# American Institute of Mathematical Sciences

May  2020, 16(3): 1481-1502. doi: 10.3934/jimo.2019012

## Solving higher order nonlinear ordinary differential equations with least squares support vector machines

 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author: Muzhou Hou

Received  April 2018 Revised  October 2018 Published  March 2019

Fund Project: The second author is supported by the National Social Science Foundation of China (13BTJ010)

In this paper, a numerical method based on least squares support vector machines has been developed to solve the initial and boundary value problems of higher order nonlinear ordinary differential equations. The numerical experiments have been performed on some nonlinear ordinary differential equations to validate the accuracy and reliability of our proposed LS–SVM model. Compared with the exact solution, the results obtained by our proposed LS–SVM model can achieve a very high accuracy. The proposed LS–SVM model could be a good tool for solving higher order nonlinear ordinary differential equations.

Citation: Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1481-1502. doi: 10.3934/jimo.2019012
##### References:

show all references

##### References:
Second-order nonlinear BVP
The logarithmic relation between σ and MSE(Example 1)
Second-order nonlinear IVP
M-order nonlinear IVP
Second order nonlinear BVP (Example 4)
M-order nonlinear BVP(Example 5)
Comparison between the exact solution and the approximate solution obtained by LS-SVM model for the training points (Example 1)
 Training data Exact solution LS-SVM Training data Exact solution LS-SVM $1.000$ 1.000000000 1.000000000 $1.550$ 1.215686275 1.215686275 $1.050$ 1.024390244 1.024390233 $1.600$ 1.230769231 1.230769232 $1.100$ 1.047619048 1.047619039 $1.650$ 1.245283019 1.245283019 $1.150$ 1.069767442 1.069767431 $1.700$ 1.259259259 1.259259258 $1.200$ 1.090909091 1.090909081 $1.750$ 1.272727273 1.272727273 $1.250$ 1.111111111 1.111111105 $1.800$ 1.285714286 1.285714289 $1.300$ 1.130434783 1.130434779 $1.850$ 1.298245614 1.298245619 $1.350$ 1.148936170 1.148936165 $1.900$ 1.310344828 1.310344831 $1.400$ 1.166666667 1.166666660 $1.950$ 1.322033898 1.322033904 $1.450$ 1.183673469 1.183673463 $2.000$ 1.333333333 1.333333333 $1.500$ 1.200000000 1.199999997 
 Training data Exact solution LS-SVM Training data Exact solution LS-SVM $1.000$ 1.000000000 1.000000000 $1.550$ 1.215686275 1.215686275 $1.050$ 1.024390244 1.024390233 $1.600$ 1.230769231 1.230769232 $1.100$ 1.047619048 1.047619039 $1.650$ 1.245283019 1.245283019 $1.150$ 1.069767442 1.069767431 $1.700$ 1.259259259 1.259259258 $1.200$ 1.090909091 1.090909081 $1.750$ 1.272727273 1.272727273 $1.250$ 1.111111111 1.111111105 $1.800$ 1.285714286 1.285714289 $1.300$ 1.130434783 1.130434779 $1.850$ 1.298245614 1.298245619 $1.350$ 1.148936170 1.148936165 $1.900$ 1.310344828 1.310344831 $1.400$ 1.166666667 1.166666660 $1.950$ 1.322033898 1.322033904 $1.450$ 1.183673469 1.183673463 $2.000$ 1.333333333 1.333333333 $1.500$ 1.200000000 1.199999997 
