# American Institute of Mathematical Sciences

December  2021, 11(4): 633-644. doi: 10.3934/naco.2021001

## Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization

 Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq

*Corresponding author

Received  March 2020 Revised  December 2020 Published  December 2021 Early access  January 2021

In this paper, Lyapunov's artificial small parameter method (LASPM) with continuous particle swarm optimization (CPSO) is presented and used for solving nonlinear differential equations. The proposed method, LASPM-CPSO, is based on estimating the $\varepsilon$ parameter in LASPM through a PSO algorithm and based on a proposed objective function. Three different examples are used to evaluate the proposed method LASPM-CPSO, and compare it with the classical method LASPM through different intervals of the domain. The results from the maximum absolute error (MAE) and mean squared error (MSE) obtained through the given examples show the reliability and efficiency of the proposed LASPM-CPSO method, compared to the classical method LASPM.

Citation: Omar Saber Qasim, Ahmed Entesar, Waleed Al-Hayani. Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 633-644. doi: 10.3934/naco.2021001
##### References:
 [1] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, New York, 1986.  doi: 10.1016/B978-0-12-044375-8.50024-1. [2] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2569-4. [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-015-8289-6. [4] K. Ahmad, A. Qaiser and A. H. Soori, Nonlinear least squares solutions of ordinary differential equations, Mitteilungen Klosterneuburg, 2 (2020). [5] Y. Aksoy and M. Pakdemirli, New perturbation solutions for Bratu-type equtions, Computers and Mathematics with Applications, 59 (2010), 2802-2808.  doi: 10.1016/j.camwa.2010.01.050. [6] W. Al-Hayani, L. Alzubaidy and A. Entesar, Solutions of singular IVP's of Lane-Emden type by Homotopy analysis method with genetic algorithm, Appl. Math. Inf. Sci., 11 (2017), 1-10. [7] N. A. Al-Thanoon, O. S. Qasim and Z. Y. Algamal, Tuning parameter estimation in SCAD-support vector machine using firefly algorithm with application in gene selection and cancer classification, Computers in Biology and Medicine, 2018. [8] I. V. Andrianov, J. Awrejcewicz and A. Ivankov, Artificial small parameter method-solving mixed boundary value problems, Mathematical Problems in Engineering, 2005 (2005), 325-340.  doi: 10.1155/MPE.2005.325. [9] M. Babaei, A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization, Applied Soft Computing, 13 (2013), 3354-3365. [10] N. Delgarm, B. Sajadi, F. Kowsary and S. Delgarm, Multi-objective optimization of the building energy performance: A simulation-based approach by means of particle swarm optimization (PSO), Applied Energy, 170 (2016), 293-303. [11] S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University-Science, 30 (2018), 91-99. [12] A. Entesar, O. Saber and W. Al-Hayani, Hybridization of genetic algorithm with homotopy analysis method for solving fractional partial differential equations, Eurasian Journal of Science & Engineering, 4 (2019). [13] J.-H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3. [14] J.-H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34 (1999), 699-708. [15] S. Liao, Beyond Perturbation-Introduction to the Homotopy Analysis Method, CRC Press LLC, 2004. [16] S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer, 2012. [17] A. M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, 1992. doi: 10.1080/00207179208934253. [18] A. M. Lyapunov, The general problem of the stability of motion, Int. J. Control, 553 (1992), 531-534.  doi: 10.1080/00207179208934253. [19] O. Moaaz, I. Dassios, O. Bazighifan and A. Muhib, Oscillation theorems for nonlinear differential equations of fourth-order, Mathematics, 8 (2020), 520. doi: 10.3934/math.2020414. [20] A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, 2000. doi: 10.1002/9783527617609. [21] A. Ouyang, Y. Zhou and Q. Luo, Hybrid particle swarm optimization algorithm for solving systems of nonlinear equations, in 2009 IEEE International Conference on Granular Computing, (2009), 460–465. [22] O. S. Qasim and Z. Y. Algamal, Feature selection using particle swarm optimization-based logistic regression model, Chemometrics and Intelligent Laboratory Systems, 182 (2018), 41-46. [23] M. S. Rawashdeh and S. Maitama, Solving nonlinear ordinary differential equations using the NDM, J. Appl. Anal. Comput., 5 (2015), 77-88.  doi: 10.11948/2015007. [24] Y. Shi, Particle swarm optimization: developments, applications and resources, in Evolutionary Computation, Proceedings of the 2001 Congress (2001), 81–86. [25] Y. Shi and R. C. Eberhart, Parameter selection in particle swarm optimization, in International Conference on Evolutionary Programming, (1998), 591–600. [26] X. Zhang and S. Liang, Adomian decomposition method is a special case of Lyapunov's artificial small parameter method, Applied Mathematics Letters, 48 (2015), 177–179. doi: 10.1016/j.aml.2015.04.011.

