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

• *Corresponding author
• 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.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Table 1.  Maximum absolute error (MAE) for example 1

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $3.60E-05$ $3.79E-05$ $\left[ -1,1\right]$ $1+2.00E-06$ $3.77E-05$ $3.79E-05$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.26E-08$ $1.27E-08$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $1.93E-09$ $1.95E-09$ $\left[ 0,1\right]$ $1+3.10E-07$ $8.34E-07$ $8.45E-07$

Table 2.  Mean squared error (MSE) for example 1

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.98E-05$ $1.26E-10$ $1.41E-10$ $\left[ -1,0.5\right]$ $1+1.98E-05$ $8.67E-11$ $9.72E-11$ $\left[ -1,1\right]$ $1+2.00E-06$ $7.32E-11$ $7.41E-11$ $\left[ -0.5,0\right]$ $1+2.60E-09$ $2.68E-17$ $2.70E-17$ $\left[ -0.5,0.5\right]$ $1+2.50E-09$ $1.50E-17$ $1.51E-17$ $\left[ -0.5,1\right]$ $1+3.10E-07$ $5.17E-14$ $5.33E-14$ $\left[ 0,0.5\right]$ $1+2.60E-09$ $6.29E-19$ $6.47E-19$ $\left[ 0,1\right]$ $1+3.10E-07$ $7.51E-14$ $7.78E-14$

Table 3.  Maximum absolute error for example 2

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ -1,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $4.23E-07$ $1.30E-06$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $1.33E-03$ $3.20E-03$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $4.17E-07$ $1.30E-06$ $\left[ 0,1\right]$ $1+1.25E-03$ $1.28E-03$ $3.20E-03$

Table 4.  Mean squared error (MSE) for example 2

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$ $\left[ -1,0.5\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ -1,1\right]$ $1+1.25E-03$ $3.27E-07$ $1.06E-06$ $\left[ -0.5,0\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ -0.5,0.5\right]$ $1+3.23E-06$ $8.31E-14$ $3.10E-13$ $\left[ -0.5,1\right]$ $1+1.22E-03$ $2.25E-07$ $6.95E-07$ $\left[ 0,0.5\right]$ $1+3.23E-06$ $7.62E-14$ $2.84E-13$ $\left[ 0,1\right]$ $1+1.25E-03$ $3.12E-07$ $1.01E-06$

Table 5.  Maximum absolute error for example 3

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MAE MAE $\left[ -1,0\right]$ $0.9969649575$ $4.77E-03$ $6.90E-03$ $\left[ -1,0.5\right]$ $0.9974640403$ $5.12E-03$ $6.90E-03$ $\left[ -1,1\right]$ $0.9960515183$ $6.36E-03$ $9.16E-03$ $\left[ -0.5,0\right]$ $1.0004736382$ $3.69E-05$ $2.44E-04$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $1.62E-04$ $2.44E-04$ $\left[ -0.5,1\right]$ $0.9960267479$ $6.34E-03$ $9.16E-03$ $\left[ 0,0.5\right]$ $0.9999618650$ $4.83E-05$ $6.52E-05$ $\left[ 0,1\right]$ $0.9951507666$ $5.72E-03$ $9.16E-03$

Table 6.  Mean squared error (MSE) for example 3

 Domain Proposed Method LASPM, $\varepsilon =1$ $\left[ -1,1\right]$ $\varepsilon$ MSE MSE $\left[ -1,0\right]$ $0.9969649575$ $3.27E-06$ $5.22E-06$ $\left[ -1,0.5\right]$ $0.9974640403$ $2.47E-06$ $3.59E-06$ $\left[ -1,1\right]$ $0.9960515183$ $4.50E-06$ $7.97E-06$ $\left[ -0.5,0\right]$ $1.0004736382$ $6.48E-10$ $1.80E-08$ $\left[ -0.5,0.5\right]$ $1.0002174260$ $6.28E-09$ $1.03E-08$ $\left[ -0.5,1\right]$ $0.9960267479$ $4.10E-06$ $6.88E-06$ $\left[ 0,0.5\right]$ $0.9999618650$ $7.32E-10$ $8.45E-10$ $\left[ 0,1\right]$ $0.9951507666$ $4.96E-06$ $9.99E-06$
• Tables(6)

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