Exact and LS-SVM results for the testing points(Example 1)
 Testing data Exact solution LS-SVM Testing data Exact solution LS-SVM $1.0250$ 1.012345679 1.012345667 $1.5250$ 1.207920792 1.207920791 $1.0750$ 1.036144578 1.036144569 $1.5750$ 1.223300971 1.223300972 $1.1250$ 1.058823529 1.058823519 $1.6250$ 1.238095238 1.238095239 $1.1750$ 1.080459770 1.080459759 $1.6750$ 1.252336449 1.252336448 $1.2250$ 1.101123596 1.101123588 $1.7250$ 1.266055046 1.266055045 $1.2750$ 1.120879121 1.120879117 $1.7750$ 1.279279279 1.279279281 $1.3250$ 1.139784946 1.139784942 $1.8250$ 1.292035398 1.292035403 $1.3750$ 1.157894737 1.157894731 $1.8750$ 1.304347826 1.304347830 $1.4250$ 1.175257732 1.175257725 $1.9250$ 1.316239317 1.316239320 $1.4750$ 1.191919192 1.191919187 $1.9750$ 1.327731092 1.327731099
 Testing data Exact solution LS-SVM Testing data Exact solution LS-SVM $1.0250$ 1.012345679 1.012345667 $1.5250$ 1.207920792 1.207920791 $1.0750$ 1.036144578 1.036144569 $1.5750$ 1.223300971 1.223300972 $1.1250$ 1.058823529 1.058823519 $1.6250$ 1.238095238 1.238095239 $1.1750$ 1.080459770 1.080459759 $1.6750$ 1.252336449 1.252336448 $1.2250$ 1.101123596 1.101123588 $1.7250$ 1.266055046 1.266055045 $1.2750$ 1.120879121 1.120879117 $1.7750$ 1.279279279 1.279279281 $1.3250$ 1.139784946 1.139784942 $1.8250$ 1.292035398 1.292035403 $1.3750$ 1.157894737 1.157894731 $1.8750$ 1.304347826 1.304347830 $1.4250$ 1.175257732 1.175257725 $1.9250$ 1.316239317 1.316239320 $1.4750$ 1.191919192 1.191919187 $1.9750$ 1.327731092 1.327731099
Comparison of the MSE with the differential number of the training points(Example 1)
 N $\sigma$ $MSE_{test}$ $11$ 1.50 $2.2982\times10^{-14}$ $21$ 0.90 $3.2973\times10^{-17}$ $41$ 0.60 $1.3527\times10^{-19}$
 N $\sigma$ $MSE_{test}$ $11$ 1.50 $2.2982\times10^{-14}$ $21$ 0.90 $3.2973\times10^{-17}$ $41$ 0.60 $1.3527\times10^{-19}$
Comparison between the exact solution and the approximate solution obtained by LS-SVM model for the training points(Example 2)
 Training data Exact solution LS-SVM Training data Exact solution LS-SVM $0.0000$ 1.000000000 1.000000000 $0.2750$ 0.7843137255 0.783960261 $0.0250$ 0.975609756 0.975507638 $0.3000$ 0.7692307692 0.768994225 $0.0500$ 0.952380952 0.952038212 $0.3250$ 0.7547169811 0.754586586 $0.0750$ 0.930232558 0.929751041 $0.3500$ 0.7407407407 0.740709174 $0.1000$ 0.909090909 0.908583285 $0.3750$ 0.7272727273 0.727354567 $0.1250$ 0.888888889 0.888395226 $0.4000$ 0.7142857143 0.714516534 $0.1500$ 0.869565217 0.869067566 $0.4250$ 0.7017543860 0.702168828 $0.1750$ 0.851063830 0.850537044 $0.4500$ 0.6896551724 0.690259179 $0.2000$ 0.833333333 0.832783164 $0.4750$ 0.6779661017 0.678730163 $0.2250$ 0.816326531 0.815793928 $0.5000$ 0.6666666667 0.667567043 $0.2500$ 0.800000000 0.799538249 
