2π(5(1+ω2)2a61+5(314ω2+11ω4)a51a3+2a31a3(8(29ω2+5ω4)+3(550ω2+109ω4)a23)+3a41(4(23ω2+ω4)+(15110ω2+159ω4)a23)+a23(8(29ω2)2+12(227ω2+81ω4)a23+5(19ω2)2a43)+a21(8(2+ω2)2+16(645ω2+59ω4)a23+3(15190ω2+559ω4)a43))=0,) (14)
    π8ω(5(314ω2+11ω4)a51+6(15110ω2+159ω4)a41a3+12(550ω2+109ω4)a31a23+2a31(8(29ω2+5ω4)+3(550ω2+109ω4)a23)+a23(24(227ω281ω4)a3+20(19ω2)2a33)+a21(32(645ω2+59ω4)a3+12(15190ω2+559ω4)a33)+2a3(8(29ω2)2+12(227ω2+81ω4)a23+5(19ω2)2a23))=0.) (15)

    For a known amplitude, the parameters of a1, a3 and angular frequency ω can be obtained by using the condition A=a1+a3 and solving Eqs (14) and (15). The computations were performed using the Mathematica software program, version 9.

    To illustrate the capacity of this method, the third order approximation is applied by using the following trial function:

    u3(t)=a1cos(ωt)+a3cos(3ωt)+a5cos(5ωt), (16)

    where A=a1+a3+a5. Bringing Eq (16) into Eq (8) results in

    E(u3)=π8ω(5a61(ω21)2+a51(ω21)(5a3(11ω23)+3a5(9ω21))+5a63(19ω2)2+2a31a3(3a23(109ω450ω2+5)+3a5a3(663ω4230ω2+15)+8(5ω49ω2+2)+10a25(251ω462ω2+3))+3a43(a25(2911ω4430ω2+15)+4(81ω427ω2+2))+2a1a23a5(a23(2894ω4620ω2+30)+15a5a3(153ω426ω2+1)+8(287ω499ω2+6)+3a25(3399ω4470ω2+15))+a25(5a45(125ω2)2+12a25(625ω475ω2+2)+8(225ω2)2)+a23(3a45(5631ω4590ω2+15)+16a25(803ω4153ω2+6)+8(29ω2)2)+a23(3a45(5631ω4590ω2+15)+16a25(803ω4153ω2+6)+8(29ω2)ω2)2+3a25(277ω490ω2+5)))+a21(3a43(559ω4190ω2+15)+18a5a33(341ω490ω2+5)+8(ω22)2)+9a45(1317ω4170ω2+5)+48a25(121ω439ω2+2)4a23(3a25(1743ω4350ω2+15)+4(59ω445ω2+6))+6a3a5(a25(2719ω4430ω2+15)+8(49ω427ω2+2))))=0.) (17)

