
-
Previous Article
A class of accelerated conjugate-gradient-like methods based on a modified secant equation
- JIMO Home
- This Issue
-
Next Article
Optimal reinsurance-investment problem with dependent risks based on Legendre transform
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3nd edition, Brooks/Cole, Pacific Grove, CA, 1991. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
H. J. Ricardo, A Modern Introduction to Differential Equations, CRC Press, Boca Raton, FL, 2010.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. R. Kincaid and E. W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3nd edition, Brooks/Cole, Pacific Grove, CA, 1991. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
H. J. Ricardo, A Modern Introduction to Differential Equations, CRC Press, Boca Raton, FL, 2010.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |






Training data | Exact solution | LS-SVM | Training data | Exact solution | LS-SVM |
1.000000000 | 1.000000000 | 1.215686275 | 1.215686275 | ||
1.024390244 | 1.024390233 | 1.230769231 | 1.230769232 | ||
1.047619048 | 1.047619039 | 1.245283019 | 1.245283019 | ||
1.069767442 | 1.069767431 | 1.259259259 | 1.259259258 | ||
1.090909091 | 1.090909081 | 1.272727273 | 1.272727273 | ||
1.111111111 | 1.111111105 | 1.285714286 | 1.285714289 | ||
1.130434783 | 1.130434779 | 1.298245614 | 1.298245619 | ||
1.148936170 | 1.148936165 | 1.310344828 | 1.310344831 | ||
1.166666667 | 1.166666660 | 1.322033898 | 1.322033904 | ||
1.183673469 | 1.183673463 | 1.333333333 | 1.333333333 | ||
1.200000000 | 1.199999997 |
Training data | Exact solution | LS-SVM | Training data | Exact solution | LS-SVM |
1.000000000 | 1.000000000 | 1.215686275 | 1.215686275 | ||
1.024390244 | 1.024390233 | 1.230769231 | 1.230769232 | ||
1.047619048 | 1.047619039 | 1.245283019 | 1.245283019 | ||
1.069767442 | 1.069767431 | 1.259259259 | 1.259259258 | ||
1.090909091 | 1.090909081 | 1.272727273 | 1.272727273 | ||
1.111111111 | 1.111111105 | 1.285714286 | 1.285714289 | ||
1.130434783 | 1.130434779 | 1.298245614 | 1.298245619 | ||
1.148936170 | 1.148936165 | 1.310344828 | 1.310344831 | ||
1.166666667 | 1.166666660 | 1.322033898 | 1.322033904 | ||
1.183673469 | 1.183673463 | 1.333333333 | 1.333333333 | ||
1.200000000 | 1.199999997 |
Testing data | Exact solution | LS-SVM | Testing data | Exact solution | LS-SVM |
1.012345679 | 1.012345667 | 1.207920792 | 1.207920791 | ||
1.036144578 | 1.036144569 | 1.223300971 | 1.223300972 | ||
1.058823529 | 1.058823519 | 1.238095238 | 1.238095239 | ||
1.080459770 | 1.080459759 | 1.252336449 | 1.252336448 | ||
1.101123596 | 1.101123588 | 1.266055046 | 1.266055045 | ||
1.120879121 | 1.120879117 | 1.279279279 | 1.279279281 | ||
1.139784946 | 1.139784942 | 1.292035398 | 1.292035403 | ||
1.157894737 | 1.157894731 | 1.304347826 | 1.304347830 | ||
1.175257732 | 1.175257725 | 1.316239317 | 1.316239320 | ||
1.191919192 | 1.191919187 | 1.327731092 | 1.327731099 |
Testing data | Exact solution | LS-SVM | Testing data | Exact solution | LS-SVM |
1.012345679 | 1.012345667 | 1.207920792 | 1.207920791 | ||
1.036144578 | 1.036144569 | 1.223300971 | 1.223300972 | ||
