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

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.

34A30, 34B60

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@Article{AAMM-5-269, author = {M. and Aminbaghai and and 20119 and and M. Aminbaghai and M. and Dorn and and 20120 and and M. Dorn and J. and Eberhardsteiner and and 20121 and and J. Eberhardsteiner and B. and Pichler and and 20122 and 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} }
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 -

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.

M. Aminbaghai, M. Dorn, J. Eberhardsteiner & B. Pichler. (1970). 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. 5 (3). 269-308. doi:10.4208/aamm.12-m1211
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