doi: 10.3934/naco.2019037

A hybrid parametrization approach for a class of nonlinear optimal control problems

1. 

Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

2. 

Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

3. 

Department of Applied Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

Received  November 2018 Revised  April 2019 Published  May 2019

In this paper, a suitable hybrid iterative scheme for solving a class of non-linear optimal control problems (NOCPs) is proposed. The technique is based upon homotopy analysis and parametrization methods. Actually an appropriate parametrization of control is applied and state variables are computed using homotopy analysis method (HAM). Then performance index is transformed by replacing new control and state variables. The results obtained from the given method are compared with the results which are obtained using the spectral homotopy analysis method (SHAM), homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM), modified variational iteration method (MVIM) and differential transformations. The existence and uniqueness of the solution are presented. The comparison and ability of the given approach is illustrated via two examples.

Citation: M. Alipour, M. A. Vali, A. H. Borzabadi. A hybrid parametrization approach for a class of nonlinear optimal control problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019037
References:
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S. Abbasbandi, Homotopy analysis method for Kawahara equations nonlinear analysis, Real World Applications, 11 (2010), 307-312. doi: 10.1016/j.nonrwa.2008.11.005. Google Scholar

[2]

S. EffatiH. Saberi Nik and M. Shirazian, Analytic-approximate solution for a class of nonlinear optimal control problems by homotopy analysis method, Asian-European Journal of Mathematics, 6 (2013), 1-22. doi: 10.1142/S1793557113500125. Google Scholar

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S. Ganjefar and S. Rezaei, Modified homotopy perturbation method for optimal control problems using Pade approximant, Applied Mathematical Modelling, 40 (2016), 7062-7081. doi: 10.1016/j.apm.2016.02.039. Google Scholar

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X. GaoK. L. Teo and G. R. Duan, An optimal control approach to spacecraft rendezvous on elliptical orbit, Optim. Control Appl. Meth., 36 (2015), 158-178. doi: 10.1002/oca.2108. Google Scholar

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C. K. Ghaddar, Rapid solution of optimal control problems by a functional spreadsheet paradigm: A practical method for the non-programmer, Mathematical and Computational Applications, 23 (2018), 54-82. doi: 10.3390/mca23040054. Google Scholar

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C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problem with general constraints, Automatica, 24 (1988), 3-18. doi: 10.1016/0005-1098(88)90003-9. Google Scholar

[7]

Q. GongI. M. RossW. Kang and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Comput. Optim. Appl., 41 (2008), 307-335. doi: 10.1007/s10589-007-9102-4. Google Scholar

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J. H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. Google Scholar

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I. Hwang, A computational approach to solve optimal control problems using differential transformation, In Proceedings of the 2007 American Control Conference, Marriott Marquis Hotel at Times Square, New York City, USA, 11–13, July 2007.Google Scholar

[10]

M. ItikM. U. Salamci and S. P. Banksa, Optimal control of drug therapy in cancer treatment, Nonlinear Analysis, 71 (2009), 1473-1486. Google Scholar

[11]

H. Jafari and M. Alipour, Solution of the Davey Stewartson equation using homotopy analysis method, Nonlinear Analysis: Modelling and Control, 15 (2010), 423-433. Google Scholar

[12]

A. JajarmiN. ParizA. Vahidian Kamyad and S. Effati, A highly computational efficient method to solve nonlinear optimal control problems, Scientia Iranica D, 19 (2012), 759-766. Google Scholar

[13]

A. JajarmiM. HajipourE. Mohammadzadeh and Du mitru Baleanu, A new approach for the nonlinear fractional optimal control problems with external persistent disturbances, Journal of the Franklin Institute, 355 (2018), 3938-3967. doi: 10.1016/j.jfranklin.2018.03.012. Google Scholar

[14]

W. JiaX. He and L. Guo, The optimal homotopy analysis method for solving linear optimal control problems, Applied Mathematical Modelling, 45 (2017), 865-880. doi: 10.1016/j.apm.2017.01.024. Google Scholar