1.058823529 | 1.058823519 | 1.238095238 | 1.238095239 | ||
1.080459770 | 1.080459759 | 1.252336449 | 1.252336448 | ||
1.101123596 | 1.101123588 | 1.266055046 | 1.266055045 | ||
1.120879121 | 1.120879117 | 1.279279279 | 1.279279281 | ||
1.139784946 | 1.139784942 | 1.292035398 | 1.292035403 | ||
1.157894737 | 1.157894731 | 1.304347826 | 1.304347830 | ||
1.175257732 | 1.175257725 | 1.316239317 | 1.316239320 | ||
1.191919192 | 1.191919187 | 1.327731092 | 1.327731099 |
N | ||
1.50 | ||
0.90 | ||
0.60 |
N | ||
1.50 | ||
0.90 | ||
0.60 |
Training data | Exact solution | LS-SVM | Training data | Exact solution | LS-SVM |
1.000000000 | 1.000000000 | 0.7843137255 | 0.783960261 | ||
0.975609756 | 0.975507638 | 0.7692307692 | 0.768994225 | ||
0.952380952 | 0.952038212 | 0.7547169811 | 0.754586586 | ||
0.930232558 | 0.929751041 | 0.7407407407 | 0.740709174 | ||
0.909090909 | 0.908583285 | 0.7272727273 | 0.727354567 | ||
0.888888889 | 0.888395226 | 0.7142857143 | 0.714516534 | ||
0.869565217 | 0.869067566 | 0.7017543860 | 0.702168828 | ||
0.851063830 | 0.850537044 | 0.6896551724 | 0.690259179 | ||
0.833333333 | 0.832783164 | 0.6779661017 | 0.678730163 | ||
0.816326531 | 0.815793928 | 0.6666666667 | 0.667567043 | ||
0.800000000 | 0.799538249 |
Training data | Exact solution | LS-SVM | Training data | Exact solution | LS-SVM |
1.000000000 | 1.000000000 | 0.7843137255 | 0.783960261 | ||
0.975609756 | 0.975507638 | 0.7692307692 | 0.768994225 | ||
0.952380952 | 0.952038212 | 0.7547169811 | 0.754586586 | ||
0.930232558 | 0.929751041 | 0.7407407407 | 0.740709174 | ||
0.909090909 | 0.908583285 | 0.7272727273 | 0.727354567 | ||
0.888888889 | 0.888395226 | 0.7142857143 | 0.714516534 | ||
0.869565217 | 0.869067566 | 0.7017543860 | 0.702168828 | ||
0.851063830 | 0.850537044 | 0.6896551724 | 0.690259179 | ||
0.833333333 | 0.832783164 | 0.6779661017 | 0.678730163 | ||
0.816326531 | 0.815793928 | 0.6666666667 | 0.667567043 | ||
0.800000000 | 0.799538249 |
Testing data | Exact solution | LS-SVM | Testing data | Exact solution | LS-SVM |
0.9876543210 | 0.9876489423 | 0.7920792079 | 0.7916686867 | ||
0.9638554217 | 0.9636280130 | 0.7766990291 | 0.7764046131 | ||
0.9411764706 | 0.9407474275 | 0.7619047619 | 0.7617230005 | ||
0.9195402299 | 0.9190356064 | 0.7476635514 | 0.7475823709 | ||
0.8988764045 | 0.8983754394 | 0.7339449541 | 0.7339666547 | ||
0.8791208791 | 0.8786291359 | 0.7207207207 | 0.7208719671 | ||
0.8602150538 | 0.8597045711 | 0.7079646018 | 0.7082841451 | ||
0.8421052632 | 0.8415635623 | 0.6956521739 | 0.6961631623 | ||
0.8247422680 | 0.8241942736 | 0.6837606838 | 0.6844497170 | ||
0.8080808089 | 0.8075774037 | 0.6722689076 | 0.6731004361 |
Testing data | Exact solution | LS-SVM | Testing data | Exact solution | LS-SVM |
0.9876543210 | 0.9876489423 | 0.7920792079 | 0.7916686867 | ||
0.9638554217 | 0.9636280130 | 0.7766990291 | 0.7764046131 | ||
0.9411764706 | 0.9407474275 | 0.7619047619 | 0.7617230005 | ||
0.9195402299 | 0.9190356064 | 0.7476635514 | 0.7475823709 | ||
0.8988764045 | 0.8983754394 | 0.7339449541 | 0.7339666547 | ||
0.8791208791 | 0.8786291359 | 0.7207207207 | 0.7208719671 | ||
0.8602150538 | 0.8597045711 | 0.7079646018 | 0.7082841451 | ||