 Training data Exact solution LS-SVM Training data Exact solution LS-SVM $0.0000$ 1.000000000 1.000000000 $0.2750$ 0.7843137255 0.783960261 $0.0250$ 0.975609756 0.975507638 $0.3000$ 0.7692307692 0.768994225 $0.0500$ 0.952380952 0.952038212 $0.3250$ 0.7547169811 0.754586586 $0.0750$ 0.930232558 0.929751041 $0.3500$ 0.7407407407 0.740709174 $0.1000$ 0.909090909 0.908583285 $0.3750$ 0.7272727273 0.727354567 $0.1250$ 0.888888889 0.888395226 $0.4000$ 0.7142857143 0.714516534 $0.1500$ 0.869565217 0.869067566 $0.4250$ 0.7017543860 0.702168828 $0.1750$ 0.851063830 0.850537044 $0.4500$ 0.6896551724 0.690259179 $0.2000$ 0.833333333 0.832783164 $0.4750$ 0.6779661017 0.678730163 $0.2250$ 0.816326531 0.815793928 $0.5000$ 0.6666666667 0.667567043 $0.2500$ 0.800000000 0.799538249 
Exact and LS-SVM results for the testing points(Example 2)
 Testing data Exact solution LS-SVM Testing data Exact solution LS-SVM $0.0125$ 0.9876543210 0.9876489423 $0.2625$ 0.7920792079 0.7916686867 $0.0375$ 0.9638554217 0.9636280130 $0.2875$ 0.7766990291 0.7764046131 $0.0625$ 0.9411764706 0.9407474275 $0.3125$ 0.7619047619 0.7617230005 $0.0875$ 0.9195402299 0.9190356064 $0.3375$ 0.7476635514 0.7475823709 $0.1125$ 0.8988764045 0.8983754394 $0.3625$ 0.7339449541 0.7339666547 $0.1375$ 0.8791208791 0.8786291359 $0.3875$ 0.7207207207 0.7208719671 $0.1625$ 0.8602150538 0.8597045711 $0.4125$ 0.7079646018 0.7082841451 $0.1875$ 0.8421052632 0.8415635623 $0.4375$ 0.6956521739 0.6961631623 $0.2125$ 0.8247422680 0.8241942736 $0.4625$ 0.6837606838 0.6844497170 $0.2375$ 0.8080808089 0.8075774037 $0.4875$ 0.6722689076 0.6731004361
 Testing data Exact solution LS-SVM Testing data Exact solution LS-SVM $0.0125$ 0.9876543210 0.9876489423 $0.2625$ 0.7920792079 0.7916686867 $0.0375$ 0.9638554217 0.9636280130 $0.2875$ 0.7766990291 0.7764046131 $0.0625$ 0.9411764706 0.9407474275 $0.3125$ 0.7619047619 0.7617230005 $0.0875$ 0.9195402299 0.9190356064 $0.3375$ 0.7476635514 0.7475823709 $0.1125$ 0.8988764045 0.8983754394 $0.3625$ 0.7339449541 0.7339666547 $0.1375$ 0.8791208791 0.8786291359 $0.3875$ 0.7207207207 0.7208719671 $0.1625$ 0.8602150538 0.8597045711 $0.4125$ 0.7079646018 0.7082841451 $0.1875$ 0.8421052632 0.8415635623 $0.4375$ 0.6956521739 0.6961631623 $0.2125$ 0.8247422680 0.8241942736 $0.4625$ 0.6837606838 0.6844497170 $0.2375$ 0.8080808089 0.8075774037 $0.4875$ 0.6722689076 0.6731004361
Comparison of the MSE with the differential number of the training points(Example 2)
 N $\sigma$ $MSE_{test}$ $11$ 0.1625 $4.1363\times10^{-7}$ $21$ 0.2235 $1.9693\times10^{-7}$ $41$ 0.4085 $1.8597\times10^{-7}$
 N $\sigma$ $MSE_{test}$ $11$ 0.1625 $4.1363\times10^{-7}$ $21$ 0.2235 $1.9693\times10^{-7}$ $41$ 0.4085 $1.8597\times10^{-7}$
Comparison between the exact solution and the approximate solution obtained by LS-SVM model for the training points(Example 3)
 Example 3 Exact solution LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.050$ 0.04997916927 0.04997915236 $0.100$ 0.09983341665 0.09983310498 $0.150$ 0.14943813247 0.14943658985 $0.200$ 0.19866933080 0.19866476297 $0.250$ 0.24740395925 0.24739375649 $0.300$ 0.29552020666 0.29550123081 $0.350$ 0.34289780746 0.34286692221 $0.400$ 0.38941834231 0.38937318159 $0.450$ 0.43496553411 0.43490549976 $0.500$ 0.47942553860 0.47935301511
