Solution of linear dynamic system with initial or boundary value conditions by shifted Legendre approximations

Abstract

The shifted Legendre polynomial approximation is employed to solve the ordinary differential equation with initial or two point boundary value problems. By appropriating transformation, the two point boundary value problem can be reduced to the initial value problem. The solution of the ordinary differential equation of the initial value problem are obtained in a series of the shifted Legendre polynomial with expansion coefficients. An effective computation algorithm is proposed in order to calculate the expansion coefficients recursively. The operational matrix for the integration of the shifted Legendre polynomials is introduced during calculation of the expansion coefficients for saving computation time. The advantage of the computation algorithm is discussed in detail. The proposed method is straightforward and effective and the computational results are accurate.