    Applying E(u3)/ω=0, E(u3)/a3=0 and E(u3)/a5=0 yields

    π8ω(20a61ω(ω21)+a51(ω21)(110a3ω+54a5ω)+2a51ω(5a3(11ω23)+3a5(9ω21))+3a5(9ω21))+2a31a3(3a23(436ω3100ω)+3a5a3(2652ω3460ω)+8(20ω318ω)+10a25(1004ω3124ω))+3a43(a25(11644ω3860ω)+4(324ω354ω))+2a1a23a5(a23(11576ω31240ω)+15a5a3(612ω352ω)+8(1148ω3198ω)+3a25(13596ω3940ω))+a25(12a25(2500ω3150ω)800ω(225ω2)500a45ω(125ω3))+a23(3a45(22524ω31180ω)+16a25(3212ω3306ω)288ω(29ω2))+a41(a23(1908ω3660ω)+4a5a3(1468ω3380ω)+3(3a25(1108ω3180ω)+4(4ω36ω)))a21(3a43(2236ω3380ω)+18a5a33(1364ω3180ω)+48a25(484ω378ω)+32ω(ω22)+9a45(5268ω3340ω)+4a23(3a25(6972ω3700ω)+4(236ω390ω))+6a3a5(a25(10876ω3860ω)+8(196ω354ω))))π8ω2(5a61(ω21)2+a51(ω21)(5a3(11ω23)+3a5(9ω21))+5a63(19ω2)2+2a31a3(3a23(109ω450ω2+5)+3a5a3(663ω4230ω2+15)+8(5ω49ω2+2)+10a25(251ω462ω2+3))+3a43(a25(2911ω4430ω2+15)+4(81ω427ω2+2))+2a1a23a5(a23(2894ω4620ω2+30)+15a5a3(153ω426ω2+1)+8(287ω499ω2+6)+3a25(3399ω4470ω2+15))+a25(5a43(125ω2)2+12a25(625ω475ω2+2)+8(225ω2)2)+a23(3a45(5631ω4590ω2+15)+16a25(803ω4153ω2+6)+8(29ω2)2)+a23(3a45(5631ω4590ω2+15)+16a25(803ω4153ω2+6)+8(29ω2)2)+3a25(277ω490ω2+5)))+a21(3a43(559ω4190ω2+15)+8(ω22)2+9a45(1317ω4170ω2+5)+48a25(121ω439ω2+2)+18a33a5(341ω490ω2+5)+4a23(3a25(1743ω4350ω2+15)+4(59ω445ω2+6))+6a3a5(a25(2719ω4430ω2+15)+8(49ω427ω2+2))))=0,) (18)
    π8ω(5a51(ω21)(11ω23)+30a53(19ω2)2+2a1a23a5(2a3(2894ω4620ω2+30)+15a5(153ω426ω2+1))+a41(2a3(477ω4330ω2+45)+4a5(367ω4190ω2+15))+2a31a3(6a3(109ω450ω2+5)+3a5(663ω4230ω2+15))+2a31(8(5ω49ω2+2)+3a23(109ω450ω2+5)+3a5a3(663ω4230ω2+15)+10a25(251ω462ω2+3))+12a33(a25(2911ω4430ω2+15)+4(81ω427ω2+2))+4a1a3a5(8(287ω499ω2+6)+a23(2894ω4620ω2+30)+15a5a3(153ω426ω2+1)+3a25(3399ω4470ω2+15))+2a3(3a45(5631ω4590ω2+15)+16a25(803ω4153ω2+6)+8(29ω2)2)+a21(12a33(559ω4190ω2+15)+54a23a5(341ω490ω2+5)+8a3(4(59ω445ω2+6)+3a25(1743ω4350ω2+15))+6a5(a25(2719ω4430ω2+15)+8(49ω427ω2+2))))=0,) (19)
    π8ω(3a51(ω21)(9ω21)+6a43a5(2911ω4430ω2+15)+2a31a3(3a3(663ω4230ω2+15)+20a5(251ω462ω2+3))+a41(4a3(367ω4190ω2+15)+18a5(277ω490ω2+5))+2a1a23a5(15a3(153ω426ω2+1)+6a5(3399ω4470ω2+15))+2a1a23(8(287ω499ω2+6)+a23(2894ω4620ω2+30)+15a5a3(153ω426ω2+1)+3a25(3399ω4470ω2+15))+a25(20a35(125ω2)2+24a5(625ω475ω2+2))+a23(32a5(803ω4153ω2+6)+12a35(5631ω4590ω2+15))+2a5(12a25(625ω475ω2+2)+8(225ω2)2+5a45(125ω2)2)+a21(18a33(341ω490ω2+5)+96a5(121ω439ω2+2)+36a35(1317ω4170ω2+5)+12a3a25(2719ω4430ω2+15)+24a23a5(1743ω4350ω2+15)+6a3(a25(2719ω4430ω2+15)+8(49ω427ω2+2))))=0.) (20)

    Now, by solving Eqs (18)–(20) and applying the condition A=a1+a3+a5, the parameters a1,a3,a5 and the angular frequency ω can be obtained for the known amplitude A, using the Mathematica software program, version 9. To examine the accuracy of the MGEMM solutions, the obtained results are compared with the frequency-amplitude formulation (FAF) [12], the energy balance method (EBM) [19], the modified harmonic balance method (MHBM) [16] and the exact solutions, as presented in Table 1 and Figure 1. We conclude that the third order approximation provides an excellent accuracy with respect to the exact numerical solutions.

