# 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 & Optimization, 2021, 11 (4) : 633-644. doi: 10.3934/naco.2021001
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##### References:
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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