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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.

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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.

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Forecasting international iron ore is a well-known issue, BIC criterion is used to select the relevant variables of iron ore price. On the basis of the traditional extreme learning machine (ELM), the regular term is introduced to control the complexity of the model, and the genetic algorithm (GA) is used to regularize the extreme learning machine. The input-layer weight matrix and the hidden-layer threshold matrix of the (RE-ELM) model are optimized to establish a BIC-based genetic algorithm and a regularization extreme learning machine (BIC-GA-RELM) iron ore price prediction model to increase the performance of the RE-ELM model. The results show that BIC-GA-RELM model has achieved the state of art performance, then a new method is provided for iron ore price prediction. Keywords-BIC criterion; genetic algorithm; regularized extreme learning machine; iron ore price forecast I.R , pred R denote the real value and estimating value. 212

PurposeThis paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).Design/methodology/approachThe proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.FindingsTo demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.Originality/valueThe H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.