    Table 1.  Comparison of the approximate analytical frequencies with the exact solutions.
    A ωFAF ωEBM ωMHBM ω3rdGEMM ωexact
    [12] [19] [16] present Exact
    0.01 1.41419 1.41419 1.41419 1.41419 1.41419
    0.1 1.41158 1.41158 1.41158 1.41158 1.41158
    0.2 1.40388 1.40389 1.40390 1.40390 1.40390
    0.4 1.37581 1.37595 1.37616 1.37616 1.37616
    0.6 1.33694 1.33743 1.33827 1.33827 1.33827
    0.8 1.29448 1.29550 1.29744 1.29743 1.29743
    1 1.25375 1.25514 1.25845 1.25840 1.25842
    5 1.02500 1.02588 1.03148 1.02945 1.03139
    10 1.00656 1.00681 1.00895 1.00790 1.00893
    100 1.00007 1.0000 1.00010 1.00010 1.00010
    1000 1.00000 1.00000 1.00000 1.00000 1.00000

     | Show Table
    DownLoad: CSV
    Figure 1.  Comparison of the approximate solution (red line) with the numerical solution (blue line).

    Now, if we put k1=0, k2=1 and k3=1 in Eq (6), we obtain the following nonlinear oscillator, in which the restoring force has a rational expression [21,26].

    ¨u+u1+u2=0,u(0)=A,˙u(0)=0. (21)

    Using the previously mentioned procedure, the solution up to a third-order approximation is calculated. Depending on the analytical approximation, first, second or third, the approximate solution is assumed in the forms of (9), (12) and (16), respectively.

    Finally, as in Case 1, the third order approximate solutions are compared with the homotopy perturbation method (HPM) [21], the Hamiltonian approach (HA) [26] and the exact solutions, as displayed in Table 2.

    Table 2.  Comparison of the approximate analytical frequencies with the exact solutions.
    A ωHPM ωHA ω3rdGEMM ωexact
    [21] [26] present Exact
    0.01 0.999963 0.9999625 0.999963 0.99999
    0.1 0.996271 0.99627403 0.996273 0.9991208
    1 0.755929 0.765366864 0.761539 0.76157808
    10 0.114708 0.13420106 0.11948 0.123322
    100 0.0115462 0.01407125 0.0120357 0.0125265

     | Show Table
    DownLoad: CSV

    In the second application, we will consider the following nonlinear Duffing-harmonic oscillator [18]:

    ¨u+k1u+k3u31+k2u2=0,u(0)=A,˙u(0)=0, (22)

    where dots denote differentiation with respect to t. Now, we consider some following cases to compare the present solutions with published solutions using different approximate analytical methods.

    First, we consider k1=1, k2=1 and k3=1 in Eq (22). Then, we have the following nonlinear Duffing-harmonic equation [18,28]:

    ¨u+u+u31+u2=0,u(0)=A,˙u(0)=0. (23)

    The minimization problem is

    E(u)=T0(¨u+u+u31+u2)2dt,T=2πω. (24)

    For the first-order approximation, assume that the trial function is given by

    u1(t)=a1cos(ωt), (25)

    where A=a1. By inserting Eq (25) into Eq (24), we obtain

    E(u1)=A2πω+3A4πω+5A6π2ω2A2πω92A4πω52A6πω+A2πω3+32A4πω3+58A6πω3=0. (26)

    The frequency can be found through the condition E(u1)/ω=0, as follows:

    ω=ω1=8+18A2+10A4+264+288A2+487A4+360A6+100A83(8+12A2+5A4). (27)

    To improve the analytical approximation, we add additional terms to the trial function:

    u2(t)=a1cos(ωt)+a3cos(3ωt). (28)

    The constraint of this minimization is A=a1+a3. Substituting the above new trial function into Eq (24), we obtain

