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doi: 10.3934/naco.2021008
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Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

Department of Mathematics, College of Education for Pure Science, University of Mosul, Mosul, Iraq

* Corresponding author: Azzam S. Y. Aladool

Received  December 2020 Revised  February 2021 Early access March 2021

Ordinary differential equations are converted into a constrained optimization problems to find their approximate solutions. In this work, an algorithm is proposed by applying particle swarm optimization (PSO) to find an approximate solution of ODEs based on an expansion approximation. Since many cases of linear and nonlinear ODEs have singularity point, Padé approximant which is fractional expansion is employed for more accurate results compare to Fourier and Taylor expansions. The fitness function is obtained by adding the discrete least square weighted function to a penalty function. The proposed algorithm is applied to 13 famous ODEs such as Lane Emden, Emden-Fowler, Riccati, Ivey, Abel, Thomas Fermi, Bernoulli, Bratu, Van der pol, the Troesch problem and other cases. The proposed algorithm offer fast and accurate results compare to the other methods presented in this paper. The results demonstrate the ability of proposed approach to solve linear and nonlinear ODEs with initial or boundary conditions.

Citation: Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021008
References:
[1]

S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Applied Mathematics and Computation, Elsevier, 172 (2006), 485–490. doi: 10.1016/j.amc.2005.02.014.  Google Scholar

[2]

E. J. Ali, New treatment of the solution of initial boundary value problems by using variational iteration method, Basrah Journal of Science, Basrah University, 30 (2012), 57–74. Google Scholar

[3]

M. Almazmumy, F. A. Hendi, H. O. Bakodah and H. Alzumi, Recent modifications of Adomian decomposition method for initial value problem in ordinary differential equations, American Journal of Computational Mathematics, Scientific Research Publishing, 2 (2012), 228–234. doi: 10.4236/ajcm.2012.23030.  Google Scholar

[4]

U. M. Ascher, R. M. M. Mattheij and R. D. Russel, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 1995. doi: 10.1137/1.9781611971231.  Google Scholar

[5]

M. Babaei, A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization, Applied Soft Computing, Elsevier, 13 (2013), 3354–3365. Google Scholar

[6] G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition: Encyclopedia of Mathematics and It's Applications, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511530074.  Google Scholar
[7] A. Borzì, Modelling with Ordinary Differential Equations: A Comprehensive Approach, CRC Press, 2020.   Google Scholar
[8]

W. E. Boyce, R. C. DiPrima and D. B. Meade, Elementary Differential Equations and Boundary Value Problems, 11$^{nd}$ edition, WILEY, 2017.  Google Scholar

[9]

R. L. Burden and J. D. Faires, Numerical Analysis, 9$^{nd}$ edition, Brooks/Cole, Cencag Learning, 2011. Google Scholar

[10] S. Chakraverty and Su smita Mall, Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations, CRC Press, 2017.  doi: 10.1201/9781315155265.  Google Scholar
[11]

J. M. Chaquet and E. Carmona, Solving differential equations with Fourier series and evolution strategies, Appl. Soft Comput., 12 (2012), 3051–3062. Google Scholar

[12]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, IEEE, 6 (2002), 58–73. doi: 10.1002/9780470612163.  Google Scholar

[13]

R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, IEEE, (1995), 39–43. Google Scholar

[14]

A. Engelbrecht, Particle swarm optimization: Velocity initialization, 2012 IEEE Congress on Evolutionary Computation, IEEE, (2012), 1–8. Google Scholar

[15]

D. Gutierrez-Navarro and S. Lopez-Aguayo, Solving ordinary differential equations using genetic algorithms and the Taylor series matrix method, Journal of Physics Communications, IOP Publishing, 2 (2018), 115010. Google Scholar

[16]

M. Hermann and M. Saravi, Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods, Springer India, 2016. doi: 10.1007/978-81-322-2812-7.  Google Scholar

[17]

E. A. Hussain and Y. M. Abdul–Abbass, Solving differential equation by modified genetic algorithms, Journal of University of Babylon for Pure and Applied Sciences, 26 (2018), 233-241.   Google Scholar

[18]

J. Kennedy, The particle swarm: social adaptation of knowledge, Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC'97), IEEE, (1997), 303–308. Google Scholar

[19]