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X. J. TangJ. L. Wei and K. Chen, A Chebyshev-Gauss pseudospectral method for solving optimal control problems, Acta Automatica Sinica, 41 (2015), 1778-1787. Google Scholar

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J. L. Junkins and J. D. Turner, Optimal Spacecraft Rotational Maneuvers, Elsevier-Amsterdam, 1986.Google Scholar

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M. El-Kady, Legendre approximations for solving optimal control problems governed by ordinary differential equations, International Journal of Control Science and Engineering, 2 (2012), 54-59. Google Scholar

[18]

B. KafashA. DelavarkhalafiS. M. Karbassi and K. Boubaker, A numerical approach for solving optimal control problems using the Boubaker polynomials expansion scheme, Journal of Interpolation and Approximation in Scientific Computing, 2014 (2014), 1-18. doi: 10.5899/2014/jiasc-00033. Google Scholar

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S. L. Kek, K. L. Teo and M. I. A. Aziz, Efficient output solution for nonlinear stochastic optimal control problem with model-reality differences, Mathematical Problems in Engineering, 2015 (2015), Article ID 659506, 9 pages. doi: 10.1155/2015/659506. Google Scholar

[20]

M. Keyanpour and M. Azizsefat, Numerical solution of optimal control problems by an iterative scheme, AMO- Advanced Modeling and Optimization, 13 (2011), 25-37. Google Scholar

[21]

R. LiaK. L. TeoK. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. Google Scholar

[22]

S. J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis- Shanghai Jiao Tong University, 1992.Google Scholar

[23]

S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press-Boca Raton, Chapman Hall, 2003. Google Scholar

[24]

S. J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer/Higher Education, 2012.Google Scholar

[25]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependents stopping criteria, Automatica, 48 (2012), 2116-2129. doi: 10.1016/j.automatica.2012.06.055. Google Scholar

[26]

Q. LinR. Loxton and K. L. Teo, Optimal control of nonlinear switched systems: Computational methods and applications, JORC, 1 (2013), 275-311. Google Scholar

[27]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Managment Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar

[28]

M. Matinfar and M. Saeidy, A new analytical method for solving a class of nonlinear optimal control problems, Optimal Control Applications and Methods, 35 (2014), 286-302. doi: 10.1002/oca.2068. Google Scholar

[29]

H. Mirinejad and T. Inanc, An RBF collocation method for solving optimal control problems, Robotics and Autonomous Systems, 87 (2017), 219-225. Google Scholar

[30]

A. NazemiS. Hesam and A. Haghbin, An application of differential transform method for solving nonlinear optimal control problems, Computational Methods for Differential Equations, 3 (2015), 200-217. Google Scholar

[31]

S. Nezhadhosein, A. Heyda and R. Ghanbari, A modified hybrid genetic algorithm for solving nonlinear optimal control problems, Mathematical Problems in Engineering, 2015, Article ID 139036, 21 pages. doi: 10.1155/2015/139036. Google Scholar

[32]

H. Saberi NikS. EffatiS. S. Motsa and M. Shirazian, Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems, Numer. Algor., 65 (2014), 171-194. doi: 10.1007/s11075-013-9700-4. Google Scholar

[33]

M. Shirazian and S. Effati, Solving a class of nonlinear optimal control problems via Hes variational iteration method, International Journal of Control, Automation, and Systems, 10 (2012), 249-256. Google Scholar

[34]

O. Y. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimazation, Annals of Operations Research, 37 (1992), 357-373. doi: 10.1007/BF02071065. Google Scholar

[35]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. Google Scholar

[36]

K. L. TeoL. S. JenningsH. W. J. Lee and V. Rehbock, Control parametrization enhancing technique for constrained optimal control problems, J. Austral. Math. Soc. B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. Google Scholar

[37]

S. Wei, M. Zefran and R. A. Decarlo, Optimal control of robotic system with logical constraints: application to UAV path planning, Q6 Proceeding(s) of the IEEE International Conference on Robotic and Automation, Pasadena. CA, USA, 2008.Google Scholar

