American Institute of Mathematical Sciences

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

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

Yanfei Lu , Qingfei Yin , Hongyi Li , Hongli Sun , Yunlei Yang and  Muzhou Hou ,

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

* Corresponding author: Muzhou Hou

Received  April 2018 Revised  October 2018 Published  May 2020 Early access  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.

Keywords: Nonlinear ordinary differential equations, initial value problems, boundary value problems, least squares support vector machines, approximate solutions.
Mathematics Subject Classification: Primary: 34A45; Secondary: 68W25, 68T99.
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:
[1]

D. O. Awoyemi, A p–stable linear multistep method for solving general third order ordinary differential equations, International Journal of Computer Mathematics, 80 (2003), 987-993.  doi: 10.1080/0020716031000079572.  Google Scholar

[2]

S. Chakraverty and S. Mall, Regression based weight generation algorithm in neural network for solution of initial and boundary value problems, Neural Computing and Applications, 25 (2014), 585-594.  doi: 10.1007/s00521-013-1526-4.  Google Scholar

[3]

E. H. Doha, A. H. Bhrawy and R. M. Hafez, On shifted Jacobi spectral method for high–order multi–point boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 3802-3810.  doi: 10.1016/j.cnsns.2012.02.027.  Google Scholar

[4]

N. M. Duy, H. See and T. T. Cong, A spectral collocation technique based on integrated chebyshev polynomials for biharmonic problems in irregular domains, Applied Mathematical Modelling, 33 (2009), 284-299.  doi: 10.1016/j.apm.2007.11.002.  Google Scholar

[5]

T. Hofmann, B. Schölkopf and A. J. Smola, Kernel methods in machine learning, Annals of Statistics, 36 (2008), 1171-1220.  doi: 10.1214/009053607000000677.  Google Scholar

[6]

G. B. Huang, Q. Y. Zhu and C. K. Siew, Extreme learning machine: Theory and applications, Neurocomputing, 70 (2006), 489-501.  doi: 10.1016/j.neucom.2005.12.126.  Google Scholar

[7]

K. Hussain, F. Ismail, N. Senu and F. Rabiei, Fourth–order improved Runge–Kutta method for directly solving special third–order ordinary differential equations, Iranian Journal of Science and Technology Transaction A–Science, 41 (2017), 429-437.  doi: 10.1007/s40995-017-0258-1.  Google Scholar

[8]

K. Hussain, F. Ismail and N. Senu, Solving directly special fourth–order ordinary differential equations using Runge–Kutta type method, Journal of Computational and Applied Mathematics, 306 (2016), 179-199.  doi: 10.1016/j.cam.2016.04.002.  Google Scholar

[9]

S. Islam, I. Aziz and B. Šarler, The numerical solution of second–order boundary value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling, 52 (2010), 1577-1590.  doi: 10.1016/j.mcm.2010.06.023.  Google Scholar

[10]

D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3nd edition, Brooks/Cole, Pacific Grove, CA, 1991.  Google Scholar

[11]

J. Kierzenka and L. F. Shampine, A BVP solver that controls residual and error, Journal of Numerical Analysis, Industrial and Applied Mathematics, 3 (2008), 27-41.   Google Scholar

[12]

M. Lakestani and M. Dehgan, The solution of a second–order nonlinear differential equation with Neumann boundary conditions using semi–orthogonal B–spline wavelets, International Journal of Computer Mathematics, 83 (2006), 685-694.  doi: 10.1080/00207160601025656.  Google Scholar

[13]

Z. A. Majid and M. Suleiman, Direct integration method implicit variable steps method for solving higher order systems of ordinary differential equations directly, Sains Malaysiana, 35 (2006), 63-68.   Google Scholar

[14]

Z. A. Majid, N. A. Azmi, M. Suleiman and Z. B. Ibrahaim, Solving directly general third order ordinary differential equations using two–point four step block method, Sains Malaysiana, 41 (2012), 623-632.   Google Scholar

[15]

A. Malek and R. S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network–optimization method, Applied Mathematics and Computation, 183 (2006), 260-271.  doi: 10.1016/j.amc.2006.05.068.  Google Scholar

[16]

S. Mall and S. Chakraverty, Application of Legendre neural network for solving ordinary differential equations, Applied Soft Computing, 43 (2016), 347-356.   Google Scholar

