scholarly journals Developing and Analyzing Newton – C’otes Quadrature Formulae for Approximating Definite Integrals- AC++ Approach

1970 ◽  
Vol 11 (2) ◽  
pp. 301-316
Author(s):  
Teka Tuemay ◽  
Assefa Dessalegn
Keyword(s):  
Approximate Solutions ◽  
Accurate Solution ◽  
Computational Effort ◽  
Trapezoidal Rule ◽  
Step Size ◽  
C Programs ◽  
Quadrature Formulae ◽  
Definite Integrals ◽  
Short Period ◽  
Number Of Iterations

In this paper, different Newton – C’otes quadrature formulae for the approximation of definite integrals and their error analysis are derived. The order of convergences of the methods is also derived and of these Newton – C’otes quadrature formulae, the Simpson’s 1/3 rule have been shown to have high order of convergence. Since the functionality of these numerical integration methods is practical only if we can use computer programs and applications to produce approximate solutions with acceptable errors within short period, C++ programs for the selected methods are written. These programs are used on the comparison of the Newton – C’otes quadrature formulae and the result obtained based on the inputs and outputs of the programs for different integrands. The results of these programs show that the convergence of the methods highly depends on the number of iterations. The results of different numerical examples show that for high accuracy of the trapezoidal rule computational effort is higher and round off errors with large number of iterations limit the accuracy. The results show that the Simpson’s 1/3 rule produces much more accurate solution than other methods even within small number of iterations. This shows that the error for Simpson’s rule 1/3 converges to zero faster than the error for the trapezoidal rule as the step size decreases. It is finally observed that Simpson’s 1/3 rule is much faster than the Trapezoidal and the Simpson’s 3/8 rules according to the results of the C++ programs.

scholarly journals Analysis of Pressure Transient Following Rapid Filling of a Vented Horizontal Pipe

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1698 ◽  
Author(s):  
Lin Li ◽  
David Zhu ◽  
Biao Huang
Keyword(s):  
Three Dimensional ◽  
Pressure Oscillation ◽  
Approximate Solutions ◽  
Periodic Wave ◽  
Oscillation Period ◽  
Maximum Pressure ◽  
Ansys Fluent ◽  
Horizontal Pipe ◽  
Sinusoidal Motion ◽  
Short Period

Rapid filling/emptying of pipes is commonly encountered in water supply and sewer systems, during which pressure transients may cause unexpected large pressure and/or geyser events. In the present study, a linearized analytical model is first developed to obtain the approximate solutions of the maximum pressure and the characteristics of pressure oscillations caused by the pressurization of trapped air in a horizontal pipe when there is no or insignificant air release. The pressure pattern is a typical periodic wave, analogous to sinusoidal motion. The oscillation period and the time when the pressure attains the peak value are significantly influenced by the driving pressure and the initial length of the entrapped air pocket. When there is air release through a venting orifice, analysis by a three-dimensional computational fluid dynamics model using ANSYS Fluent was also conducted to furnish insights and details of air–water interactions. Flow features associated with the pressurization and air release were examined, and an air–water interface deformation that one-dimensional models are incapable of predicating was presented. Modelling results indicate that the residual air in the system depends on the relative position of the venting orifice. There are mainly two types of pressure oscillation patterns: namely, long or short-period oscillations and waterhammer. The latter can be observed when the venting orifice is located near the end of the pipe where the air is trapped.

scholarly journals Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 130 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Iterative Procedure ◽  
Accurate Solution ◽  
Final Solution ◽  
Worst Case ◽  
Flow Friction ◽  
One Step ◽  
Newton Raphson ◽  
Number Of Iterations

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

scholarly journals A Dai-Liao Hybrid Hestenes-Stiefel and Fletcher-Revees Methods for Unconstrained Optimization

2021 ◽  
Vol 2 (1) ◽  
pp. 33
Author(s):  
Nasiru Salihu ◽  
Mathew Remilekun Odekunle ◽  
Also Mohammed Saleh ◽  
Suraj Salihu
Keyword(s):  
Unconstrained Optimization ◽  
Line Search ◽  
Optimization Problems ◽  
Convex Combination ◽  
Gradient Methods ◽  
Approximate Solutions ◽  
Step Size ◽  
Conjugacy Condition ◽  
Inexact Line Search ◽  
Wolfe Line Search

Some problems have no analytical solution or too difficult to solve by scientists, engineers, and mathematicians, so the development of numerical methods to obtain approximate solutions became necessary. Gradient methods are more efficient when the function to be minimized continuously in its first derivative. Therefore, this article presents a new hybrid Conjugate Gradient (CG) method to solve unconstrained optimization problems. The method requires the first-order derivatives but overcomes the steepest descent method’s shortcoming of slow convergence and needs not to save or compute the second-order derivatives needed by the Newton method. The CG update parameter is suggested from the Dai-Liao conjugacy condition as a convex combination of Hestenes-Stiefel and Fletcher-Revees algorithms by employing an optimal modulating choice parameterto avoid matrix storage. Numerical computation adopts an inexact line search to obtain the step-size that generates a decent property, showing that the algorithm is robust and efficient. The scheme converges globally under Wolfe line search, and it’s like is suitable in compressive sensing problems and M-tensor systems.