F. Mirzaee, Differential transform method for solving linear and nonlinear systems of ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3465-3472.   Google Scholar

[20] T. RadhikaT. Iyengar and T. Rani, Approximate Analytical Methods for Solving Ordinary Differential Equations, CRC Press, 2014.   Google Scholar
[21]

A. Sadollah, H. Eskandar, D. G. Yoo and J. H. Kim, Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms, Engineering Applications of Artificial Intelligence, Elsevier, 40 (2015), 117–132. Google Scholar

[22]

A. Sahu, S. K. Panigrahi and S. Pattnaik, Fast convergence particle swarm optimization for functions optimization, Procedia Technology, Elsevier, 4 (2012), 319–324. Google Scholar

[23]

M. Shehab, A. T. Khader and M. Al-Betar, New selection schemes for particle swarm optimization, IEEJ Transactions on Electronics, Information and Systems, The Institute of Electrical Engineers of Japan, 136 (2016), 1706–1711. Google Scholar

[24]

S. Talukder, Mathematicle Modelling and Applications of Particle Swarm Optimization, Mc. S. Thesis, Blekinge Institute of Technology, School of Engineering, 2011. Google Scholar

[25]

W. F. Trench, Elementary Differential Equations with Boundary Value Problems, Brooks/Cole Thomson Learning, 2013. Google Scholar

[26]

W. Van Assche, Padé and Hermite-Padé approximation and orthogonality, arXiv: math/0609094, 2006.  Google Scholar

[27]

Z. Zhang, Y. Cai and D. Zhang, Solving ordinary differential equations with adaptive differential evolution, IEEE Access, IEEE, 8 (2020), 128908–128922. Google Scholar

[28]

P. Zitnan, Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary, Appl. Soft Comput., 217 (2011), 8973-8982.  doi: 10.1016/j.amc.2011.03.103.  Google Scholar

show all references

References:
[1]

S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Applied Mathematics and Computation, Elsevier, 172 (2006), 485–490. doi: 10.1016/j.amc.2005.02.014.  Google Scholar

[2]

E. J. Ali, New treatment of the solution of initial boundary value problems by using variational iteration method, Basrah Journal of Science, Basrah University, 30 (2012), 57–74. Google Scholar

[3]

M. Almazmumy, F. A. Hendi, H. O. Bakodah and H. Alzumi, Recent modifications of Adomian decomposition method for initial value problem in ordinary differential equations, American Journal of Computational Mathematics, Scientific Research Publishing, 2 (2012), 228–234. doi: 10.4236/ajcm.2012.23030.  Google Scholar

[4]

U. M. Ascher, R. M. M. Mattheij and R. D. Russel, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 1995. doi: 10.1137/1.9781611971231.  Google Scholar

[5]

M. Babaei, A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization, Applied Soft Computing, Elsevier, 13 (2013), 3354–3365. Google Scholar

[6] G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition: Encyclopedia of Mathematics and It's Applications, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511530074.  Google Scholar
[7] A. Borzì, Modelling with Ordinary Differential Equations: A Comprehensive Approach, CRC Press, 2020.   Google Scholar
[8]

W. E. Boyce, R. C. DiPrima and D. B. Meade, Elementary Differential Equations and Boundary Value Problems, 11$^{nd}$ edition, WILEY, 2017.  Google Scholar

[9]

R. L. Burden and J. D. Faires, Numerical Analysis, 9$^{nd}$ edition, Brooks/Cole, Cencag Learning, 2011. Google Scholar

[10] S. Chakraverty and Su smita Mall, Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations, CRC Press, 2017.  doi: 10.1201/9781315155265.  Google Scholar
[11]

J. M. Chaquet and E. Carmona, Solving differential equations with Fourier series and evolution strategies, Appl. Soft Comput., 12 (2012), 3051–3062. Google Scholar

[12]

M. Clerc and J. Kennedy, The particle swarm-explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation, IEEE, 6 (2002), 58–73. doi: 10.1002/9780470612163.  Google Scholar

[13]

R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, IEEE, (1995), 39–43. Google Scholar

[14]

A. Engelbrecht, Particle swarm optimization: Velocity initialization, 2012 IEEE Congress on Evolutionary Computation, IEEE, (2012), 1–8. Google Scholar

[15]

