Abstract
This paper introduces the improved LS-SVM algorithms for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations. To demonstrate the reliability and powerfulness of the improved LS-SVM algorithms, some numerical experiments for third-order, fourth-order linear and nonlinear ordinary differential equations with two-point and multi-point boundary conditions are performed. The idea can be extended to other complicated ordinary differential equations.
Highlights
High-order boundary value problems for ordinary differential equations are used to model different problems in some fields such as biology, economics, and engineering
The improved LS-SVM algorithms to the solution of two-point boundary value problems of high-order linear and nonlinear ordinary differential equations are described
6 Conclusion In this paper, the improved LS-SVM algorithms have been developed for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations
Summary
High-order boundary value problems for ordinary differential equations are used to model different problems in some fields such as biology, economics, and engineering. To the best of our knowledge, there are not too many results on LS-SVM algorithms for solving two-point and multipoint boundary value problems of high-order linear and nonlinear ordinary differential equations. The improved LS-SVM algorithms to the solution of two-point boundary value problems of high-order linear and nonlinear ordinary differential equations are described. The improved LS-SVM algorithms to the solution of multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations are described. YN–1]T ; [Θqj,l ]M,N–M = [Θq0:qM–1,M]M,N–M + [Θ q0:qM–1,l ]M,N–MDTal ; β = [β0, β1, β2, . The proposed LS-SVM algorithm for two-point boundary value problems of high-order linear ordinary differential equation has been trained with 11 equidistant points in the given interval [–1, 1]. The proposed LS-SVM algorithm obtains a satisfactory result for multi-point boundary value problems of third-order nonlinear ordinary differential equation.
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