Use of complete gamma function in accurate evaluation of Einstein integrals

2008 ◽  
Vol 39 (3) ◽  
pp. 223-227
Author(s):  
I. I. Guseinov ◽  
B. A. Mamedov
Keyword(s):  
Sediment Transport ◽  
Series Expansion ◽  
Gamma Function ◽  
Numerical Calculations ◽  
Computational Time ◽  
Expansion Theorem ◽  
Accurate Evaluation ◽  
Modern Sediment ◽  
Binomial Expansion

By the use of the binomial expansion theorem, the series expansion relations in terms of the complete gamma function are obtained for Einstein integrals arising in the hydraulic and modern sediment transport mechanics. The approach presented for Einstein integrals is accurate enough over the whole range of parameters. The computational time for calculation of the series with respect to the literature is fast. Furthermore, the comparison of the method with numerical calculations demonstrates the applicability and accuracy of the method.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals Comparison of Local and Global Optimization Methods for Calibration of a 3D Morphodynamic Model of a Curved Channel

Water ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 1333
Author(s):  
Vahid Shoarinezhad ◽  
Silke Wieprecht ◽  
Stefan Haun
Keyword(s):  
Global Optimization ◽  
Sediment Transport ◽  
Numerical Model ◽  
Numerical Models ◽  
Optimization Algorithms ◽  
Flow Characteristics ◽  
Optimization Methods ◽  
Computational Time ◽  
Curved Channel ◽  
Time Model

In curved channels, the flow characteristics, sediment transport mechanisms, and bed evolution are more complex than in straight channels, owing to the interaction between the centrifugal force and the pressure gradient, which results in the formation of secondary currents. Therefore, using an appropriate numerical model that considers this fully three-dimensional effect, and subsequently, the model calibration are substantial tasks for achieving reliable simulation results. The calibration of numerical models as a subjective approach can become challenging and highly time-consuming, especially for inexperienced modelers, due to dealing with a large number of input parameters with respect to hydraulics and sediment transport. Using optimization methods can notably facilitate and expedite the calibration procedure by reducing the user intervention, which results in a more objective selection of parameters. This study focuses on the application of four different optimization algorithms for calibration of a 3D morphodynamic numerical model of a curved channel. The performance of a local gradient-based method is compared with three global optimization algorithms in terms of accuracy and computational time (model runs). The outputs of the optimization methods demonstrate similar sets of calibrated parameters and almost the same degree of accuracy according to the achieved minimum of the objective function. Accordingly, the most efficient method concerning the number of model runs (i.e., local optimization method) is selected for further investigation by setting up additional numerical models using different sediment transport formulae and various discharge rates. The comparisons of bed topography changes in several longitudinal and cross-sections between the measured data and the results of the calibrated numerical models are presented. The outcomes show an acceptable degree of accuracy for the automatically calibrated models.

scholarly journals Improving Accuracy of Coarse Grid Numerical Solution of Solid-Solid Reactions by Taylor Series Expansion of the Reaction Term

2009 ◽  
Vol 2009 ◽  
pp. 1-13
Author(s):  
Hassan Hassanzadeh ◽  
Mehran Pooladi-Darvish ◽  
Jalal Abedi
Keyword(s):  
Numerical Simulations ◽  
Series Expansion ◽  
Taylor Series ◽  
Large Scale ◽  
Mesh Size ◽  
Taylor Series Expansion ◽  
Computational Time ◽  
Time Saving ◽  
Reaction Term ◽  
Reaction Fronts

Exothermic solid-solid reactions lead to sharp reaction fronts that cannot be captured by coarse spatial mesh size numerical simulations that are often required for large-scale simulations. We present a coarse-scale formulation with high accuracy by using a Taylor series expansion of the reaction term. Results show that such expansion could adequately maintain the accuracy of fine-scale behavior of a constant pattern reaction front while using a smaller number of numerical grid cells. Results for a one-dimensional solid-solid reacting system reveal reasonable computational time saving. The presented formulation improves our capabilities for conducting fast and accurate numerical simulations of industrial-scale solid-solid reactions.

GAUSS SUMMATION AND RAMANUJAN-TYPE SERIES FOR 1/π

2012 ◽  
Vol 08 (02) ◽  
pp. 289-297 ◽  
Author(s):  
ZHI-GUO LIU
Keyword(s):  
Series Expansion ◽  
Gamma Function ◽  
Hypergeometric Series ◽  
Summation Formula ◽  
Expansion Formula ◽  
Type Series

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan-type series for 1/π.

Sign in / Sign up

Export Citation Format

Share Document