Approximate Solutions of the Lane-Emden Equations by LS-SVM Method

Year 2026, Volume: 18 Issue: 1, 11 - 23, 23.02.2026

Süleyman Şengül , Hasan Halit Tali

https://doi.org/10.47000/tjmcs.1609938

https://izlik.org/JA65CC27SY

Abstract

In this study, approximate solutions of the Lane-Emden differential equation, which plays an important
role in the literature, were obtained using the Least Squares Support Vector Machines (LS-SVM) method for both linear and nonlinear cases. The Collocation method was employed to define the constraints in the solution process. The system of equations obtained in the linear case was solved directly to determine the unknown parameters, while the Newton-Raphson method was used to solve the nonlinear equation system. The approximate solutions obtained in the applications considered in this study were compared with the exact solution for the linear case; with the analytical solution for the nonlinear case; and with the Adomian Decomposition Method (ADM) in the final application where no analytical solution exists. The results show that the numerical solutions obtained using the LS-SVM method are highly accurate and consistent with the reference results.

Keywords

Least square method , support vector machine method , Collocation method , Lane-Emden differential equation

References

• Ala’yed, O., Saadeh, R., Qazza, A., Numerical solution for the system of Lane-Emden type equations using cubic B-spline method arising in engineering, AIMS Mathematics, 6(2023), 14747–14766.
• Anitescu, C., Atroshchenko, E., et al., Artificial neural network methods for the solution of second order boundary value problems, Computers, Materials & Continua, 1(2019), 345–359.
• Kourosh, P., Nikarya, M., Rad, J.A., Solving non-linear Lane–Emden type equations using Bessel orthogonal functions collocation method, Celestial Mechanics and Dynamical Astronomy, 1(2013), 97–107.
• Lagaris, I., Likas, A., Fotiadis, D.I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw., 5(1998), 987–1000.
• Lee, H., Kang, I., Neural algorithms for solving differential equations, J. Comput. Phys., 1(1990), 110–117.
• Mall, S., Chakraverty, S., Application of Legendre neural network for solving ordinary differential equations, Applied Soft Computing, 43(2016), 347–356.
• McFall, K.S., Mahan, J.R., Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions, IEEE Trans. Neural Netw., 8(2009), 1221–1233.
• Meade, A.J., Fernadez, A.A., The numerical solution of linear ordinary differential equations by feedforward neural networks, Math. Comput. Model., 12(1994), 1–25.
• Mehrkanoon, S., Suykens, J.A.K., LS-SVM approximate solution to linear time varying descriptor systems, Automatica, 48(2012), 2502–2511.
• Mehrkanoon, S., Falck, T., Suykens, J.A.K., Approximate solutions to ordinary differential equations using least squares support vector machines, IEEE Trans. Neural Netw. Learn. Syst., 9(2012), 1356–1367.
• Mehrkanoon, S., Suykens, J.A.K., Learning solutions to partial differential equations using LS-SVM, Neurocomputing, 159(2015), 105–116.
• Momoniat, E., Harley, C., Approximate implicit solution of a Lane-Emden equation, New Astronomy, 7(2006), 520–526.
• Parand, K., Aghaei, A.A., et al., A neural network approach for solving nonlinear differential equations of Lane–Emden type, Engineering with Computers, 40(2024), 1–17.
• Ramuhalli, P., Udpa, L., Udpa, S.S., Finite-element neural networks for solving differential equations, IEEE Trans. Neural Netw., 6(2005), 1381–1392.
• Razzaghi, M., Shekarpaz, S., Rajabi, A., Solving ordinary differential equations by LS-SVM, in Learning with Fractional Orthogonal Kernel Classifiers in Support Vector Machines: Theory, Algorithms and Applications, Springer Nature Singapore, 147–170, (2023).
• Sch¨olkopf, B., Smola, A.J., Learning with Kernels, MA: MIT Press, Cambridge, 2002.
• Suykens, J.A.K., Vandewalle, J., Least squares support vector machine classifiers, Neural Processing Letters, 3(1999), 293–300.
• Suykens, J.A.K., Vandewalle, J., De Moor, B., Optimal control by least squares support vector machines, Neural Networks, 1(2001), 23–35.
• Suykens, J.A.K., Van Gestel, T., et al., Least Squares Support Vector Machines, World Scientific Publishing, Singapore, 2002.
• Suykens, J.A.K., Least Squares Support Vector Machines, PowerPoint slides, 2003.
• Tsoulos, I.G., Gavrilis, D., Glavas, E., Solving differential equations with constructed neural networks, Neurocomputing, 10–12(2009), 2385–2391.
• Vanani, S.K., Aminataei, A., On the numerical solution of differential equations of Lane–Emden type, Computers & Mathematics with Applications, 8(2010), 2815–2820.
• Vapnik, V., Statistical Learning Theory, Wiley, New York, 1998.
• Wen, Y., Chaolu, T., Wang, X., Solving the initial value problem of ordinary differential equations by Lie group based neural network method, PLoS One, 4(2022), e0265992, (2022).
• Yadav, N., Yadav, A., Kumar, M., An Introduction to Neural Network Methods for Differential Equations, Springer, Warsaw, 2015.
• Yazdi, H.S., Pakdaman, M., Modaghegh, H., Unsupervised kernel least mean square algorithm for solving ordinary differential equations, Neurocomputing, 12–13(2011), 2062–2071.