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##### References:
 [1] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, New York, 1986.  doi: 10.1016/B978-0-12-044375-8.50024-1. [2] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-009-2569-4. [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. doi: 10.1007/978-94-015-8289-6. [4] K. Ahmad, A. Qaiser and A. H. Soori, Nonlinear least squares solutions of ordinary differential equations, Mitteilungen Klosterneuburg, 2 (2020). [5] Y. Aksoy and M. Pakdemirli, New perturbation solutions for Bratu-type equtions, Computers and Mathematics with Applications, 59 (2010), 2802-2808.  doi: 10.1016/j.camwa.2010.01.050. [6] W. Al-Hayani, L. Alzubaidy and A. Entesar, Solutions of singular IVP's of Lane-Emden type by Homotopy analysis method with genetic algorithm, Appl. Math. Inf. Sci., 11 (2017), 1-10. [7] N. A. Al-Thanoon, O. S. Qasim and Z. Y. Algamal, Tuning parameter estimation in SCAD-support vector machine using firefly algorithm with application in gene selection and cancer classification, Computers in Biology and Medicine, 2018. [8] I. V. Andrianov, J. Awrejcewicz and A. Ivankov, Artificial small parameter method-solving mixed boundary value problems, Mathematical Problems in Engineering, 2005 (2005), 325-340.  doi: 10.1155/MPE.2005.325. [9] M. Babaei, A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization, Applied Soft Computing, 13 (2013), 3354-3365. [10] N. Delgarm, B. Sajadi, F. Kowsary and S. Delgarm, Multi-objective optimization of the building energy performance: A simulation-based approach by means of particle swarm optimization (PSO), Applied Energy, 170 (2016), 293-303. [11] S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University-Science, 30 (2018), 91-99. [12] A. Entesar, O. Saber and W. Al-Hayani, Hybridization of genetic algorithm with homotopy analysis method for solving fractional partial differential equations, Eurasian Journal of Science & Engineering, 4 (2019). [13] J.-H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999), 257-262.  doi: 10.1016/S0045-7825(99)00018-3. [14] J.-H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34 (1999), 699-708. [15] S. Liao, Beyond Perturbation-Introduction to the Homotopy Analysis Method, CRC Press LLC, 2004. [16] S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer, 2012. [17] A. M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, 1992. doi: 10.1080/00207179208934253. [18] A. M. Lyapunov, The general problem of the stability of motion, Int. J. Control, 553 (1992), 531-534.  doi: 10.1080/00207179208934253. [19] O. Moaaz, I. Dassios, O. Bazighifan and A. Muhib, Oscillation theorems for nonlinear differential equations of fourth-order, Mathematics, 8 (2020), 520. doi: 10.3934/math.2020414. [20] A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, 2000. doi: 10.1002/9783527617609. [21] A. Ouyang, Y. Zhou and Q. Luo, Hybrid particle swarm optimization algorithm for solving systems of nonlinear equations, in 2009 IEEE International Conference on Granular Computing, (2009), 460–465. [22] O. S. Qasim and Z. Y. Algamal, Feature selection using particle swarm optimization-based logistic regression model, Chemometrics and Intelligent Laboratory Systems, 182 (2018), 41-46. [23] M. S. Rawashdeh and S. Maitama, Solving nonlinear ordinary differential equations using the NDM, J. Appl. Anal. Comput., 5 (2015), 77-88.  doi: 10.11948/2015007. [24] Y. Shi, Particle swarm optimization: developments, applications and resources, in Evolutionary Computation, Proceedings of the 2001 Congress (2001), 81–86. [25] Y. Shi and R. C. Eberhart, Parameter selection in particle swarm optimization, in International Conference on Evolutionary Programming, (1998), 591–600. [26] X. Zhang and S. Liang, Adomian decomposition method is a special case of Lyapunov's artificial small parameter method, Applied Mathematics Letters, 48 (2015), 177–179. doi: 10.1016/j.aml.2015.04.011.
Maximum absolute error (MAE) for example 1
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,1\right]$ $1+2.00E-06$ $3.77E-05$ $3.79E-05$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $1.93E-09$ $1.95E-09$ $\left[ 0,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,1\right]$ $1+2.00E-06$ $3.77E-05$ $3.79E-05$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $1.93E-09$ $1.95E-09$ $\left[ 0,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$