D. Gutierrez-Navarro and S. Lopez-Aguayo, Solving ordinary differential equations using genetic algorithms and the Taylor series matrix method, Journal of Physics Communications, IOP Publishing, 2 (2018), 115010. Google Scholar

[16]

M. Hermann and M. Saravi, Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods, Springer India, 2016. doi: 10.1007/978-81-322-2812-7.  Google Scholar

[17]

E. A. Hussain and Y. M. Abdul–Abbass, Solving differential equation by modified genetic algorithms, Journal of University of Babylon for Pure and Applied Sciences, 26 (2018), 233-241.   Google Scholar

[18]

J. Kennedy, The particle swarm: social adaptation of knowledge, Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC'97), IEEE, (1997), 303–308. Google Scholar

[19]

F. Mirzaee, Differential transform method for solving linear and nonlinear systems of ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3465-3472.   Google Scholar

[20] T. RadhikaT. Iyengar and T. Rani, Approximate Analytical Methods for Solving Ordinary Differential Equations, CRC Press, 2014.   Google Scholar
[21]

A. Sadollah, H. Eskandar, D. G. Yoo and J. H. Kim, Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms, Engineering Applications of Artificial Intelligence, Elsevier, 40 (2015), 117–132. Google Scholar

[22]

A. Sahu, S. K. Panigrahi and S. Pattnaik, Fast convergence particle swarm optimization for functions optimization, Procedia Technology, Elsevier, 4 (2012), 319–324. Google Scholar

[23]

M. Shehab, A. T. Khader and M. Al-Betar, New selection schemes for particle swarm optimization, IEEJ Transactions on Electronics, Information and Systems, The Institute of Electrical Engineers of Japan, 136 (2016), 1706–1711. Google Scholar

[24]

S. Talukder, Mathematicle Modelling and Applications of Particle Swarm Optimization, Mc. S. Thesis, Blekinge Institute of Technology, School of Engineering, 2011. Google Scholar

[25]

W. F. Trench, Elementary Differential Equations with Boundary Value Problems, Brooks/Cole Thomson Learning, 2013. Google Scholar

[26]

W. Van Assche, Padé and Hermite-Padé approximation and orthogonality, arXiv: math/0609094, 2006.  Google Scholar

[27]

Z. Zhang, Y. Cai and D. Zhang, Solving ordinary differential equations with adaptive differential evolution, IEEE Access, IEEE, 8 (2020), 128908–128922. Google Scholar

[28]

P. Zitnan, Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary, Appl. Soft Comput., 217 (2011), 8973-8982.  doi: 10.1016/j.amc.2011.03.103.  Google Scholar

