Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines

Siamak Mehrkanoon; Tillmann Falck; Johan A. K. Suykens

doi:10.1109/TNNLS.2012.2202126

Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines

Siamak Mehrkanoon^*, Tillmann Falck, Johan A. K. Suykens

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.

Original language	English
Pages (from-to)	1356-1367
Number of pages	12
Journal	IEEE Transactions on Neural Networks and Learning Systems
Volume	23
Issue number	9
DOIs	https://doi.org/10.1109/TNNLS.2012.2202126
Publication status	Published - Sept 2012
Externally published	Yes

Keywords

Closed-form approximate solution
collocation method
least squares support vector machines (LS-SVMs)
ordinary differential equations (ODEs)
BOUNDARY-VALUE-PROBLEMS
NEURAL-NETWORK METHOD
SOLVING ORDINARY
ALGORITHM

Access to Document

10.1109/TNNLS.2012.2202126

Cite this

@article{800962f24a8a4a72aa495839d79d6957,

title = "Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines",

abstract = "In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.",

keywords = "Closed-form approximate solution, collocation method, least squares support vector machines (LS-SVMs), ordinary differential equations (ODEs), BOUNDARY-VALUE-PROBLEMS, NEURAL-NETWORK METHOD, SOLVING ORDINARY, ALGORITHM",

author = "Siamak Mehrkanoon and Tillmann Falck and Suykens, {Johan A. K.}",

year = "2012",

month = sep,

doi = "10.1109/TNNLS.2012.2202126",

language = "English",

volume = "23",

pages = "1356--1367",

journal = "IEEE Transactions on Neural Networks and Learning Systems",

issn = "2162-237X",

publisher = "IEEE",

number = "9",

}

TY - JOUR

T1 - Approximate Solutions to Ordinary Differential Equations Using Least Squares Support Vector Machines

AU - Mehrkanoon, Siamak

AU - Falck, Tillmann

AU - Suykens, Johan A. K.

PY - 2012/9

Y1 - 2012/9

N2 - In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.

AB - In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.

KW - Closed-form approximate solution

KW - collocation method

KW - least squares support vector machines (LS-SVMs)

KW - ordinary differential equations (ODEs)

KW - BOUNDARY-VALUE-PROBLEMS

KW - NEURAL-NETWORK METHOD

KW - SOLVING ORDINARY

KW - ALGORITHM

U2 - 10.1109/TNNLS.2012.2202126

DO - 10.1109/TNNLS.2012.2202126

M3 - Article

C2 - 24807921

SN - 2162-237X

VL - 23

SP - 1356

EP - 1367

JO - IEEE Transactions on Neural Networks and Learning Systems

JF - IEEE Transactions on Neural Networks and Learning Systems

IS - 9

ER -