Efficient Propagation of Error Through System Models for Functions Common in Engineering

2012 ◽  
Vol 134 (1) ◽  
Author(s):  
Travis V. Anderson ◽  
Christopher A. Mattson ◽  
Brad J. Larson ◽  
David T. Fullwood
Keyword(s):  
Taylor Series ◽  
Error Propagation ◽  
Truncation Error ◽  
System Modeling ◽  
Kinematic Model ◽  
Computational Cost ◽  
Taylor Series Expansion ◽  
Design Decisions ◽  
First Order ◽  
Higher Order Terms

System modeling can help designers make and verify design decisions early in the design process if the model’s accuracy can be determined. The formula typically used to analytically propagate error is based on a first-order Taylor series expansion. Consequently, this formula can be wrong by one or more orders of magnitude for nonlinear systems. Clearly, adding higher-order terms increases the accuracy of the approximation but it also requires higher computational cost. This paper shows that truncation error can be reduced and accuracy increased without additional computational cost by applying a predictable correction factor to lower-order approximations. The efficiency of this method is demonstrated in the kinematic model of a flapping wing. While Taylor series error propagation is typically applicable only to closed-form equations, the procedure followed in this paper may be used with other types of models, provided that model outputs can be determined from model inputs, derivatives can be calculated, and truncation error is predictable.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

Income and factor substitution: an investigation on the Solow growth model under the constant elasticity of substitution

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sedat Alatas
Keyword(s):  
Developing Countries ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Elasticity Of Substitution ◽  
Data Driven ◽  
Constant Elasticity ◽  
Content Type ◽  
Constant Elasticity Of Substitution ◽  
First Order

PurposeThe purpose of this study is to examine whether the elasticity of substitution (ES) varies between developed and developing countries.Design/methodology/approachThe author derives the growth regressions from the Solow model under the constant elasticity of substitution production function by using the first-order Taylor series expansion and estimate them for each country group classified based on time-varying behavior of income per worker using the data-driven algorithm.FindingsThe ES is not unitary and varies among country groups. Developed countries generally have a higher ES than developing countries.Originality/valueFor the first time, the author uses the first-order Taylor series expansion to linearize the steady-state value of income per worker, as the author considers this approach to be relatively more straight-forward and tractable. Furthermore, the author estimates the equations using both cross-section and panel data techniques and employs the data-driven algorithm proposed by Phillips and Sul (2007) to classify countries.

scholarly journals Generalized Quadratic Linearization of Machine Models

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Parvathy Ayalur Krishnamoorthy ◽  
Kamaraj Vijayarajan ◽  
Devanathan Rajagopalan
Keyword(s):  
Nonlinear System ◽  
Taylor Series ◽  
State Feedback ◽  
Taylor Series Expansion ◽  
Higher Order ◽  
Exact Linearization ◽  
Linearization Technique ◽  
Practical Applications ◽  
Higher Order Terms ◽  
Approximate Linearization

In the exact linearization of involutive nonlinear system models, the issue of singularity needs to be addressed in practical applications. The approximate linearization technique due to Krener, based on Taylor series expansion, apart from being applicable to noninvolutive systems, allows the singularity issue to be circumvented. But approximate linearization, while removing terms up to certain order, also introduces terms of higher order than those removed into the system. To overcome this problem, in the case of quadratic linearization, a new concept called “generalized quadratic linearization” is introduced in this paper, which seeks to remove quadratic terms without introducing third- and higher-order terms into the system. Also, solution of generalized quadratic linearization of a class of control affine systems is derived. Two machine models are shown to belong to this class and are reduced to only linear terms through coordinate and state feedback. The result is applicable to other machine models as well.

HEGDE'S INTERFACE NUMERICAL TECHNIQUE FOR RESOLVING QUASI-ONE-DIMENSIONAL NOZZLE FLOWS

2009 ◽  
Vol 06 (01) ◽  
pp. 75-91
Author(s):  
GANESH S. HEGDE ◽  
G. M. MADHU
Keyword(s):  
Truncation Error ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Computational Cost ◽  
Optimal Solution ◽  
Taylor Series Expansion ◽  
Computational Effort ◽  
Navier Stokes Equation ◽  
One Dimensional ◽  
Interface Theory

Faster convergence, better accuracy and improved stability of the solutions to fluid flow and heat transfer problems in CFD reduce the computational cost and time. The numerical solutions to partial differential equations governing the physical flow and heat phenomena, using computer software and hardware, have been obtained by various techniques which have been refined over the years. The numerical techniques have obtained the base in finite difference method (FDM) approximations derived from Taylor series expansion. Because of linearization, FDM approximations have truncation error creeping into the values of the partial derivatives, which projects an unrealistic picture of the final outcome of results in terms of accuracy, convergence and stability. As the prime objective of this paper, the minimization of truncation error is attempted with the aid of the interface theory (briefly described in the appendix) used as a computational treatment tool. In simple terms, the interface theory provides an optimal solution to all variables in a linear indeterminate system with redundancy in unknowns. The effort has converged in the form of Hegde's interface numerical technique (HINT), which is demonstrated on a quasi-one-dimensional nozzle flow, the physical behavior of which is described by the Navier–Stokes equation considered specific to the said case. HINT could successfully match the results of MacCormack's predictor–corrector method as far as the accuracy is concerned, but with less computational effort and higher productivity. To the knowledge of the authors, HINT may be considered both original and different for its kind in the vast developments in CFD.

Sign in / Sign up

Export Citation Format

Share Document