Figure 1.  shows a plot of the resulting approximate solution with the corresponding exact solutions.
Figure 2.  Shows numerical error between the approximate solution and exact solution arising from the proposed algorithm in four examples.in NLIVP5, NLIVP6, NBVP1 and NBVP3 respectively.
Figure 3.  Show the convergence of PSO-PEA-DLSWF algorithm in 500 iteration.
Figure 4.  Comparison of expansions: PEA, TEA and FEA to consumed time.
Figure 5.  shows the simulating of example (NLBVP3) in terms of errors, convergence and stability.
Figure 6.  Comparison between the fitness function: WRF, LSWF and DLSWF with consumed time.
Figure 7.  shows the simulating of example (NLBVP1) in terms of errors, convergence and stability.
Table 1.  Illustrates different type of the ordinary differential equations, with its conditions, domain and exact solution.
ODE NAME ODE EQUATION I.C. or B.C
DOMAIN
EXACT SOLUTION
Simple ODEs1
LIVP1
$y'=\frac{y}{x}+1$ $y(1)=0$
$x\in[1,2]$
$y=xln(x)$
Simple ODEs2
LIVP2
$y''-y=e^x$ $y(0)=0$
$y'(0)=1$
$x\in[0,1]$
$y=0.25(e^x+2xe^x-e^{-x})$
Simple Harmonic
LBVP1
$y''+25y=0$ $y(0)=1$
$y(\frac{\pi}{10})=0$
$x\in[0,\frac{\pi}{10}]$
$y=\cos(5x)$
Riccati
NLIVP1
$y'=1+2y-y^2$ $y(0)=0$
$x\in[0,1]$
$y=1+\sqrt{2} \tanh{(\sqrt{2}x}+\frac{1}{2} ln(\frac{\sqrt{2}-1}{\sqrt{2}+1}))$
Abel
NLIVP2
$y'=1-x^2 y+y^3$ $y(0)=0$
$x\in[0,1]$
$y=x$
Ivey
NLIVP3
$y''=\frac{(y' )^2}{y}-\frac{2y'}{x}-2y^2$ $y(1)=1$
$y'(1)=-2$
$x\in[1,2]$
$y=\frac{1}{x^2}$
Lane- Emden
NLIVP4
$y''+2\frac{y'}{x}=-4(2e^y+e^{\frac{y}{2}})$ $y(0)=0$
$y'(0)=0$
$x\in[0,1]$
$y=-2ln(1+x^2)$
Emden-Fowler
NLIVP5
$y''+6\frac{y'}{x}+14y+4ylny=0$ $y(0)=1$
$y'(0)=0$
$x\in[0,1]$
$y=e^{-x^2}$
Van Der Pol
NLIVP6
$y''+y=0.05(1-y^2 )y'$ $y(0)=0$
$y'(0)=0.5$
$x\in[0,1]$
Sol. By ode45
solver in MATLAB
Bratu-Gelfand
NLBVP1
$y''-\pi^2 e^y=0$ $y(0)=0$
$y(1)=0$
$x\in[0,1]$
$y=-2ln(\sqrt{2}\cos(\pi(\frac{x}{2}-\frac{1}{4})))$
Thomas-Fermi
NLBVP2
$y''=\sqrt{\frac{y^3}{x}}$ $y(1)=144$
$y(2)=18$
$x\in[1,2]$
$y=\frac{144}{x^3}$
Bernoulli
NLBVP3
$y''+(y')^2=2e^{-y}$ $y(0)=0$
$y(1)=0$
$x\in[0,1]$
$y=\log{(0.75+(x-0.5)^2)}$
Troesch Problem
NLBVP4
$y''=\sinh{y}$ $y(0)=0$
$y(1)=1$
$x\in[0,1]$
Sol. By bvp4c
solver in MATLAB
ODE NAME ODE EQUATION I.C. or B.C
DOMAIN
EXACT SOLUTION
Simple ODEs1
LIVP1
$y'=\frac{y}{x}+1$ $y(1)=0$
$x\in[1,2]$
$y=xln(x)$
Simple ODEs2
LIVP2
$y''-y=e^x$ $y(0)=0$
$y'(0)=1$
$x\in[0,1]$
$y=0.25(e^x+2xe^x-e^{-x})$
Simple Harmonic
LBVP1
$y''+25y=0$ $y(0)=1$
$y(\frac{\pi}{10})=0$
$x\in[0,\frac{\pi}{10}]$
$y=\cos(5x)$
Riccati
NLIVP1
$y'=1+2y-y^2$ $y(0)=0$
$x\in[0,1]$
$y=1+\sqrt{2} \tanh{(\sqrt{2}x}+\frac{1}{2} ln(\frac{\sqrt{2}-1}{\sqrt{2}+1}))$
Abel
NLIVP2
