A Matrix-Vector Operation-Based Numerical Solution Method for Linear $m$-th Order Ordinary Differential Equations: Application to Engineering Problems
Year: 2013
Author: M. Aminbaghai, M. Dorn, J. Eberhardsteiner, B. Pichler
Advances in Applied Mathematics and Mechanics, Vol. 5 (2013), Iss. 3 : pp. 269–308
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.12-m1211
Advances in Applied Mathematics and Mechanics, Vol. 5 (2013), Iss. 3 : pp. 269–308
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 40
Keywords: Numerical integration polynomial approximation ODE variable coefficients initial conditions boundary conditions stiff equation.
Author Details
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