 Example 3 Exact solution LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.050$ 0.04997916927 0.04997915236 $0.100$ 0.09983341665 0.09983310498 $0.150$ 0.14943813247 0.14943658985 $0.200$ 0.19866933080 0.19866476297 $0.250$ 0.24740395925 0.24739375649 $0.300$ 0.29552020666 0.29550123081 $0.350$ 0.34289780746 0.34286692221 $0.400$ 0.38941834231 0.38937318159 $0.450$ 0.43496553411 0.43490549976 $0.500$ 0.47942553860 0.47935301511
Exact and LS-SVM results for the testing points(Example 3)
 Example 3 Exact solution LS-SVM $0.0250$ 0.02499739591 0.02499739536 $0.0750$ 0.07492970727 0.07492961138 $0.1250$ 0.12467473339 0.12467397503 $0.1750$ 0.17410813759 0.17410536148 $0.2250$ 0.22310636213 0.22309934567 $0.2750$ 0.27154693696 0.27153275506 $0.3250$ 0.31930878586 0.31928421995 $0.3750$ 0.36627252909 0.36623471736 $0.4250$ 0.41232078174 0.41226810347 $0.4750$ 0.45733844718 0.45727163071
 Example 3 Exact solution LS-SVM $0.0250$ 0.02499739591 0.02499739536 $0.0750$ 0.07492970727 0.07492961138 $0.1250$ 0.12467473339 0.12467397503 $0.1750$ 0.17410813759 0.17410536148 $0.2250$ 0.22310636213 0.22309934567 $0.2750$ 0.27154693696 0.27153275506 $0.3250$ 0.31930878586 0.31928421995 $0.3750$ 0.36627252909 0.36623471736 $0.4250$ 0.41232078174 0.41226810347 $0.4750$ 0.45733844718 0.45727163071
Comparison of the MSE with the differential number of the training points(Example 3)
 N $\sigma$ $MSE_{test}$ $11$ 2.1200 $9.5313\times10^{-10}$ $21$ 4.3750 $1.2437\times10^{-10}$ $41$ 4.4595 $1.1478\times10^{-13}$
 N $\sigma$ $MSE_{test}$ $11$ 2.1200 $9.5313\times10^{-10}$ $21$ 4.3750 $1.2437\times10^{-10}$ $41$ 4.4595 $1.1478\times10^{-13}$
Comparison between bvp4c and the approximate solution obtained by LS-SVM model for the training points(Example 4)
 Example 4 Bvp4c LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.100$ 0.08208961709 0.07269777630 $0.200$ 0.16500909145 0.15151008989 $0.300$ 0.24963062540 0.23642312027 $0.400$ 0.33691524509 0.32742078522 $0.500$ 0.42796806735 0.42448474566 $0.600$ 0.52410852807 0.52759441177 $0.700$ 0.62696468677 0.63672695027 $0.800$ 0.73860596417 0.75185729280 $0.900$ 0.86173796851 0.87295814556 $1.000$ 1.00000000000 1.00000190735
 Example 4 Bvp4c LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.100$ 0.08208961709 0.07269777630 $0.200$ 0.16500909145 0.15151008989 $0.300$ 0.24963062540 0.23642312027 $0.400$ 0.33691524509 0.32742078522 $0.500$ 0.42796806735 0.42448474566 $0.600$ 0.52410852807 0.52759441177 $0.700$ 0.62696468677 0.63672695027 $0.800$ 0.73860596417 0.75185729280 $0.900$ 0.86173796851 0.87295814556 $1.000$ 1.00000000000 1.00000190735
Bvp4c and LS-SVM results for the testing points(Example 4)