$y'=1-x^2 y+y^3$ $y(0)=0$
$x\in[0,1]$
$y=x$
Ivey
NLIVP3
$y''=\frac{(y' )^2}{y}-\frac{2y'}{x}-2y^2$ $y(1)=1$
$y'(1)=-2$
$x\in[1,2]$
$y=\frac{1}{x^2}$
Lane- Emden
NLIVP4
$y''+2\frac{y'}{x}=-4(2e^y+e^{\frac{y}{2}})$ $y(0)=0$
$y'(0)=0$
$x\in[0,1]$
$y=-2ln(1+x^2)$
Emden-Fowler
NLIVP5
$y''+6\frac{y'}{x}+14y+4ylny=0$ $y(0)=1$
$y'(0)=0$
$x\in[0,1]$
$y=e^{-x^2}$
Van Der Pol
NLIVP6
$y''+y=0.05(1-y^2 )y'$ $y(0)=0$
$y'(0)=0.5$
$x\in[0,1]$
Sol. By ode45
solver in MATLAB
Bratu-Gelfand
NLBVP1
$y''-\pi^2 e^y=0$ $y(0)=0$
$y(1)=0$
$x\in[0,1]$
$y=-2ln(\sqrt{2}\cos(\pi(\frac{x}{2}-\frac{1}{4})))$
Thomas-Fermi
NLBVP2
$y''=\sqrt{\frac{y^3}{x}}$ $y(1)=144$
$y(2)=18$
$x\in[1,2]$
$y=\frac{144}{x^3}$
Bernoulli
NLBVP3
$y''+(y')^2=2e^{-y}$ $y(0)=0$
$y(1)=0$
$x\in[0,1]$
$y=\log{(0.75+(x-0.5)^2)}$
Troesch Problem
NLBVP4
$y''=\sinh{y}$ $y(0)=0$
$y(1)=1$
$x\in[0,1]$
Sol. By bvp4c
solver in MATLAB
Table 2.  the values of parameters and inputs used in all example.
parameter $ \phi_1 $ $ \phi_2 $ $ \kappa $ $ \rho $ $ \sigma $ $ h $ $ TOL $ $ K_m \forall m $ $ Maxit $ $ nPop $
value 2.05 2.05 1 0.8 0.4 0.01 1e-03 1 or 10 500 200
parameter $ \phi_1 $ $ \phi_2 $ $ \kappa $ $ \rho $ $ \sigma $ $ h $ $ TOL $ $ K_m \forall m $ $ Maxit $ $ nPop $
value 2.05 2.05 1 0.8 0.4 0.01 1e-03 1 or 10 500 200
Table 3.  Search space and nVar sitting of each example.
Equation PEA TEA FEA
VarMin VarMax nVar VarMin VarMax nVar VarMin VarMax nVar
LIVP1 -1 1 10 -2 2 10 -2 2 9
LIVP2 -2 2 10 -2 2 10 -2 2 11
LBVP1 -3 3 10 -3 3 10 -1 1 11
NLIVP1 -2 2 10 -2 2 10 -2 2 9
NLIVP2 0 1 10 0 1 10 -1 1 11
NLIVP3 0 1 10 -3 3 10 -2 2 9
NLIVP4 -2 2 10 -2 2 10 -1 1 11
NLIVP5 -1 1 10 -1 1 10 -1 1 9
NLIVP6 -1 1 10 -2 2 10 -0.5 0.5 9
NLBVP1 -1 1 10 -3 3 10 -1 1 9
NLBVP2 0 1 10 -900 900 10 -200 200 9
NLBVP3 -2 2 10 -3 3 10 -1 1 9
NLBVP4 -4 4 10 -1 1 10 -1 1 9
Equation PEA TEA FEA
VarMin VarMax nVar VarMin VarMax nVar VarMin VarMax nVar
LIVP1 -1 1 10 -2 2 10 -2 2 9
LIVP2 -2 2 10 -2 2 10 -2 2 11
LBVP1 -3 3 10 -3 3 10 -1 1 11
NLIVP1 -2 2 10 -2 2 10 -2 2 9
NLIVP2 0 1 10 0 1 10 -1 1 11
NLIVP3 0 1 10 -3 3 10 -2 2 9
NLIVP4 -2 2 10 -2 2 10 -1 1 11
NLIVP5 -1 1 10 -1 1 10 -1 1 9
NLIVP6 -1 1 10 -2 2 10 -0.5 0.5 9
NLBVP1 -1 1 10 -3 3 10 -1 1 9
NLBVP2 0 1 10 -900 900 10 -200 200 9
NLBVP3 -2 2 10 -3 3 10 -1 1 9
NLBVP4 -4 4 10 -1 1 10 -1 1 9
Table 4.  shows the values of variables founded by using the PSO-PEA-DLSWF algorithm.
Eq. Coff, $ m=1 $ $ m=2 $ $ m=3 $ $ m=4 $ $ m=5 $
LIVP1 $ \alpha_m $ 0.449646621 -0.955885566 -0.662451518 0.5990158869 0.5696745764
$ \beta_m $ -0.383029549 1.000000000 0.9928258613 0.2090703809 -0.01223876
LIVP2 $ \alpha_m $ 1.5905e-08 -2.000000000 -1.99993129 -1.999993802 1.320889355