Mean squared error (MSE) for example 1
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.98E-05$ $1.26E-10$ $1.41E-10$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $8.67E-11$ $9.72E-11$ $\left[ -1,1\right]$ $1+2.00E-06$ $7.32E-11$ $7.41E-11$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $2.68E-17$ $2.70E-17$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.50E-17$ $1.51E-17$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $5.17E-14$ $5.33E-14$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $6.29E-19$ $6.47E-19$ $\left[ 0,1\right]$ $1+3.10E-07$ $7.51E-14$ $7.78E-14$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.98E-05$ $1.26E-10$ $1.41E-10$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $8.67E-11$ $9.72E-11$ $\left[ -1,1\right]$ $1+2.00E-06$ $7.32E-11$ $7.41E-11$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $2.68E-17$ $2.70E-17$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.50E-17$ $1.51E-17$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $5.17E-14$ $5.33E-14$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $6.29E-19$ $6.47E-19$ $\left[ 0,1\right]$ $1+3.10E-07$ $7.51E-14$ $7.78E-14$
Maximum absolute error for example 2
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ -1,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $4.17E-07$ $1.30E-06$ $\left[ 0,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ -1,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $4.17E-07$ $1.30E-06$ $\left[ 0,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$
Mean squared error (MSE) for example 2
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ -1,1\right]$ $1+1.25E-03$ $3.27E-07$ $1.06E-06$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $8.31E-14$ $3.10E-13$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ 0,1\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ -1,1\right]$ $1+1.25E-03$ $3.27E-07$ $1.06E-06$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $8.31E-14$ $3.10E-13$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ 0,1\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$
Maximum absolute error for example 3
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $0.9969649575$ $4.77E-03$ $6.90E-03$ $\left[ -1,0.5\right]$ $0.9974640403$ $5.12E-03$ $6.90E-03$ $\left[ -1,1\right]$ $0.9960515183$ $6.36E-03$ $9.16E-03$ $\left[ -0.5,0\right]$ $1.0004736382$ $3.69E-05$ $2.44E-04$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $1.62E-04$ $2.44E-04$ $\left[ -0.5,1\right]$ $0.9960267479$ $6.34E-03$ $9.16E-03$ $\left[ 0,0.5\right]$ $0.9999618650$ $4.83E-05$ $6.52E-05$ $\left[ 0,1\right]$ $0.9951507666$ $5.72E-03$ $9.16E-03$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $0.9969649575$ $4.77E-03$ $6.90E-03$ $\left[ -1,0.5\right]$ $0.9974640403$ $5.12E-03$ $6.90E-03$ $\left[ -1,1\right]$ $0.9960515183$ $6.36E-03$ $9.16E-03$ $\left[ -0.5,0\right]$ $1.0004736382$ $3.69E-05$ $2.44E-04$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $1.62E-04$ $2.44E-04$ $\left[ -0.5,1\right]$ $0.9960267479$ $6.34E-03$ $9.16E-03$ $\left[ 0,0.5\right]$ $0.9999618650$ $4.83E-05$ $6.52E-05$ $\left[ 0,1\right]$ $0.9951507666$ $5.72E-03$ $9.16E-03$
Mean squared error (MSE) for example 3
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $0.9969649575$ $3.27E-06$ $5.22E-06$ $\left[ -1,0.5\right]$ $0.9974640403$ $2.47E-06$ $3.59E-06$ $\left[ -1,1\right]$ $0.9960515183$ $4.50E-06$ $7.97E-06$ $\left[ -0.5,0\right]$ $1.0004736382$ $6.48E-10$ $1.80E-08$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $6.28E-09$ $1.03E-08$ $\left[ -0.5,1\right]$ $0.9960267479$ $4.10E-06$ $6.88E-06$ $\left[ 0,0.5\right]$ $0.9999618650$ $7.32E-10$ $8.45E-10$ $\left[ 0,1\right]$ $0.9951507666$ $4.96E-06$ $9.99E-06$
 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $0.9969649575$ $3.27E-06$ $5.22E-06$ $\left[ -1,0.5\right]$ $0.9974640403$ $2.47E-06$ $3.59E-06$ $\left[ -1,1\right]$ $0.9960515183$ $4.50E-06$ $7.97E-06$ $\left[ -0.5,0\right]$ $1.0004736382$ $6.48E-10$ $1.80E-08$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $6.28E-09$ $1.03E-08$ $\left[ -0.5,1\right]$ $0.9960267479$ $4.10E-06$ $6.88E-06$ $\left[ 0,0.5\right]$ $0.9999618650$ $7.32E-10$ $8.45E-10$ $\left[ 0,1\right]$ $0.9951507666$ $4.96E-06$ $9.99E-06$
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