 Example 4 Bvp4c LS-SVM $0.050$ 0.04099337478 0.03558379567 $0.150$ 0.12339244872 0.11134042665 $0.250$ 0.20704941350 0.19320496766 $0.350$ 0.29287463208 0.28116246679 $0.450$ 0.38189348425 0.37519571270 $0.550$ 0.47530819486 0.47528524009 $0.650$ 0.57457869449 0.58140933646 $0.750$ 0.68153387549 0.69354404985 $0.850$ 0.79853157226 0.81166319791 $0.950$ 0.92869807782 0.93573837804
 Example 4 Bvp4c LS-SVM $0.050$ 0.04099337478 0.03558379567 $0.150$ 0.12339244872 0.11134042665 $0.250$ 0.20704941350 0.19320496766 $0.350$ 0.29287463208 0.28116246679 $0.450$ 0.38189348425 0.37519571270 $0.550$ 0.47530819486 0.47528524009 $0.650$ 0.57457869449 0.58140933646 $0.750$ 0.68153387549 0.69354404985 $0.850$ 0.79853157226 0.81166319791 $0.950$ 0.92869807782 0.93573837804
Comparison between the exact solution and the approximate solution obtained by LS-SVM model for the training points(Example 5)
 Example 5 Exact solution LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.100$ 0.00072900000 0.00072904178 $0.200$ 0.00409600000 0.00409607260 $0.300$ 0.00926100000 0.01382410696 $0.400$ 0.01382400000 0.01390350516 $0.500$ 0.01562500000 0.01562510374 $0.600$ 0.01382400000 0.01382410574 $0.700$ 0.00926100000 0.00926110345 $0.800$ 0.00409600000 0.00409607300 $0.900$ 0.00072900000 0.00072904190 $1.000$ 0.00000000000 0.00000000000
 Example 5 Exact solution LS-SVM $0.000$ 0.00000000000 0.00000000000 $0.100$ 0.00072900000 0.00072904178 $0.200$ 0.00409600000 0.00409607260 $0.300$ 0.00926100000 0.01382410696 $0.400$ 0.01382400000 0.01390350516 $0.500$ 0.01562500000 0.01562510374 $0.600$ 0.01382400000 0.01382410574 $0.700$ 0.00926100000 0.00926110345 $0.800$ 0.00409600000 0.00409607300 $0.900$ 0.00072900000 0.00072904190 $1.000$ 0.00000000000 0.00000000000
Exact and LS-SVM results for the testing points(Example 5)
 Example 5 Exact solution LS-SVM $0.050$ 0.000107171875 0.000107186927 $0.150$ 0.002072671875 0.002072728701 $0.250$ 0.006591796875 0.006591885737 $0.350$ 0.011774546875 0.011774653640 $0.450$ 0.015160921875 0.015161024270 $0.550$ 0.015160921875 0.015161026227 $0.650$ 0.011774546875 0.011774654858 $0.750$ 0.006591796875 0.006591885679 $0.850$ 0.002072671875 0.002072728375 $0.950$ 0.000107171875 0.000107189695
 Example 5 Exact solution LS-SVM $0.050$ 0.000107171875 0.000107186927 $0.150$ 0.002072671875 0.002072728701 $0.250$ 0.006591796875 0.006591885737 $0.350$ 0.011774546875 0.011774653640 $0.450$ 0.015160921875 0.015161024270 $0.550$ 0.015160921875 0.015161026227 $0.650$ 0.011774546875 0.011774654858 $0.750$ 0.006591796875 0.006591885679 $0.850$ 0.002072671875 0.002072728375 $0.950$ 0.000107171875 0.000107189695
Comparison of the MSE with the differential number of the training points(Example 5)
 N $\sigma$ $MSE_{test}$ $11$ 1.8000 $6.7181\times10^{-15}$ $21$ 0.7000 $6.9489\times10^{-18}$ $41$ 0.2001 $1.6352\times10^{-19}$
 N $\sigma$ $MSE_{test}$ $11$ 1.8000 $6.7181\times10^{-15}$ $21$ 0.7000 $6.9489\times10^{-18}$ $41$ 0.2001 $1.6352\times10^{-19}$
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