$ \beta_m $ -2.000000000 -1.028027457 -0.640309044 1.893006788 -0.627961135
LBVP1 $ \alpha_m $ -0.23700822 0.2939052397 1.7497711122 -0.815400817 -0.28105974
$ \beta_m $ -0.237008298 0.2940587929 -1.172283084 1.983810204 -2.525130497
NLIVP1 $ \alpha_m $ -6.595e-07 -2.000000000 -1.969510526 -2.000000000 -1.99980317
$ \beta_m $ -1.999999998 0.1334264437 -1.999346637 0.2828697917 -1.132644395
NLIVP2 $ \alpha_m $ 0.00000000 1.00000000 1.00000000 0.9159523195 1.00000000
$ \beta_m $ 1.00000000 1.00000000 0.9159523195 1.00000000 0.00000000
NLIVP3 $ \alpha_m $ 1.00000000 0.501789937 0.3963523812 0.00000000 0.00000000
$ \beta_m $ 0.00000000 0.00000000 1.00000000 0.501789937 0.3963523812
NLIVP4 $ \alpha_m $ -3.556e-10 1.1605e-09 1.371544377 1.705016485 1.664659425
$ \beta_m $ -0.699013232 -0.795001437 -1.237894199 -0.487132482 -0.20116878
NLIVP5 $ \alpha_m $ -1.0000000 -0.999999714 0.9996994842 1.00000000 -0.670316746
$ \beta_m $ -0.999999761 -1.0000000 0.0694451202 -0.363172581 0.4710444283
NLIVP6 $ \alpha_m $ 9.8817e-08 0.499999967 -0.128335779 -0.225730262 0.4574883022
$ \beta_m $ 1.00000000 -0.334983343 0.0548712929 0.1919886974 0.4900184664
NLBVP1 $ \alpha_m $ 6.0988e-07 -1.0000000 -0.168001861 0.1680024128 1.00000000
$ \beta_m $ 0.3182717396 0.5556535208 0.2683350684 0.1086186654 -0.242687286
NLBVP2 $ \alpha_m $ 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$ \beta_m $ 0.00000000 0.00000000 0.00000000 0.0069444444 0.00000000
NLBVP3 $ \alpha_m $ -2.979e-11 -1.512033065 0.5287945264 -0.376563712 1.359802251
$ \beta_m $ 1.512152004 0.2390046775 1.385502167 0.2230003398 0.4953153062
NLBVP4 $ \alpha_m $ 2.6966e-06 -3.398796238 -0.916386583 -3.997792287 -2.079546692
$ \beta_m $ -4.00000000 -1.420908971 -2.843010022 -3.999816103 1.8712207261
Eq. Coff, $ m=1 $ $ m=2 $ $ m=3 $ $ m=4 $ $ m=5 $
LIVP1 $ \alpha_m $ 0.449646621 -0.955885566 -0.662451518 0.5990158869 0.5696745764
$ \beta_m $ -0.383029549 1.000000000 0.9928258613 0.2090703809 -0.01223876
LIVP2 $ \alpha_m $ 1.5905e-08 -2.000000000 -1.99993129 -1.999993802 1.320889355
$ \beta_m $ -2.000000000 -1.028027457 -0.640309044 1.893006788 -0.627961135
LBVP1 $ \alpha_m $ -0.23700822 0.2939052397 1.7497711122 -0.815400817 -0.28105974
$ \beta_m $ -0.237008298 0.2940587929 -1.172283084 1.983810204 -2.525130497
NLIVP1 $ \alpha_m $ -6.595e-07 -2.000000000 -1.969510526 -2.000000000 -1.99980317
$ \beta_m $ -1.999999998 0.1334264437 -1.999346637 0.2828697917 -1.132644395
NLIVP2 $ \alpha_m $ 0.00000000 1.00000000 1.00000000 0.9159523195 1.00000000
$ \beta_m $ 1.00000000 1.00000000 0.9159523195 1.00000000 0.00000000
NLIVP3 $ \alpha_m $ 1.00000000 0.501789937 0.3963523812 0.00000000 0.00000000
