A Matrix-Vector Operation-Based Numerical Solution Method for Linear $m$-th Order Ordinary Differential Equations: Application to Engineering Problems (Advances in Applied Mathematics and Mechanics, 2013)
Many problems in engineering sciences can be described by linear,inhomogeneous, $m$-th order ordinary differential equations (ODEs) withvariable coefficients. For this wide class of problems, we here present a new, simple, flexible,and robust solution method, based on piecewise exact integration of localapproximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficientenvironments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can beexpected to perform (i) similar to the Runge-Kutta method, when applied tostiff initial value problems, and (ii) significantly better than the finitedifference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineeringproblems including the oscillation of a modulated torsional springpendulum, steady-state heat transfer through a cooling web, and thestructural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfyingcharacteristics of the proposed solution scheme.