[38]

X. S. Chen, X. K. Li, L. L. Zhang, and S. T. Cai, A new spectral method for the nonlinear optimal control, Proceedings of the 36th Chinese Control Conference, July 26–28, 2017, Dalian, China.Google Scholar

show all references

References:
[1]

S. Abbasbandi, Homotopy analysis method for Kawahara equations nonlinear analysis, Real World Applications, 11 (2010), 307-312. doi: 10.1016/j.nonrwa.2008.11.005. Google Scholar

[2]

S. EffatiH. Saberi Nik and M. Shirazian, Analytic-approximate solution for a class of nonlinear optimal control problems by homotopy analysis method, Asian-European Journal of Mathematics, 6 (2013), 1-22. doi: 10.1142/S1793557113500125. Google Scholar

[3]

S. Ganjefar and S. Rezaei, Modified homotopy perturbation method for optimal control problems using Pade approximant, Applied Mathematical Modelling, 40 (2016), 7062-7081. doi: 10.1016/j.apm.2016.02.039. Google Scholar

[4]

X. GaoK. L. Teo and G. R. Duan, An optimal control approach to spacecraft rendezvous on elliptical orbit, Optim. Control Appl. Meth., 36 (2015), 158-178. doi: 10.1002/oca.2108. Google Scholar

[5]

C. K. Ghaddar, Rapid solution of optimal control problems by a functional spreadsheet paradigm: A practical method for the non-programmer, Mathematical and Computational Applications, 23 (2018), 54-82. doi: 10.3390/mca23040054. Google Scholar

[6]

C. J. Goh and K. L. Teo, Control parameterization: a unified approach to optimal control problem with general constraints, Automatica, 24 (1988), 3-18. doi: 10.1016/0005-1098(88)90003-9. Google Scholar

[7]

Q. GongI. M. RossW. Kang and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Comput. Optim. Appl., 41 (2008), 307-335. doi: 10.1007/s10589-007-9102-4. Google Scholar

[8]

J. H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73-79. doi: 10.1016/S0096-3003(01)00312-5. Google Scholar

[9]

I. Hwang, A computational approach to solve optimal control problems using differential transformation, In Proceedings of the 2007 American Control Conference, Marriott Marquis Hotel at Times Square, New York City, USA, 11–13, July 2007.Google Scholar

[10]

M. ItikM. U. Salamci and S. P. Banksa, Optimal control of drug therapy in cancer treatment, Nonlinear Analysis, 71 (2009), 1473-1486. Google Scholar

[11]

H. Jafari and M. Alipour, Solution of the Davey Stewartson equation using homotopy analysis method, Nonlinear Analysis: Modelling and Control, 15 (2010), 423-433. Google Scholar

[12]

A. JajarmiN. ParizA. Vahidian Kamyad and S. Effati, A highly computational efficient method to solve nonlinear optimal control problems, Scientia Iranica D, 19 (2012), 759-766. Google Scholar

[13]

A. JajarmiM. HajipourE. Mohammadzadeh and Du mitru Baleanu, A new approach for the nonlinear fractional optimal control problems with external persistent disturbances, Journal of the Franklin Institute, 355 (2018), 3938-3967. doi: 10.1016/j.jfranklin.2018.03.012. Google Scholar

[14]

W. JiaX. He and L. Guo, The optimal homotopy analysis method for solving linear optimal control problems, Applied Mathematical Modelling, 45 (2017), 865-880. doi: 10.1016/j.apm.2017.01.024. Google Scholar

[15]

X. J. TangJ. L. Wei and K. Chen, A Chebyshev-Gauss pseudospectral method for solving optimal control problems, Acta Automatica Sinica, 41 (2015), 1778-1787. Google Scholar

[16]

J. L. Junkins and J. D. Turner, Optimal Spacecraft Rotational Maneuvers, Elsevier-Amsterdam, 1986.Google Scholar

[17]