$ \beta_m $ 0.00000000 0.00000000 1.00000000 0.501789937 0.3963523812
NLIVP4 $ \alpha_m $ -3.556e-10 1.1605e-09 1.371544377 1.705016485 1.664659425
$ \beta_m $ -0.699013232 -0.795001437 -1.237894199 -0.487132482 -0.20116878
NLIVP5 $ \alpha_m $ -1.0000000 -0.999999714 0.9996994842 1.00000000 -0.670316746
$ \beta_m $ -0.999999761 -1.0000000 0.0694451202 -0.363172581 0.4710444283
NLIVP6 $ \alpha_m $ 9.8817e-08 0.499999967 -0.128335779 -0.225730262 0.4574883022
$ \beta_m $ 1.00000000 -0.334983343 0.0548712929 0.1919886974 0.4900184664
NLBVP1 $ \alpha_m $ 6.0988e-07 -1.0000000 -0.168001861 0.1680024128 1.00000000
$ \beta_m $ 0.3182717396 0.5556535208 0.2683350684 0.1086186654 -0.242687286
NLBVP2 $ \alpha_m $ 1.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$ \beta_m $ 0.00000000 0.00000000 0.00000000 0.0069444444 0.00000000
NLBVP3 $ \alpha_m $ -2.979e-11 -1.512033065 0.5287945264 -0.376563712 1.359802251
$ \beta_m $ 1.512152004 0.2390046775 1.385502167 0.2230003398 0.4953153062
NLBVP4 $ \alpha_m $ 2.6966e-06 -3.398796238 -0.916386583 -3.997792287 -2.079546692
$ \beta_m $ -4.00000000 -1.420908971 -2.843010022 -3.999816103 1.8712207261
Table 5.  Comparison of expansions: PEA, TEA and FEA. The capital letters P, T and F denotes when PEA, TEA and FEA is the best results respectively.
Equation PEA TEA FEA
No. It RMSE Time
(s)
No. It RMSE Time
(s)
No. It RMSE Time
(s)
LIVP1 047 6.98e-04 26.79 113 9.70e-04 37.70 349 3.28e-02 090.10 P
LIVP2 017 3.35e-04 10.37 062 8.50e-04 21.39 500 4.68e-02 130.30 P
LBVP1 019 6.49e-04 09.25 081 9.87e-04 21.64 128 9.69e-04 027.38 P
NLIVP1 049 9.53e-04 25.92 122 9.90e-04 39.58 500 1.50e-02 093.13 P
NLIVP2 006 4.95e-04 03.61 003 0.00e-00 02.06 500 2.90e-02 193.67 T
NLIVP3 006 6.83e-04 09.67 500 1.36e-03 247.7 500 9.90e-03 233.29 P
NLIVP4 028 9.80e-04 28.07 500 8.60e-03 189.1 500 6.00e-02 181.89 P
NLIVP5 008 6.70e-04 7.122 500 2.60e-03 238.3 500 2.30e-02 201.20 P
NLIVP6 014 7.95e-04 18.14 040 9.90e-04 36.58 500 9.10e-03 339.87 P
NLBVP1 014 9.01e-04 08.89 500 2.10e-02 151.0 500 4.80e-03 152.02 P
NLBVP2 013 8.69e-04 12.08 500 2.14e-01 143.2 500 6.30e-01 126.00 P
NLBVP3 014 8.59e-04 16.40 500 5.80e-03 158.5 500 1.49e-03 226.66 P
NLBVP4 013 4.58e-04 10.62 500 2.30e-03 200.7 500 2.15e-03 121.30 P
Equation PEA TEA FEA
No. It RMSE Time
(s)
No. It RMSE Time
(s)
No. It RMSE Time
(s)
LIVP1 047 6.98e-04 26.79 113 9.70e-04 37.70 349 3.28e-02 090.10 P
LIVP2 017 3.35e-04 10.37 062 8.50e-04 21.39 500 4.68e-02 130.30 P
LBVP1 019 6.49e-04 09.25 081 9.87e-04 21.64 128 9.69e-04 027.38 P
NLIVP1 049 9.53e-04 25.92 122 9.90e-04 39.58 500 1.50e-02 093.13 P
NLIVP2 006 4.95e-04 03.61 003 0.00e-00 02.06 500 2.90e-02 193.67 T