    E(u2)=π8ω(5a61(ω22)2+5a51a3(11ω428ω2+12)+3a41(4(ω43ω2+2)+a23(159ω4220ω2+60))+2a31a3(a23(327ω4300ω2+60)+8(5ω49ω2+2))+a23(5a43(29ω2)2+12a23(81ω427ω2+2)+8(19ω2)2)+a21(3a43(559ω4380ω2+60)+16a23(59ω445ω2+6)+8(ω21)2))=0.) (29)

    By using E(u2)/ω=0 and E(u2)/a3=0, it follows that

    π8ω(5a3a51(44ω356ω)+3a41(a23(636ω3440ω)+4(4ω36ω))+20a61ω(ω22)+2a31a3(a23(1308ω3600ω)+8(20ω318ω))+a23(12a23(324ω354ω)288ω(19ω3)180a43ω(29ω3))+a21(3a43(2236ω3760ω)+16a23(236ω390ω)+32ω(ω21)))π8ω2(5a61(ω22)2+5a3a51(11ω428ω2+12)+3a41(a23(159ω4220ω2+60)+4(ω43ω2+2))+2a31a3(a23(327ω4300ω2+60)+8(5ω49ω2+2))+a23(12a23(81ω427ω2+2)+8(19ω2)2+5a43(29ω2)2)+a21(3a43(559ω4380ω2+60)+16a23(59ω445ω2+6)+8(ω21)2))=0,) (30)
    π8ω(5a51(11ω428ω2+12)+6a3a41(159ω4220ω2+60)+4a23a31(327ω4300ω2+60)+2a31(a23(327ω4300ω2+60)+8(5ω49ω2+2))+a23(24a3(81ω427ω2+2)+20a33(29ω2)2)+a21(12a33(559ω4380ω2+60)+32a3(59ω445ω2+6))+2a3(5a43(29ω2)2+12a23(81ω427ω2+2)+8(19ω2)2))=0.) (31)

    The minimization problem's conditions can be easily achieved by replacing a1=Aa3, and the parameters a1, a3 and angular frequency ω can be obtained for a known amplitude A.

    To show the accuracy of the MGEM method in higher order approximations, we apply the third order approximation and consider the following trial function:

    u3(t)=a1cos(ωt)+a3cos(3ωt)+a5cos(5ωt). (32)

    Using Eq (32) as the trial function in Eq (24), where A=a1+a3+a5, leads to

    E(u3)=πω8(5a61(ω24)+a51(5a3(11ω228)+3a5(9ω220))+45a63(9ω24)+2a31a3(a23(327ω2300)+10a25(251ω2124)+3a3a5(663ω2460)+40ω2)+a41(a23(477ω2660)+9a25(277ω2180)+4a3a5(367ω2380)+12ω2)+3a43(a25(2911ω2860)+324ω2)+2a1a23a5(2a23(1447ω2620)+2296ω2+3a25(3399ω2940)+15a3a5(153ω252))+125a25(60a25ω2+40ω2+a45(25ω24))+a21(3a43(559ω2380)+944a23ω2+8ω2+6a3a5(3a23(341ω2180)+392ω2)+12a25(7a23(249ω2100)+484ω2)+9a45(1317ω2340)+6a3a35(2719ω2860))+a23(12848a25ω2+648ω2+3a45(5631ω21180)))=0.) (33)