M. El-Kady, Legendre approximations for solving optimal control problems governed by ordinary differential equations, International Journal of Control Science and Engineering, 2 (2012), 54-59. Google Scholar

[18]

B. KafashA. DelavarkhalafiS. M. Karbassi and K. Boubaker, A numerical approach for solving optimal control problems using the Boubaker polynomials expansion scheme, Journal of Interpolation and Approximation in Scientific Computing, 2014 (2014), 1-18. doi: 10.5899/2014/jiasc-00033. Google Scholar

[19]

S. L. Kek, K. L. Teo and M. I. A. Aziz, Efficient output solution for nonlinear stochastic optimal control problem with model-reality differences, Mathematical Problems in Engineering, 2015 (2015), Article ID 659506, 9 pages. doi: 10.1155/2015/659506. Google Scholar

[20]

M. Keyanpour and M. Azizsefat, Numerical solution of optimal control problems by an iterative scheme, AMO- Advanced Modeling and Optimization, 13 (2011), 25-37. Google Scholar

[21]

R. LiaK. L. TeoK. H. Wong and G. R. Duan, Control parameterization enhancing transform for optimal control of switched systems, Mathematical and Computer Modelling, 43 (2006), 1393-1403. doi: 10.1016/j.mcm.2005.08.012. Google Scholar

[22]

S. J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis- Shanghai Jiao Tong University, 1992.Google Scholar

[23]

S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press-Boca Raton, Chapman Hall, 2003. Google Scholar

[24]

S. J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer/Higher Education, 2012.Google Scholar

[25]

Q. LinR. LoxtonK. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependents stopping criteria, Automatica, 48 (2012), 2116-2129. doi: 10.1016/j.automatica.2012.06.055. Google Scholar

[26]

Q. LinR. Loxton and K. L. Teo, Optimal control of nonlinear switched systems: Computational methods and applications, JORC, 1 (2013), 275-311. Google Scholar

[27]

Q. LinR. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Managment Optimization, 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar

[28]

M. Matinfar and M. Saeidy, A new analytical method for solving a class of nonlinear optimal control problems, Optimal Control Applications and Methods, 35 (2014), 286-302. doi: 10.1002/oca.2068. Google Scholar

[29]

H. Mirinejad and T. Inanc, An RBF collocation method for solving optimal control problems, Robotics and Autonomous Systems, 87 (2017), 219-225. Google Scholar

[30]

A. NazemiS. Hesam and A. Haghbin, An application of differential transform method for solving nonlinear optimal control problems, Computational Methods for Differential Equations, 3 (2015), 200-217. Google Scholar

[31]

S. Nezhadhosein, A. Heyda and R. Ghanbari, A modified hybrid genetic algorithm for solving nonlinear optimal control problems, Mathematical Problems in Engineering, 2015, Article ID 139036, 21 pages. doi: 10.1155/2015/139036. Google Scholar

[32]

H. Saberi NikS. EffatiS. S. Motsa and M. Shirazian, Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems, Numer. Algor., 65 (2014), 171-194. doi: 10.1007/s11075-013-9700-4. Google Scholar

[33]

M. Shirazian and S. Effati, Solving a class of nonlinear optimal control problems via Hes variational iteration method, International Journal of Control, Automation, and Systems, 10 (2012), 249-256. Google Scholar

[34]

O. Y. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimazation, Annals of Operations Research, 37 (1992), 357-373. doi: 10.1007/BF02071065. Google Scholar

[35]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 1991. Google Scholar

[36]

K. L. TeoL. S. JenningsH. W. J. Lee and V. Rehbock, Control parametrization enhancing technique for constrained optimal control problems, J. Austral. Math. Soc. B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. Google Scholar

[37]

S. Wei, M. Zefran and R. A. Decarlo, Optimal control of robotic system with logical constraints: application to UAV path planning, Q6 Proceeding(s) of the IEEE International Conference on Robotic and Automation, Pasadena. CA, USA, 2008.Google Scholar