NLIVP3 006 6.83e-04 09.67 500 1.36e-03 247.7 500 9.90e-03 233.29 P
NLIVP4 028 9.80e-04 28.07 500 8.60e-03 189.1 500 6.00e-02 181.89 P
NLIVP5 008 6.70e-04 7.122 500 2.60e-03 238.3 500 2.30e-02 201.20 P
NLIVP6 014 7.95e-04 18.14 040 9.90e-04 36.58 500 9.10e-03 339.87 P
NLBVP1 014 9.01e-04 08.89 500 2.10e-02 151.0 500 4.80e-03 152.02 P
NLBVP2 013 8.69e-04 12.08 500 2.14e-01 143.2 500 6.30e-01 126.00 P
NLBVP3 014 8.59e-04 16.40 500 5.80e-03 158.5 500 1.49e-03 226.66 P
NLBVP4 013 4.58e-04 10.62 500 2.30e-03 200.7 500 2.15e-03 121.30 P
Table 6.  Comparison of fitness function: WRF, LSWF and DLSWF. W, L and D symbols indicates that method PEA-PSO-WRF, PEA-PSO-LSWF and PEA-PSO-DLSWF is the best results respectively.
Equation WRF LSWF DLSWF
No. It RMSE Time
(s)
No. It RMSE Time
(s)
No. It RMSE Time
(s)
LIVP1 012 8.03e-04 14.13 014 5.91e-03 22.88 039 1.24e-03 58.56 W
LIVP2 012 8.98e-04 15.68 016 9.57e-04 29.54 030 6.09e-04 22.33 W
LBVP1 031 6.76e-04 23.92 027 9.14e-04 25.76 018 9.81e-04 11.42 D
NLIVP1 212 8.38e-04 270.9 150 9.94e-04 182.0 057 9.51e-04 77.09 D
NLIVP2 004 1.42e-17 07.64 003 1.44e-17 04.26 003 0.00e-00 07.48 L
NLIVP3 003 3.79e-17 08.43 003 3.79e-17 08.77 003 3.79e-17 07.71 D
NLIVP4 077 9.81e-04 172.9 246 8.19e-04 486.1 016 8.96e-04 39.77 D
NLIVP5 013 7.60e-04 40.40 018 8.52e-04 36.23 009 6.52e-04 25.20 D
NLIVP6 027 7.14e-04 85.23 018 7.86e-04 39.10 012 9.79e-04 23.62 D
NLBVP1 034 6.13e-04 54.26 074 9.14e-04 114.2 018 8.97e-04 33.88 D
NLBVP2 019 1.86e-04 35.08 027 7.55e-05 48.71 012 2.52e-04 17.02 D
NLBVP3 016 3.75e-04 27.01 011 8.91e-04 18.40 018 9.11e-04 28.41 L
NLBVP4 015 8.11e-04 25.80 016 7.19e-04 22.62 021 8.84e-04 24.08 L
Equation WRF LSWF DLSWF
No. It RMSE Time
(s)
No. It RMSE Time
(s)
No. It RMSE Time
(s)
LIVP1 012 8.03e-04 14.13 014 5.91e-03 22.88 039 1.24e-03 58.56 W
LIVP2 012 8.98e-04 15.68 016 9.57e-04 29.54 030 6.09e-04 22.33 W
LBVP1 031 6.76e-04 23.92 027 9.14e-04 25.76 018 9.81e-04 11.42 D
NLIVP1 212 8.38e-04 270.9 150 9.94e-04 182.0 057 9.51e-04 77.09 D
NLIVP2 004 1.42e-17 07.64 003 1.44e-17 04.26 003 0.00e-00 07.48 L
NLIVP3 003 3.79e-17 08.43 003 3.79e-17 08.77 003 3.79e-17 07.71 D
NLIVP4 077 9.81e-04 172.9 246 8.19e-04 486.1 016 8.96e-04 39.77 D
NLIVP5 013 7.60e-04 40.40 018 8.52e-04 36.23 009 6.52e-04 25.20 D
NLIVP6 027 7.14e-04 85.23 018 7.86e-04 39.10 012 9.79e-04 23.62 D
NLBVP1 034 6.13e-04 54.26 074 9.14e-04 114.2 018 8.97e-04 33.88 D
NLBVP2 019 1.86e-04 35.08 027 7.55e-05 48.71 012 2.52e-04 17.02 D
NLBVP3 016 3.75e-04 27.01 011 8.91e-04 18.40 018 9.11e-04 28.41 L
NLBVP4 015 8.11e-04 25.80 016 7.19e-04 22.62 021 8.84e-04 24.08 L
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