0.8421052632 | 0.8415635623 | 0.6956521739 | 0.6961631623 | ||
0.8247422680 | 0.8241942736 | 0.6837606838 | 0.6844497170 | ||
0.8080808089 | 0.8075774037 | 0.6722689076 | 0.6731004361 |
N | ||
0.1625 | ||
0.2235 | ||
0.4085 |
N | ||
0.1625 | ||
0.2235 | ||
0.4085 |
Example 3 | Exact solution | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.04997916927 | 0.04997915236 | |
0.09983341665 | 0.09983310498 | |
0.14943813247 | 0.14943658985 | |
0.19866933080 | 0.19866476297 | |
0.24740395925 | 0.24739375649 | |
0.29552020666 | 0.29550123081 | |
0.34289780746 | 0.34286692221 | |
0.38941834231 | 0.38937318159 | |
0.43496553411 | 0.43490549976 | |
0.47942553860 | 0.47935301511 |
Example 3 | Exact solution | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.04997916927 | 0.04997915236 | |
0.09983341665 | 0.09983310498 | |
0.14943813247 | 0.14943658985 | |
0.19866933080 | 0.19866476297 | |
0.24740395925 | 0.24739375649 | |
0.29552020666 | 0.29550123081 | |
0.34289780746 | 0.34286692221 | |
0.38941834231 | 0.38937318159 | |
0.43496553411 | 0.43490549976 | |
0.47942553860 | 0.47935301511 |
Example 3 | Exact solution | LS-SVM |
0.02499739591 | 0.02499739536 | |
0.07492970727 | 0.07492961138 | |
0.12467473339 | 0.12467397503 | |
0.17410813759 | 0.17410536148 | |
0.22310636213 | 0.22309934567 | |
0.27154693696 | 0.27153275506 | |
0.31930878586 | 0.31928421995 | |
0.36627252909 | 0.36623471736 | |
0.41232078174 | 0.41226810347 | |
0.45733844718 | 0.45727163071 |
Example 3 | Exact solution | LS-SVM |
0.02499739591 | 0.02499739536 | |
0.07492970727 | 0.07492961138 | |
0.12467473339 | 0.12467397503 | |
0.17410813759 | 0.17410536148 | |
0.22310636213 | 0.22309934567 | |
0.27154693696 | 0.27153275506 | |
0.31930878586 | 0.31928421995 | |
0.36627252909 | 0.36623471736 | |
0.41232078174 | 0.41226810347 | |
0.45733844718 | 0.45727163071 |
N | ||
2.1200 | ||
4.3750 | ||
4.4595 |
N | ||
2.1200 | ||
4.3750 | ||
4.4595 |
Example 4 | Bvp4c | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.08208961709 | 0.07269777630 | |
0.16500909145 | 0.15151008989 | |
0.24963062540 | 0.23642312027 | |
0.33691524509 | 0.32742078522 | |
0.42796806735 | 0.42448474566 | |
0.52410852807 | 0.52759441177 | |
0.62696468677 | 0.63672695027 | |
0.73860596417 | 0.75185729280 | |
0.86173796851 | 0.87295814556 | |
1.00000000000 | 1.00000190735 |
Example 4 | Bvp4c | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.08208961709 | 0.07269777630 | |
0.16500909145 | 0.15151008989 | |
0.24963062540 | 0.23642312027 | |
0.33691524509 | 0.32742078522 | |
0.42796806735 | 0.42448474566 | |
0.52410852807 | 0.52759441177 | |
0.62696468677 | 0.63672695027 | |
0.73860596417 | 0.75185729280 | |
0.86173796851 | 0.87295814556 | |
1.00000000000 | 1.00000190735 |
Example 4 | Bvp4c | LS-SVM |
0.04099337478 | 0.03558379567 | |
0.12339244872 | 0.11134042665 | |
0.20704941350 | 0.19320496766 | |
0.29287463208 | 0.28116246679 | |
0.38189348425 | 0.37519571270 | |
0.47530819486 | 0.47528524009 | |
0.57457869449 | 0.58140933646 | |
0.68153387549 | 0.69354404985 | |
0.79853157226 | 0.81166319791 | |
0.92869807782 | 0.93573837804 |
Example 4 | Bvp4c | LS-SVM |
0.04099337478 | 0.03558379567 | |
0.12339244872 | 0.11134042665 | |
