Fig 1.
Graphical illustration of our soft computing procedure for doubly singular non-linear ODEs and Porous fin model.
Fig 2.
Pseudo-code of our soft computing technique.
Fig 3.
Neural network architecture.
Fig 4.
Best weights obtained, convergence of error values and step sizes used to reach the best solution for problem 1 case 1, 2, 3 using feed-forward ANNs based on FO-DPSO algorithm.
Fig 5.
Best weights obtained, convergence of error values and step sizes used to reach the best solution for problem 2 case 1, 2, 3 using feed-forward ANNs based on FO-DPSO algorithm.
Fig 6.
Best weights obtained, convergence of error values and step sizes used to reach the best solution for problem 3 case 1, 2, 3 using feed-forward ANNs based on FO-DPSO algorithm.
Fig 7.
Graphical illustration of absolute errors in best solutions, for problem 1, 2 and 3 (Case 1, 2, 3), obtained by FO-DPSO and GA.
Table 1.
Empirical solutions for problem 1 (Case 1, 2, 3) achieved by FO-DPSO and GA.
Which are compared with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.05.
Table 2.
Absolute errors in results for problem 1 (Case 1, 2, 3) achieved by FO-DPSO and GA-SQP.
Which are matched with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.2.
Table 3.
Empirical solutions for problem 2 (Case 1, 2, 3) achieved by FO-DPSO and GA.
Which are compared with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.05.
Table 4.
Absolute errors in results for problem 2 (Case 1, 2, 3) achieved by FO-DPSO and GA.
Which are matched with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.2.
Table 5.
Empirical solutions for problem 3 (Case 1, 2, 3) achieved by FO-DPSO and GA.
Which are compared with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.05.
Table 6.
Absolute errors in results for problem 3 (Case 1, 2, 3) achieved by FO-DPSO and GA-SQP.
Which are matched with exact solutions for inputs x varying from 0 to 1 with a step size h = 0.2.
Table 7.
Comparison of solution obtained for the problem 4 of porous fin designed model using FO-DPSO.
Fig 8.
Design of a porous fin.
Fig 9.
Solution obtained by our proposed method and other comparative algorithms for porous fin model.
Fig 10.
Graphical illustration of sorted absolute errors in solutions, for problem 1 (Case 1, 2, 3), obtained by FO-DPSO during 100 runs.
Fig 11.
Graphical illustration of sorted absolute errors in solutions, for problem 2 (Case 1, 2, 3), obtained by FO-DPSO during 100 runs.
Fig 12.
Graphical illustration of sorted absolute errors in solutions, for problem 3 (Case 1, 2, 3), obtained by FO-DPSO during 100 runs.
Table 8.
Performance indicators based on proposed results for Problem 1.
Table 9.
Performance indicators based on proposed results for Problem 2.
Table 10.
Performance indicators based on proposed results for Problem 3.
Fig 13.
Normal plots of MAE obtained by FO-DPSO during 100 runs.
Fig 14.
Normal plots of MAE obtained by FO-DPSO during 100 runs.