TY - JOUR
T1 - A Matrix-Vector Operation-Based Numerical Solution Method for Linear $m$-th Order Ordinary Differential Equations: Application to Engineering Problems
AU - Aminbaghai , M.
AU - Dorn , M.
AU - Eberhardsteiner , J.
AU - Pichler , B.
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 269
EP - 308
PY - 2013
DA - 2013/05
SN - 5
DO - http://doi.org/10.4208/aamm.12-m1211
UR - https://global-sci.org/intro/article_detail/aamm/70.html
KW - Numerical integration, polynomial approximation, ODE, variable coefficients, initial
conditions, boundary conditions, stiff equation.
AB - <p style="text-align: justify;">Many problems in engineering sciences can be described by linear,
inhomogeneous, $m$-th order ordinary differential equations (ODEs) with
variable coefficients. For this wide class of problems, we here present a new, simple, flexible,
and robust solution method, based on piecewise exact integration of local
approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient
environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be
expected to perform (i) similar to the Runge-Kutta method, when applied to
stiff initial value problems, and (ii) significantly better than the finite
difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering
problems including the oscillation of a modulated torsional spring
pendulum, steady-state heat transfer through a cooling web, and the
structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying
characteristics of the proposed solution scheme.</p>