[17]

S. Mall and S. Chakraverty, Chebyshev neural network based model for solving Lane–Emden type equations, Applied Mathematics and Computation, 247 (2014), 100-114.  doi: 10.1016/j.amc.2014.08.085.  Google Scholar

[18]

S. Mehrkanoon, A direct variable step block multistep method for solving general third–order ODEs, Numerical Algorithms, 57 (2011), 53-66.  doi: 10.1007/s11075-010-9413-x.  Google Scholar

[19]

S. Mehrkanoon, T. Falck and J. A. K. Suykens, Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1356-1367.  doi: 10.1109/TNNLS.2012.2202126.  Google Scholar

[20]

S. Mehrkanoon and J. A. K. Suykens, Learning solutions to partial differential equations using LS–SVM, Neurocomputing, 159 (2015), 105-116.  doi: 10.1016/j.neucom.2015.02.013.  Google Scholar

[21]

S. Mehrkanoon and J. A. K. Suykens, LS–SVM approximate solution to linear time varying descriptor systems, Automatica, 48 (2012), 2502-2511.  doi: 10.1016/j.automatica.2012.06.095.  Google Scholar

[22]

J. Mercer, Functions of positive and negative type and their connection with the theory of integral equations, Philosophical Transactions of the Royal Society of London, 209 (1909), 415-446.   Google Scholar

[23]

R. Noberg, Differential equations for moments of present values in life insurance, Mathematics and Economics, 17 (1995), 171-180.  doi: 10.1016/0167-6687(95)00019-O.  Google Scholar

[24]

K. Parand and M. Hemami, Numerical study of astrophysics equations by meshless collocation method based on compactly supported radial basis function, International Journal of Applied and Computational Mathematics, 3 (2017), 1053-1075.  doi: 10.1007/s40819-016-0161-z.  Google Scholar

[25] H. J. Ricardo, A Modern Introduction to Differential Equations, CRC Press, Boca Raton, FL, 2010.   Google Scholar
[26]

P. P. See, Z. A. Majid and M. Suleiman, Three–step block method for solving nonlinear boundary value problems, Abstract and Applied Analysis, 2014 (2014), Art. ID 379829, 8 pp. doi: 10.1155/2014/379829.  Google Scholar

[27]

P. K. Srivastava, M. Kumar and R. N. Mohapatra, Quintic nonpolynomial spline method for the solution of a second–order boundary value problem with engineering applications, Computers and Mathematics with Applications, 62 (2011), 1707-1714.  doi: 10.1016/j.camwa.2011.06.012.  Google Scholar

[28]

H. Sun, M. Hou, Y. Yang, T. Zhang, F. Weng and F. Han, Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm, Neural Processing Letters, (2018), 1–20. doi: 10.1007/s11063-018-9911-8.  Google Scholar

[29]

J. A. K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural Processing Letters, 9 (1999) 293–300. Google Scholar

[30]

I. A. Tirmizi and E. H. Twizell, Higher–order finite difference methods for nonlinear second–order two point boundary value problems, Applied Mathematics Letter, 15 (2002), 897-902.  doi: 10.1016/S0893-9659(02)00060-5.  Google Scholar

[31]

V. N. Vapnik, The Nature of Statistical Learning Theory, 1nd edition, Springer–Verlag, New York, 1995. doi: 10.1007/978-1-4757-2440-0.  Google Scholar

[32]

Q. Wang, K. Wang and S. Chen, Least squares approximation method for the solution of Volterra–Fredholm integral equations, Journal of Computational and Applied Mathematics, 272 (2014), 141-147.  doi: 10.1016/j.cam.2014.05.010.  Google Scholar

[33]

H. S. Yazdi, M. Pakdaman and H. Modaghegh, Unsupervised kernel least mean square algorithm for solving ordinary differential equations, Neurocomputing, 74 (2011), 2062-2071.   Google Scholar

[34]

G. Zhang, S. Wang and Y. Wang, LS–SVM approximate solution for affine nonlinear systems with partially unknown functions, Journal of Industrial and Management Optimization, 10 (2014), 621-636.  doi: 10.3934/jimo.2014.10.621.  Google Scholar

show all references

References:
[1]