De-Noising of Binary Image Using Accelerated Local Median-Filtering Approach

2019 ◽  
pp. 164-185
Author(s):  
Amit Khan ◽  
Dipankar Majumdar
Keyword(s):  
Time Complexity ◽  
Window Size ◽  
Analytical Techniques ◽  
Approximate Solutions ◽  
Median Filtering ◽  
Binary Images ◽  
Noise Characteristics ◽  
Varied Size ◽  
Number Of Iterations ◽  
Filtering Approach

In the last few decades huge amounts and diversified work has been witnessed in the domain of de-noising of binary images through the evolution of the classical techniques. These principally include analytical techniques and approaches. Although the scheme was working well, the principal drawback of these classical and analytical techniques are that the information regarding the noise characteristics is essential beforehand. In addition to that, time complexity of analytical works amounts to beyond practical applicability. Consequently, most of the recent works are based on heuristic-based techniques conceding to approximate solutions rather than the best ones. In this chapter, the authors propose a solution using an iterative neural network that applies iterative spatial filtering technology with critically varied size of the computation window. With critical variation of the window size, the authors are able to show noted acceleration in the filtering approach (i.e., obtaining better quality filtration with lesser number of iterations).

scholarly journals Weak convergence of explicit extragradient algorithms for solving equilibirum problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Yeol Je Cho ◽  
Pasakorn Yordsorn
Keyword(s):  
Weak Convergence ◽  
Line Search ◽  
Equilibrium Problems ◽  
Search Procedure ◽  
Step Size ◽  
Convergence Results ◽  
Size Evaluation ◽  
New Algorithms ◽  
Number Of Iterations ◽  
Over Time

Abstract This paper aims to propose two new algorithms that are developed by implementing inertial and subgradient techniques to solve the problem of pseudomonotone equilibrium problems. The weak convergence of these algorithms is well established based on standard assumptions of a cost bi-function. The advantage of these algorithms was that they did not need a line search procedure or any information on Lipschitz-type bifunction constants for step-size evaluation. A practical explanation for this is that they use a sequence of step-sizes that are updated at each iteration based on some previous iterations. For numerical examples, we discuss two well-known equilibrium models that assist our well-established convergence results, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.

The Rapid Series Method for Solving Differential Equation of the Odd Power Oscillator

2012 ◽  
Vol 226-228 ◽  
pp. 138-141
Author(s):  
Song Lin He ◽  
Yan Huang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Series Method ◽  
Approximate Methods ◽  
Harmonic Term ◽  
Power Oscillator

The new rapid series method to solve the differential equation of the periodic vibration of the strongly odd power nonlinear oscillator has been put forward in this paper. By adding the exponentially decaying factor to each harmonic term of the Fourier series of the periodic solution, the high accurate solution can be obtained with a few harmonic terms. The number of truncated terms is determined by the requirement of accuracy. Comparing with other approximate methods, the calculation of rapid series method is very easy and the accurate degrees of solution can be control. By comparing the analytical approximate solutions obtained by this method with numerical solutions of the cubic and fifth power oscillators, it is proven that this method is valid.

scholarly journals AN ACCURATE SOLUTION TO THE LOTKA-VOLTERRA EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD

2012 ◽  
Vol 09 ◽  
pp. 326-333 ◽  
Author(s):  
M. S. H. CHOWDHURY ◽  
M. M. RAHMAN
Keyword(s):  
Perturbation Method ◽  
Numerical Solution ◽  
Nonlinear Equations ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Volterra Equations ◽  
Algebraic Equations ◽  
Time Intervals ◽  
Homotopy Perturbation

In this paper, we suggest a method to solve the multispecies Lotka-Voltera equations. The suggested method, which we call modified homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of differential and algebraic equations. The HPM is modified in order to obtain the approximate solutions of Lotka-Voltera equation response in a sequence of time intervals. In particular, the example of two species is considered. The accuracy of this method is examined by comparison with the numerical solution of the Runge-Kutta-Verner method. The results prove that the modified HPM is a powerful tool for the solution of nonlinear equations.

scholarly journals Design analysis of Hub, Rim and Drum in Brake Assembly

2013 ◽  
Vol 3 (1) ◽  
pp. 170
Author(s):  
Ramamurti V. ◽  
Sukumar T. ◽  
Mithun S. ◽  
Prabhakar N. ◽  
Hudson P. V.
Keyword(s):  
Rear Axle ◽  
Computational Effort ◽  
Heavy Vehicles ◽  
Brake Shoe ◽  
Short Period ◽  
Rotationally Symmetric ◽  
Braking Torque ◽  
Brick Element ◽  
Machine Elements ◽  
The One

Stress analysis connected with the brake assembly of heavy vehicles is a complicated problem in view of the machine elements involved. The hub (on the rear axle), the rim (holding the wheel) and the drum (holding the brake shoe) experience severity of loads. While the vehicle is being driven the power is transmitted from the hub to the rim. When the brake is applied, the brake drum receives the braking torque and communicates it to the rim. Analysis associated with braking is actually transient since the braking torque varies with time in a short period of time whereas the one associated with driving is predominantly steady while the vehicle moves with uniform speed. None of them can be considered rotationally symmetric. Even though 3D brick element can be used for modelling all the three members, the computational effort needed to handle the problem of braking becomes extremely cumbersome. Hence a compromise solution is presented in this paper.

scholarly journals A stroboscopic averaging algorithm for highly oscillatory delay problems

2018 ◽  
Vol 39 (3) ◽  
pp. 1110-1133 ◽  
Author(s):  
J M Sanz-Serna ◽  
Beibei Zhu
Keyword(s):  
Delay Differential Equations ◽  
Averaging Method ◽  
Computational Effort ◽  
Constant Delay ◽  
Periodic Forcing ◽  
Step Size ◽  
Efficient Integration ◽  
Highly Oscillatory ◽  
Heterogeneous Multiscale ◽  
Delay Differential

Abstract We propose and analyse a heterogeneous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method suggested here may provide approximations with $\mathscr{O}\big (H^{2}+1/\varOmega ^{2}\big )$ errors with a computational effort that grows like $H^{-1}$ (the inverse of the step size), uniformly in the forcing frequency $\varOmega $.

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