A Taylor-Splitting Collocation approach and applications to linear and nonlinear engineering models

Abstract

A novel matrix method based on the Taylor series called Taylor-Splitting Collocation Method is presented to solve linear and nonlinear ordinary differential equations. Unlike the previous approaches, the fundamental matrix equation is reformulated using interval splitting. The residual error estimation algorithm is presented. Convergence analysis is given in a general form. Four different mechanical models are analyzed:1.Forced oscillations of a linear spring-mass model2.Forced oscillations of a nonlinear spring-mass model3.Free oscillations of a cubic nonlinear spring-dashpot-mass model4.Forced oscillations of a damped nonlinear pendulum modelDisplacement-time and velocity-time dependencies are plotted for each model. Phase portraits of nonlinear models are presented. Appropriate absolute or residual error analyses are obtained for the proposed application models. The results of the new solution approach are compared with exact, numerical, and approximate solutions from previous works. Consistent results are found.