D. O. Awoyemi, A p–stable linear multistep method for solving general third order ordinary differential equations, International Journal of Computer Mathematics, 80 (2003), 987-993.  doi: 10.1080/0020716031000079572.  Google Scholar

[2]

S. Chakraverty and S. Mall, Regression based weight generation algorithm in neural network for solution of initial and boundary value problems, Neural Computing and Applications, 25 (2014), 585-594.  doi: 10.1007/s00521-013-1526-4.  Google Scholar

[3]

E. H. Doha, A. H. Bhrawy and R. M. Hafez, On shifted Jacobi spectral method for high–order multi–point boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 3802-3810.  doi: 10.1016/j.cnsns.2012.02.027.  Google Scholar

[4]

N. M. Duy, H. See and T. T. Cong, A spectral collocation technique based on integrated chebyshev polynomials for biharmonic problems in irregular domains, Applied Mathematical Modelling, 33 (2009), 284-299.  doi: 10.1016/j.apm.2007.11.002.  Google Scholar

[5]

T. Hofmann, B. Schölkopf and A. J. Smola, Kernel methods in machine learning, Annals of Statistics, 36 (2008), 1171-1220.  doi: 10.1214/009053607000000677.  Google Scholar

[6]

G. B. Huang, Q. Y. Zhu and C. K. Siew, Extreme learning machine: Theory and applications, Neurocomputing, 70 (2006), 489-501.  doi: 10.1016/j.neucom.2005.12.126.  Google Scholar

[7]

K. Hussain, F. Ismail, N. Senu and F. Rabiei, Fourth–order improved Runge–Kutta method for directly solving special third–order ordinary differential equations, Iranian Journal of Science and Technology Transaction A–Science, 41 (2017), 429-437.  doi: 10.1007/s40995-017-0258-1.  Google Scholar

[8]

K. Hussain, F. Ismail and N. Senu, Solving directly special fourth–order ordinary differential equations using Runge–Kutta type method, Journal of Computational and Applied Mathematics, 306 (2016), 179-199.  doi: 10.1016/j.cam.2016.04.002.  Google Scholar

[9]

S. Islam, I. Aziz and B. Šarler, The numerical solution of second–order boundary value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling, 52 (2010), 1577-1590.  doi: 10.1016/j.mcm.2010.06.023.  Google Scholar

[10]

D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3nd edition, Brooks/Cole, Pacific Grove, CA, 1991.  Google Scholar

[11]

J. Kierzenka and L. F. Shampine, A BVP solver that controls residual and error, Journal of Numerical Analysis, Industrial and Applied Mathematics, 3 (2008), 27-41.   Google Scholar

[12]

M. Lakestani and M. Dehgan, The solution of a second–order nonlinear differential equation with Neumann boundary conditions using semi–orthogonal B–spline wavelets, International Journal of Computer Mathematics, 83 (2006), 685-694.  doi: 10.1080/00207160601025656.  Google Scholar

[13]

Z. A. Majid and M. Suleiman, Direct integration method implicit variable steps method for solving higher order systems of ordinary differential equations directly, Sains Malaysiana, 35 (2006), 63-68.   Google Scholar

[14]

Z. A. Majid, N. A. Azmi, M. Suleiman and Z. B. Ibrahaim, Solving directly general third order ordinary differential equations using two–point four step block method, Sains Malaysiana, 41 (2012), 623-632.   Google Scholar

[15]

A. Malek and R. S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network–optimization method, Applied Mathematics and Computation, 183 (2006), 260-271.  doi: 10.1016/j.amc.2006.05.068.  Google Scholar

[16]

S. Mall and S. Chakraverty, Application of Legendre neural network for solving ordinary differential equations, Applied Soft Computing, 43 (2016), 347-356.   Google Scholar

[17]

S. Mall and S. Chakraverty, Chebyshev neural network based model for solving Lane–Emden type equations, Applied Mathematics and Computation, 247 (2014), 100-114.  doi: 10.1016/j.amc.2014.08.085.  Google Scholar

[18]

S. Mehrkanoon, A direct variable step block multistep method for solving general third–order ODEs, Numerical Algorithms, 57 (2011), 53-66.  doi: 10.1007/s11075-010-9413-x.  Google Scholar

[19]

