A New Stochastic Technique for Painlevé Equation-I Using Neural Network Optimized with Swarm Intelligence

Table 4

Comparison of the results for solution of Painlevé I.

T	𝑢𝑘(𝑡) for VIM			𝑢𝑘(𝑥) for HPM				DE-NN
	𝑘=1	𝑘=2	𝑘=3	𝑘=2	𝑘=3	𝑘=4	𝑘=5	PSO	PSO-ASA
0.1	1.35𝐸−08	8.00𝐸−10	8.00𝐸−10	7.96𝐸−10	8.00𝐸−10	8.00𝐸−10	8.00𝐸−10	1.05𝐸−03	5.01𝐸−06
0.2	1.85𝐸−06	1.21𝐸−09	1.19𝐸−09	4.88𝐸−09	1.19𝐸−09	1.19𝐸−09	1.19𝐸−09	8.05𝐸−04	1.73𝐸−06
0.3	3.20𝐸−05	9.34𝐸−09	5.85𝐸−09	2.22𝐸−07	6.96𝐸−09	5.62𝐸−09	5.61𝐸−09	6.71𝐸−04	8.38𝐸−07
0.4	2.45𝐸−04	1.44𝐸−07	2.11𝐸−08	3.94𝐸−06	6.87𝐸−08	1.12𝐸−08	1.04𝐸−08	6.39𝐸−04	9.98𝐸−07
0.5	1.20𝐸−03	2.23𝐸−06	2.30𝐸−07	3.79𝐸−05	1.12𝐸−06	5.31𝐸−08	2.34𝐸−08	6.79𝐸−04	1.67𝐸−06
0.6	4.50𝐸−03	2.23𝐸−05	2.47𝐸−06	2.45𝐸−04	1.24𝐸−05	6.38𝐸−07	7.10𝐸−08	7.72𝐸−04	5.17𝐸−06
0.7	1.40𝐸−02	1.62𝐸−04	2.03𝐸−05	1.21𝐸−03	9.75𝐸−05	7.55𝐸−06	6.33𝐸−07	9.10𝐸−04	4.77𝐸−06
0.8	3.84𝐸−02	9.20𝐸−04	1.34𝐸−04	4.97𝐸−03	5.98𝐸−04	6.89𝐸−05	7.82𝐸−06	1.07𝐸−03	9.40𝐸−06
0.9	9.63𝐸−02	4.39𝐸−03	7.46𝐸−04	1.78𝐸−02	3.05𝐸−03	5.02𝐸−04	8.06𝐸−05	1.29𝐸−03	6.62𝐸−06
1.0	2.27𝐸−01	1.84𝐸−02	3.71𝐸−03	5.74𝐸−02	1.36𝐸−02	3.07𝐸−03	6.80𝐸−04	1.98𝐸−03	7.58𝐸−05
Mean	3.82𝐸−02	2.39𝐸−03	4.61𝐸−04	8.16𝐸−03	1.73𝐸−03	3.65𝐸−04	7.69𝐸−05	9.86𝐸−04	1.12𝐸−05