Solving Ordinary Differential Equations Based On LS-SVM

As the connection between abstract mathematical theory and applications ofnatural sciences, differential equations (DEs) promote the development of manydisciplines such as linear algebra, the theory of Lie groups, and control theory alongwith their growth. Theory of DEs involves ideas of various theories and methods ofdifferent practical applications, in which solving ordinary differential equations(ODEs) and studying the properties of ODEs solutions are throughout important basicwork for researchers. However, most ODEs are very complex in the actual productionand scientific research, which means analytical solutions are not available in manycases, so commonly we have to use numerical method for solving them.In recent years, researchers tried to find easier ways instead of traditionalnumerical methods that cover defects of computational complexity and discreteformed solutions. New approaches such as neural network algorithm, least meansquare method and support vector machines have been applied for solving ODEs.This paper studies a new method to solve ordinary differential equations using leastsquares support vector machines.Firstly, the computational domain is discretized, after which a transition is madefrom the ODEs to an optimization problem with conditions. Then the problem istransformed to a derivative formed LS-SVM regression by using the differentiableRBF kernel, finally the solution is obtained after solving equations.The method is used for solving non-stiff linear ODEs with initial or boundaryconditions, and one order nonlinear ODEs. The high-accuracy differentiableapproximate solution with simple and fixed structure is presented in closed form,which creates conditions for the qualitative analysis of the solution.Finally, the paper studies the parameter selection ways and large interval ofproblem solving, showing that the relationships between accuracy of the approximatesolution and parameter selection, and resolves the problem of computationally cost forsolving ODEs in large interval.