S. Mehrkanoon, T. Falck and J. A. K. Suykens, Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1356-1367.  doi: 10.1109/TNNLS.2012.2202126.  Google Scholar

[20]

S. Mehrkanoon and J. A. K. Suykens, Learning solutions to partial differential equations using LS–SVM, Neurocomputing, 159 (2015), 105-116.  doi: 10.1016/j.neucom.2015.02.013.  Google Scholar

[21]

S. Mehrkanoon and J. A. K. Suykens, LS–SVM approximate solution to linear time varying descriptor systems, Automatica, 48 (2012), 2502-2511.  doi: 10.1016/j.automatica.2012.06.095.  Google Scholar

[22]

J. Mercer, Functions of positive and negative type and their connection with the theory of integral equations, Philosophical Transactions of the Royal Society of London, 209 (1909), 415-446.   Google Scholar

[23]

R. Noberg, Differential equations for moments of present values in life insurance, Mathematics and Economics, 17 (1995), 171-180.  doi: 10.1016/0167-6687(95)00019-O.  Google Scholar

[24]

K. Parand and M. Hemami, Numerical study of astrophysics equations by meshless collocation method based on compactly supported radial basis function, International Journal of Applied and Computational Mathematics, 3 (2017), 1053-1075.  doi: 10.1007/s40819-016-0161-z.  Google Scholar

[25] H. J. Ricardo, A Modern Introduction to Differential Equations, CRC Press, Boca Raton, FL, 2010.   Google Scholar
[26]

P. P. See, Z. A. Majid and M. Suleiman, Three–step block method for solving nonlinear boundary value problems, Abstract and Applied Analysis, 2014 (2014), Art. ID 379829, 8 pp. doi: 10.1155/2014/379829.  Google Scholar

[27]

P. K. Srivastava, M. Kumar and R. N. Mohapatra, Quintic nonpolynomial spline method for the solution of a second–order boundary value problem with engineering applications, Computers and Mathematics with Applications, 62 (2011), 1707-1714.  doi: 10.1016/j.camwa.2011.06.012.  Google Scholar

[28]

H. Sun, M. Hou, Y. Yang, T. Zhang, F. Weng and F. Han, Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm, Neural Processing Letters, (2018), 1–20. doi: 10.1007/s11063-018-9911-8.  Google Scholar

[29]

J. A. K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural Processing Letters, 9 (1999) 293–300. Google Scholar

[30]

I. A. Tirmizi and E. H. Twizell, Higher–order finite difference methods for nonlinear second–order two point boundary value problems, Applied Mathematics Letter, 15 (2002), 897-902.  doi: 10.1016/S0893-9659(02)00060-5.  Google Scholar

[31]

V. N. Vapnik, The Nature of Statistical Learning Theory, 1nd edition, Springer–Verlag, New York, 1995. doi: 10.1007/978-1-4757-2440-0.  Google Scholar

[32]

Q. Wang, K. Wang and S. Chen, Least squares approximation method for the solution of Volterra–Fredholm integral equations, Journal of Computational and Applied Mathematics, 272 (2014), 141-147.  doi: 10.1016/j.cam.2014.05.010.  Google Scholar

[33]

H. S. Yazdi, M. Pakdaman and H. Modaghegh, Unsupervised kernel least mean square algorithm for solving ordinary differential equations, Neurocomputing, 74 (2011), 2062-2071.   Google Scholar

[34]

G. Zhang, S. Wang and Y. Wang, LS–SVM approximate solution for affine nonlinear systems with partially unknown functions, Journal of Industrial and Management Optimization, 10 (2014), 621-636.  doi: 10.3934/jimo.2014.10.621.  Google Scholar

Figure 1.  Second-order nonlinear BVP
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Figure 2.  The logarithmic relation between σ and MSE(Example 1)
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Figure 3.  Second-order nonlinear IVP
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Figure 4.  M-order nonlinear IVP
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Figure 5.  Second order nonlinear BVP (Example 4)
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Figure 6.  M-order nonlinear BVP(Example 5)
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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 $ $
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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 $ $
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
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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
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} $
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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} $
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 $ $
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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 $ $
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
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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
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} $
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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} $
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
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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
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
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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
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} $
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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} $
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
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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
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
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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
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
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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
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
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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
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} $
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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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