    By setting E(u3)/ω=0, E(u3)/a3=0 and E(u3)/a5=0, we obtain

    18ωπ(20a61ω(ω22)180a63ω(29ω2)+a51(ω22)(110a3ω+54a5ω)+2ω3a51(5a3(11ω26)+3a5(9ω22))+3a43(a25(11644ω31720ω)+4(324ω354ω))+2a1a23a5(8(1148ω3198ω)+2a23(5788ω31240ω)+15a5a3(612ω3104ω)+3a25(13596ω31880ω))+a25(800ω(125ω2)+12a25(2500ω3150ω)500a45ω(225ω2))+a23(16a25(3212ω3306ω)288ω(19ω2)+3a45(22524ω32360ω))+2a31a3(a23(1308ω3600ω)+3a5a3(2652ω3920ω)+2(5a25(1004ω3248ω)+4(20ω318ω)))+a41(3a23(636ω3440ω)+4a5a3(1468ω3760ω)+3(a25(3324ω31080ω)+4(4ω36ω)))+a21(3a43(2236ω3760ω)+32ω(ω21)+18a33a5(1364ω3360ω)+48a25(484ω378ω)+9a45(5268ω3680ω)+4a23(4(236ω390ω)+18a33a5(1364ω3360ω)+48a25(484ω378ω)+9a45(5268ω3680ω)+4a23(4(236ω390ω)1π8ω2)5a61(ω22)2+a51(ω22)(5a3(11ω26)+3a5(9ω22))+5a63(29ω2)2+3a43(a25(2911ω4860ω2+60)+4(81ω427ω2+2))+2a1a23a5(8(287ω499ω2+6)+2a23(1447ω4620ω2+60)+15a5a3(153ω452ω2+4)+3a25(3399ω4940ω2+60))+a25(5a45(225ω2)2+12a25(625ω475ω2+2)+8(125ω2)2)+a23(8(19ω2)2+3a45(5631ω41180ω2+60)+16a25(803ω4153ω2+6))+2a31a3(a23(327ω4300ω2+60)+3a3a5(663ω4460ω2+60)+2(5a25(251ω4124ω2+12)+4(5ω49ω2+2)))+a41(3a23(159ω4220ω2+60)+4a5a3(367ω4380ω2+60)+3(4(ω43ω2+2)a25+(831ω4540ω2+60)))+a21(3a43(559ω4380ω2+60)+8(ω21)2+9a45(1317ω4340ω2+20)+48a25(121ω439ω2+2)+18a33a5(341ω4180ω2+20)+4a23(3a25(1743ω4700ω2+60)+4(59ω445ω2+6))+6a3a5(8(49ω427ω2+2)+a25+(2719ω4860ω2+60))))=0,) (34)
    π8ω(3a51(ω22)(9ω22)+6a43a5(2911ω4860ω2+60)+2a31a3(3a3(663ω4460ω2+60)+20a5(251ω4124ω2+12))+a41(4a3(367ω4380ω2+60)+6a5(831ω4540ω2+60))+2a1a23a5(15a3(153ω452ω2+4)+6a5(3399ω4940ω2+60))+2a1a23(8(287ω499ω2+6)+2a23(1447ω4620ω2+60)+15a5a3(153ω452ω2+4)+3a25(3399ω4940ω2+60))+a25(20a35(225ω2)2+24a5(625ω475ω2+2))+a23(32a5(803ω4153ω2+6)+12a35(5631ω41180ω2+60))+2a5(12a25(625ω475ω2+2)+8(125ω2)2+5a45(225ω2)2)+a21(18a33(341ω4180ω2+20)+96a5(121ω439ω2+2)+24a5a23(1743ω4700ω2+60)+12a25a3(2719ω4860ω2+60)+36a35(1317ω4340ω2+20)+6a3(a25(2719ω4860ω2+60)+8(49ω427ω2+2))))=0,) (35)
    18ωπ(5a51(ω22)(11ω26)+30a53(29ω2)2+2a1a23a5(4a3(1447ω4620ω2+60)+15a5(153ω452ω2+4))+a41(6a3(159ω4220ω2+60)+4a5(367ω4380ω2+60))+2a31a3(2a3(327ω4300ω2+60)+3a5(663ω4460ω2+60))+12a33(4(81ω427ω2+2)+a25(2911ω4860ω2+60))+4a1a3a5(2a23(1447ω4620ω2+60)+8(287ω499ω2+6)+3a25(3399ω4940ω2+60)+15a3a5(153ω452ω2+4))+2a3(8(19ω2)2+3a45(5631ω41180ω2+60)+16a25(803ω4153ω2+6))+2a31(a23(327ω43300ω2+60)+3a3a5(663ω4460ω2+60)+2(5a25(251ω4124ω2+12)+4(5ω49ω2+2)))+a21(12a33(559ω4380ω2+60)+54a23a5(341ω4180ω2+20)+8a3(4(59ω445ω2+6)+3a25(1743ω4700ω2+60))+6a5(a25(2719ω4860ω2+60)+8(49ω427ω2+2))))=0.) (36)

    Putting a1=Aa3a5 in the minimization problem changes the constraint minimization problem to an unconstrained minimization problem, which is easier to find the solution of Eq (33) by using the conditions E(u3)/ω=0, E(u3)/a3=0 and E(u3)/a5=0.

