A class of linearization-based collocation methods for initial value and boundary value engineering problems

Abstract

Nonlinear initial value and boundary value problems are generally classified as two different categories of problems and the methods utilized to solve them are also different. This paper proposed a uniform linearization-collocation framework through which the initial value and boundary value problems can be solved in a similar way. An optional pre-conditioning procedure is provided to guarantee the computational accuracy. Rigorous analysis about the convergence rate is given. Different from the commonly seen convergence rate analysis based on one-dimensional system and mean-value theorem, this analysis is based on multidimensional system and polynomial approximation. A perturbed orbit propagation problem, a radiation heat transfer problem and an orbit optimization problem are solved to verify the validity and efficiency of the proposed methods. The proposed methods not only outperform the highly optimized MATLAB built-in integrator ODE45 in orbit propagation, but also outperform the bvp4c and bvp5c in solving the boundary value problem. Most importantly, the proposed methods are capable of solving some problems where the bvp4c and bvp5c are invalid. The quasi-linearization-collocation method proved to be a highly accurate, efficient and stable solver for ordinary differential equations with various types of constraints.