Volume 5, Issue 3
A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems

M. Aminbaghai, M. Dorn, J. Eberhardsteiner & B. Pichler

Adv. Appl. Math. Mech., 5 (2013), pp. 269-308

Published online: 2013-05

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  • 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.

  • Keywords

Numerical integration polynomial approximation ODE variable coefficients initial conditions boundary conditions stiff equation

  • AMS Subject Headings

34A30 34B60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-5-269, author = {M. Aminbaghai, M. Dorn, J. Eberhardsteiner and B. Pichler}, title = {A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {3}, pages = {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.}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-m1211}, url = {http://global-sci.org/intro/article_detail/aamm/70.html} }
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