    We plot the analytical solutions obtained from MGEMM (red line) Eq (32) and compare them with numerical solutions of Eq (23) obtained using the fourth order Runge-Kutta method (blue line). It is observed that for all different values of the amplitude A, the approximate solutions match extremely well with the numerical solutions (see Figure 2).

    Figure 2.  Comparison of the approximate solution (red line) with the numerical solution (blue line).

    Second, we consider k1=0, k2=1 and k3=1 in Eq (6). Hence, we have the one-dimensional nonlinear oscillator governed by [21,22,23,24,25,26]

    ¨u+u31+u2=0,u(0)=A,˙u(0)=0. (37)

    Finally, we can obtain the first and second-order approximations to Eq (37) given by Eqs (25) and (28), respectively. We remark that the third-order approximation in Eq (32) can be given by MGEMM in a similar manner. Generally, after three steps of MGEMM, one can obtain the approximated solutions to Eq (37) with sufficient accuracy. The analytical results of Eq (37) are compared with the iterative homotopy harmonic balance method (IHHBM) [34], the energy balance method (EBM) [20], the max-min approach (MMA) [32], the global residue harmonic balance method (GRHBM) [29], the Hamiltonian approach (HA) [26] and the exact solutions, as shown in Table 3.

    Table 3.  Comparison of the approximate analytical frequencies with the exact solutions.
    A ωEBM ωIHHBM ωMMA ωGRHBM ωHA ω3rdGEMM ωexact
    [20] [34] [32] [29] [26] present Exact
    0.01 0.00866 0.008478 0.00866 0.008472 0.00865 0.00847 0.00847
    0.1 0.08627 0.084418 0.08627 0.084394 0.08624 0.08439 0.08439
    1 0.65164 0.63136 0.65465 0.636795 0.64359 0.636783 0.63678
    5 0.97343 0.96667 0.97435 0.968107 0.96731 0.969202 0.96698
    10 0.99314 0.99090 0.99340 0.991591 0.99095 0.992005 0.99092
    50 0.99973 0.99961 0.99973 0.999657 0.999608 0.999676 0.99961
    100 0.99999 0.999901 0.99993 0.999914 0.99990 0.999919 0.99990

     | Show Table
    DownLoad: CSV

    In this paper, we test the analytical solutions of strongly nonlinear Duffing-harmonic oscillators to show the effectiveness of MGEMM. Comparisons of the analytical solutions and the exact numerical solutions of the Duffing-harmonic oscillators for small and large values of the amplitude have been illustrated in Figures 1 and 2 and Tables 13. In these Figures, the analytical solution is indicated by a red line, while the numerical solution is represented by a blue line. There is good compatibility between the analytical and numerical solutions, which confirms the accuracy of our results, and excellent matching is observed in these calculations. Moreover, as shown in Tables 13, the results produced using the MGEMM are in better agreement with those obtained using exact solutions than other existing ones in the literature. The calculations of the above applications were done using Mathematica software.

    In this paper, the modified global error minimization method has been presented successfully to obtain higher order approximate periodic solutions of strongly nonlinear Duffing-harmonic oscillators. The present method, which is proved to be a powerful mathematical tool to study nonlinear oscillators, can be easily extended to any nonlinear equation because of its efficiency and convenient applicability. We demonstrated the accuracy and efficiency of the proposed method by solving some examples. We showed that the obtained solutions are valid for the whole domain. Comparisons of the obtained solutions, whether numerical or analytical, revealed a clear match, highlighting the precision of the modified GEMM.

    M. Zayed and G. M. Ismail extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through the research groups program under grant R.G.P.2/207/43.

    The authors declare that they have no conflict of interest.



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