[38]

X. S. Chen, X. K. Li, L. L. Zhang, and S. T. Cai, A new spectral method for the nonlinear optimal control, Proceedings of the 36th Chinese Control Conference, July 26–28, 2017, Dalian, China.Google Scholar

Figure 1.  Approximate solution of $ x_1(t) $ and $ u_1(t) $ for (m = 4, k = 2)
Figure 2.  Approximate solution of x2(t) and u2(t) for (m = 4, k = 2)
Figure 3.  Approximate solution of x3(t) and u3(t) for (m = 4, k = 2)
Figure 4.  h-curve at 4-order of approximation of $ x_1(t) $ and $ x_2(t) $
Figure 5.  h-curve at 4-order of approximation of $ x_3(t) $
Figure 6.  Approximate solution of $ x_1(t) $ and $ x_2(t) $ for (m = 7, k = 3)
Figure 7.  Approximate solution of $ u(t) $
Figure 8.  h-curve at 7-order of approximation of x1(t) and x2(t)
Table 1.  Minimum of performance index value $ J_k $ of the proposed method
Itr CPU time (sec.) HAM and parametrization approaches
m=4, k=1 $ 0.109 $ $ 0.00468778 $
m=4, k=2 $ 0.121 $ $ 0.00468778 $
Itr CPU time (sec.) HAM and parametrization approaches
m=4, k=1 $ 0.109 $ $ 0.00468778 $
m=4, k=2 $ 0.121 $ $ 0.00468778 $
Table 2.  The Max error of the proposed method for $ x_1(t) $ that $ k = 2 $ and $ h = -1 $ in comparison to SHAM and HPM
Method CPU time (sec.) Max error
proposed method (m=4, k=2) $ 0.155 $ $ 2.93152 * 10^{-17} $
SHAM (Legendre) (m=6, N=50, h=-1.2) $ 0.224 $ $ 1.0589* 10^{-9} $
SHAM (Chebyshev) (m=6, N=50, h=-1.2) $ 0.224 $ $ 1.0586* 10^{-9} $
HPM (m=6) $ 46.401 $ $ 3.1420* 10^{-8} $
Method CPU time (sec.) Max error
proposed method (m=4, k=2) $ 0.155 $ $ 2.93152 * 10^{-17} $
SHAM (Legendre) (m=6, N=50, h=-1.2) $ 0.224 $ $ 1.0589* 10^{-9} $
SHAM (Chebyshev) (m=6, N=50, h=-1.2) $ 0.224 $ $ 1.0586* 10^{-9} $
HPM (m=6) $ 46.401 $ $ 3.1420* 10^{-8} $
Table 3.  Minimum of performance index value $ J $ of the proposed method and other methods
Method Cost function CPU time (sec.)
Proposed Method (m=4, k=2, h=-1) $ 0.00468778 $ $ 0.141 $
SHAM Chebyshev (m=6, N=50, h=-1.2) $ 0.0046877944625923 $ $ 0.226 $
SHAM Legendre (m=6, N=50, h=-1.2) $ 0.0046877944625906 $ $ 0.227 $
HPM (m=3) $ 0.004687795533 $ $ 10.821 $
OHPM (m=1) $ 0.004688009428 $ $ - $
MVIM (m=3) $ 0.004687986656 $ $ - $
Method Cost function CPU time (sec.)
Proposed Method (m=4, k=2, h=-1) $ 0.00468778 $ $ 0.141 $
SHAM Chebyshev (m=6, N=50, h=-1.2) $ 0.0046877944625923 $ $ 0.226 $
SHAM Legendre (m=6, N=50, h=-1.2) $ 0.0046877944625906 $ $ 0.227 $
HPM (m=3) $ 0.004687795533 $ $ 10.821 $
OHPM (m=1) $ 0.004688009428 $ $ - $
MVIM (m=3) $ 0.004687986656 $ $ - $
Table 4.  Minimum of performance index value $ J_k $ of the proposed method
Itr CPU time (sec.) HAM and parametrization approaches
