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Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

  • * Corresponding author: Azzam S. Y. Aladool

    * Corresponding author: Azzam S. Y. Aladool
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  • 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.

    Mathematics Subject Classification: Primary: 34Bxx, 65Lxx, 65Kxx, 41A21; .

    Citation:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [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.
    [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.
    [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.
    [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.
    [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.
    [6] G. A. Baker and  P. Graves-MorrisPadé Approximants, Second edition: Encyclopedia of Mathematics and It's Applications, Cambridge University Press, 1996.  doi: 10.1017/CBO9780511530074.
    [7] A. BorzìModelling with Ordinary Differential Equations: A Comprehensive Approach, CRC Press, 2020. 
    [8] W. E. Boyce, R. C. DiPrima and D. B. Meade, Elementary Differential Equations and Boundary Value Problems, 11$^{nd}$ edition, WILEY, 2017.
    [9] R. L. Burden and J. D. Faires, Numerical Analysis, 9$^{nd}$ edition, Brooks/Cole, Cencag Learning, 2011.
    [10] S. Chakraverty and  Su smita MallArtificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations, CRC Press, 2017.  doi: 10.1201/9781315155265.
    [11] J. M. Chaquet and E. Carmona, Solving differential equations with Fourier series and evolution strategies, Appl. Soft Comput., 12 (2012), 3051–3062.
    [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.
    [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.
    [14] A. Engelbrecht, Particle swarm optimization: Velocity initialization, 2012 IEEE Congress on Evolutionary Computation, IEEE, (2012), 1–8.
    [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.
    [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.
    [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. 
    [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.
    [19] F. Mirzaee, Differential transform method for solving linear and nonlinear systems of ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3465-3472. 
    [20] T. RadhikaT. Iyengar and  T. RaniApproximate Analytical Methods for Solving Ordinary Differential Equations, CRC Press, 2014. 
    [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.
    [22] A. Sahu, S. K. Panigrahi and S. Pattnaik, Fast convergence particle swarm optimization for functions optimization, Procedia Technology, Elsevier, 4 (2012), 319–324.
    [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.
    [24] S. Talukder, Mathematicle Modelling and Applications of Particle Swarm Optimization, Mc. S. Thesis, Blekinge Institute of Technology, School of Engineering, 2011.
    [25] W. F. Trench, Elementary Differential Equations with Boundary Value Problems, Brooks/Cole Thomson Learning, 2013.
    [26] W. Van Assche, Padé and Hermite-Padé approximation and orthogonality, arXiv: math/0609094, 2006.
    [27] Z. Zhang, Y. Cai and D. Zhang, Solving ordinary differential equations with adaptive differential evolution, IEEE Access, IEEE, 8 (2020), 128908–128922.
    [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.
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