0.20704941350 | 0.19320496766 | |
0.29287463208 | 0.28116246679 | |
0.38189348425 | 0.37519571270 | |
0.47530819486 | 0.47528524009 | |
0.57457869449 | 0.58140933646 | |
0.68153387549 | 0.69354404985 | |
0.79853157226 | 0.81166319791 | |
0.92869807782 | 0.93573837804 |
Example 5 | Exact solution | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.00072900000 | 0.00072904178 | |
0.00409600000 | 0.00409607260 | |
0.00926100000 | 0.01382410696 | |
0.01382400000 | 0.01390350516 | |
0.01562500000 | 0.01562510374 | |
0.01382400000 | 0.01382410574 | |
0.00926100000 | 0.00926110345 | |
0.00409600000 | 0.00409607300 | |
0.00072900000 | 0.00072904190 | |
0.00000000000 | 0.00000000000 |
Example 5 | Exact solution | LS-SVM |
0.00000000000 | 0.00000000000 | |
0.00072900000 | 0.00072904178 | |
0.00409600000 | 0.00409607260 | |
0.00926100000 | 0.01382410696 | |
0.01382400000 | 0.01390350516 | |
0.01562500000 | 0.01562510374 | |
0.01382400000 | 0.01382410574 | |
0.00926100000 | 0.00926110345 | |
0.00409600000 | 0.00409607300 | |
0.00072900000 | 0.00072904190 | |
0.00000000000 | 0.00000000000 |
Example 5 | Exact solution | LS-SVM |
0.000107171875 | 0.000107186927 | |
0.002072671875 | 0.002072728701 | |
0.006591796875 | 0.006591885737 | |
0.011774546875 | 0.011774653640 | |
0.015160921875 | 0.015161024270 | |
0.015160921875 | 0.015161026227 | |
0.011774546875 | 0.011774654858 | |
0.006591796875 | 0.006591885679 | |
0.002072671875 | 0.002072728375 | |
0.000107171875 | 0.000107189695 |
Example 5 | Exact solution | LS-SVM |
0.000107171875 | 0.000107186927 | |
0.002072671875 | 0.002072728701 | |
0.006591796875 | 0.006591885737 | |
0.011774546875 | 0.011774653640 | |
0.015160921875 | 0.015161024270 | |
0.015160921875 | 0.015161026227 | |
0.011774546875 | 0.011774654858 | |
0.006591796875 | 0.006591885679 | |
0.002072671875 | 0.002072728375 | |
0.000107171875 | 0.000107189695 |
N | ||
1.8000 | ||
0.7000 | ||
0.2001 |
N | ||
1.8000 | ||
0.7000 | ||
0.2001 |
[1] |
M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072 |
[2] |
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 |
[3] |
Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 |
[4] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[5] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[6] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[7] |
Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 |
[8] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[9] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[10] |
Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151 |
[11] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[12] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
[13] |
Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91 |
[14] |
Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 |
[15] |
Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 |
[16] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[17] |
Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 |
[18] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[19] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[20] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]