m=7, k=1 $ 0.016 $ $ 1.07504 $
m=7, k=2 $ 0.031 $ $ 1.0136 $
m= 7, k=3 $ 0.032 $ $ 1.01184 $
Itr CPU time (sec.) HAM and parametrization approaches
m=7, k=1 $ 0.016 $ $ 1.07504 $
m=7, k=2 $ 0.031 $ $ 1.0136 $
m= 7, k=3 $ 0.032 $ $ 1.01184 $
Table 5.  The Max error of our method of $ x_1(t) $ in comparison to SHAM and HPM
Itr Max error
Proposed Method (m=7, k=3, h=-0.9) $ 3.16673\times10^{-5} $
SHAM Chebyshev (m=15, N=50, h=-0.5) $ 4.2749\times10^{-4} $
SHAM Legendre (m=15, N=50, h=-0.5) $ 4.2749\times10^{-4} $
DT (m=15) $ 4.4380\times10^{-4} $
Itr Max error
Proposed Method (m=7, k=3, h=-0.9) $ 3.16673\times10^{-5} $
SHAM Chebyshev (m=15, N=50, h=-0.5) $ 4.2749\times10^{-4} $
SHAM Legendre (m=15, N=50, h=-0.5) $ 4.2749\times10^{-4} $
DT (m=15) $ 4.4380\times10^{-4} $
Table 6.  Minimum of performance index value J of the proposed method and other methods
Method Cost function CPU time (sec.)
Proposed Method (m=7, k=3, h=-0.9) $ 1.01184 $ $ 0.032 $
SHAM Chebyshev (m=15, N=50, h=-0.5) $ 1.0472 $ $ 0.200 $
SHAM Legendre (m=15, N=50, h=-0.5) $ 1.0472 $ $ 0.188 $
DT (m=15) $ 1.0478 $ $ 87.74 $
Method Cost function CPU time (sec.)
Proposed Method (m=7, k=3, h=-0.9) $ 1.01184 $ $ 0.032 $
SHAM Chebyshev (m=15, N=50, h=-0.5) $ 1.0472 $ $ 0.200 $
SHAM Legendre (m=15, N=50, h=-0.5) $ 1.0472 $ $ 0.188 $
DT (m=15) $ 1.0478 $ $ 87.74 $
Table 7.  Minimum of performance index value Jk of the proposed method
Itr CPU time (sec.) HAM and parametrization approaches
m=7, k=1 0.016 1.07504
m=7, k=2 0.031 1.0136
m= 7, k=3 0.032 1.01184
Itr CPU time (sec.) HAM and parametrization approaches
m=7, k=1 0.016 1.07504
m=7, k=2 0.031 1.0136
m= 7, k=3 0.032 1.01184
Table 8.  The Max error of our method of x1(t) in comparison to SHAM and HPM
Itr Max error
Proposed Method (m=7, k=3, h=-0.9) $5.36319\times10^{-5}$
SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$
SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$
DT (m=15) $4.4380\times10^{-4}$
Itr Max error
Proposed Method (m=7, k=3, h=-0.9) $5.36319\times10^{-5}$
SHAM Chebyshev (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$
SHAM Legendre (m=15, N=50, h=-0.5) $4.2749\times10^{-4}$
DT (m=15) $4.4380\times10^{-4}$
Table 9.  Minimum of performance index value J of the proposed method and other methods
Method Cost function
Proposed Method (m=4, k=2) 1.04483
SHAM Chebyshev (m=15, N=50, h=-0.5) 1.0472
SHAM Legendre (m=15, N=50, h=-0.5) 1.0472
DT (m=15) 1.0478
Method Cost function
Proposed Method (m=4, k=2) 1.04483
SHAM Chebyshev (m=15, N=50, h=-0.5) 1.0472
SHAM Legendre (m=15, N=50, h=-0.5) 1.0472
DT (m=15) 1.0478
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