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

2013 ◽  
Vol 5 (03) ◽  
pp. 269-308 ◽  
Author(s):  
M. Aminbaghai ◽  
M. Dorn ◽  
J. Eberhardsteiner ◽  
B. Pichler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Solution Method ◽  
Variable Coefficients ◽  
Exact Integration ◽  
Robust Solution ◽  
Vector Operation ◽  
Engineering Sciences ◽  
Engineering Problems ◽  
Matrix Vector

AbstractMany 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.

PROGRAM-BASED LOCALITY MEASURES FOR SCIENTIFIC COMPUTING

2004 ◽  
Vol 15 (03) ◽  
pp. 535-554 ◽  
Author(s):  
THOMAS RAUBER ◽  
GUDULA RÜNGER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Parallel Machines ◽  
Solution Method ◽  
Transformation Process ◽  
Program Transformations ◽  
Memory Hierarchies ◽  
Numerical Software ◽  
Scientific Libraries ◽  
With Memory

Numerical software for sequential or parallel machines with memory hierarchies can benefit from locality optimizations which are usually achieved by program restructuring or program transformations. In this paper we introduce and discuss program-based locality measures which can be used to guide the restructuring and transformation process at program level. As example we consider a popular solution method to solve ordinary differential equations provided in many scientific libraries.

Elasticity Solution of Laminated Cylindrical Shell with Piezoelectric Layer under Local Ring Load

2007 ◽  
Vol 334-335 ◽  
pp. 917-920 ◽  
Author(s):  
M.H. Yas ◽  
Morteza Shakeri ◽  
M.R. Saviz
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Local Ring ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Elasticity Solution ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
External Forces

Elasticity solution is presented for simply-supported, orthotropic, piezoelectric cylindrical shell with finite length under local ring load in the middle of shell and electrostatic excitation. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations(o.d.e.) with variable coefficients by means of trigonometric function expansion in longitudinal direction for displacement and external forces. The resulting ordinary differential equations are solved by Galerkin finite element method. Numerical examples are presented for [0/90/P] lamination with sensor and actuator for different thicknesses.

MESHFREE METHODS AND THEIR COMPARISONS

2005 ◽  
Vol 02 (04) ◽  
pp. 477-515 ◽  
Author(s):  
Y. T. GU
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Future Development ◽  
Meshless Method ◽  
Meshfree Methods ◽  
Shape Functions ◽  
Engineering Problems ◽  
Practical Engineering ◽  
Technical Issues

In recent years, one of the hottest topics in computational mechanics is the meshfree or meshless method. Increasing number of researchers are devoting themselves to the research of the meshfree methods, and a group of meshfree methods have been proposed and used to solve the ordinary differential equations (ODEs) or the partial differential equations (PDE). In the meantime, meshfree methods are being applied to a growing number of practical engineering problems. In this paper, a detailed discussion will be provided on the development of meshfree methods. First, categories of meshfree methods are introduced. Second, the methods for constructing meshfree shape functions are discussed, and the interpolation qualities of them are also studied using the surface fitting. Third, several typical meshfree methods are introduced and compared with each others in terms of their accuracy, convergence and effectivity. Finally, the major technical issues in meshfree methods are discussed, and the future development of meshfree methods is addressed.

scholarly journals On the Mohand Transform and Ordinary Differential Equations with Variable Coefficients

2020 ◽  
Vol 35 (1) ◽  
pp. 01-06
Author(s):  
Mohamed E. Attaweel ◽  
Haneen Almassry
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Integral Transform ◽  
Variable Coefficients ◽  
Sumudu Transforms ◽  
Differential And Integral Equations

The Mohand transform is a new integral transform introduced by Mohand M. Abdelrahim Mahgoub to facilitate the solution of differential and integral equations. In this article, a new integral transform, namely Mohand transform was applied to solve ordinary differential equations with variable coefficients by using the modified version of Laplace and Sumudu transforms.

Three-dimensional elasticity solution for laminated cross-ply panel under localized moment

2007 ◽  
Vol 221 (8) ◽  
pp. 859-866
Author(s):  
A Alibeigloo ◽  
M Shakeri
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Axial Direction ◽  
Variable Coefficients ◽  
Cylindrical Panel ◽  
Fourier Series Expansion ◽  
Simply Supported ◽  
Elasticity Solutions ◽  
Three Dimensional Elasticity

Three-dimensional elasticity solutions have been presented for thick laminated crossply circular cylindrical panel. The panel is under localized patch moment in axial direction and is simply supported at all edges with finite length. Ordinary differential equations with variable coefficients are obtained by means of Fourier series expansion for displacement field and loading in the circumferential and axial directions. Resulting ordinary differential equations are solved using Taylor series. Numerical results are presented for (0/90°) and (0/90/0°) lay-up, and compared with the results for simple form of loading published in literatures.

The Response Analysis of the Piezoelectric Shell Panel Actuators Based on the Theory of Elasticity

Volume 2 ◽  
2004 ◽  
Author(s):  
A. Daneshmehr ◽  
M. Shakeri
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Variable Coefficients ◽  
Theory Of Elasticity ◽  
Response Analysis ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Piezoelectric Layers ◽  
Shell Panel ◽  
Piezoelectric Shell

A study on the elasticity solution of shell panel piezoelectric actuators is presented. In this paper, the structure is infinitely long, simply-supported, orthotropic and under pressure and electrostatic excitation. The equations of equilibrium, which are coupled partial differential equations, are reduced to ordinary differential equations with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by Galerkin finite element method. Numerical results are presented for [0/90/P] lamination. Finally the results are compared with the assumption of piezoelectric layers in published results.

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