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

  • * Corresponding author: Muzhou Hou

    * Corresponding author: Muzhou Hou

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

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  • 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.

    Mathematics Subject Classification: Primary: 34A45; Secondary: 68W25, 68T99.

    Citation:

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  • Figure 1.  Second-order nonlinear BVP

    Figure 2.  The logarithmic relation between σ and MSE(Example 1)

    Figure 3.  Second-order nonlinear IVP

    Figure 4.  M-order nonlinear IVP

    Figure 5.  Second order nonlinear BVP (Example 4)

    Figure 6.  M-order nonlinear BVP(Example 5)

    Table 1.  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 $
     | Show Table
    DownLoad: CSV

    Table 2.  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
     | Show Table
    DownLoad: CSV

    Table 3.  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} $
     | Show Table
    DownLoad: CSV

    Table 4.  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 $
     | Show Table
    DownLoad: CSV

    Table 5.  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
     | Show Table
    DownLoad: CSV

    Table 6.  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} $
     | Show Table
    DownLoad: CSV

    Table 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
     | Show Table
    DownLoad: CSV

    Table 8.  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
     | Show Table
    DownLoad: CSV

    Table 9.  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} $
     | Show Table
    DownLoad: CSV

    Table 10.  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
     | Show Table
    DownLoad: CSV

    Table 11.  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
     | Show Table
    DownLoad: CSV

    Table 12.  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
     | Show Table
    DownLoad: CSV

    Table 13.  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
     | Show Table
    DownLoad: CSV

    Table 14.  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} $
     | Show Table
    DownLoad: CSV
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