scholarly journals Approximation of Modified Baskakov Operators Based on Parameter s

2021 ◽  
pp. 588-593
Author(s):  
Ali J. Mohammad ◽  
S. A. Abdul-Hammed ◽  
T. A. Abdul-Qader
Keyword(s):  
Numerical Results ◽  
Order Of Approximation ◽  
Test Functions ◽  
Baskakov Operators ◽  
Type Formula ◽  
Better Than

In this article, we define and study a family of modified Baskakov type operators based on a parameter . This family is a generalization of the classical Baskakov sequence. First, we prove that it converges to the function being approximated. Then, we find a Voronovsky-type formula and obtain that the order of approximation of this family is . This order is better than the order of the classical Baskakov sequence  whenever . Finally, we apply our sequence to approximate two test functions and analyze the numerical results obtained.

scholarly journals Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Purshottam Narain Agrawal ◽  
Jitendra Kumar Singh
Keyword(s):  
Rate Of Convergence ◽  
Maximal Function ◽  
Modulus Of Smoothness ◽  
Order Of Approximation ◽  
Test Functions ◽  
Approximation Properties ◽  
Baskakov Operators ◽  
Convergence Behaviour ◽  
Durrmeyer Variant ◽  
Lipschitz Type Maximal Function

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

The Semi-Discrete Finite Element Method for the Cauchy Equation

2014 ◽  
Vol 668-669 ◽  
pp. 1130-1133
Author(s):  
Lei Hou ◽  
Xian Yan Sun ◽  
Lin Qiu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Results ◽  
Cauchy Equation ◽  
The Time Domain ◽  
Nicolson Scheme ◽  
Discrete Finite Element ◽  
Better Than ◽  
Element Method ◽  
Crank Nicolson

In this paper, we employ semi-discrete finite element method to study the convergence of the Cauchy equation. The convergent order can reach. In numerical results, the space domain is discrete by Lagrange interpolation function with 9-point biquadrate element. The time domain is discrete by two difference schemes: Euler and Crank-Nicolson scheme. Numerical results show that the convergence of Crank-Nicolson scheme is better than that of Euler scheme.

A NEW NUMEROV-TYPE METHOD FOR COMPUTING EIGENVALUES AND RESONANCES OF THE RADIAL SCHRÖDINGER EQUATION

1996 ◽  
Vol 07 (01) ◽  
pp. 33-41 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
New Method ◽  
Resonance Problem ◽  
Step Method ◽  
Type Method ◽  
Radial Schrödinger Equation ◽  
Better Than

A two-step method is developed for computing eigenvalues and resonances of the radial Schrödinger equation. Numerical results obtained for the integration of the eigenvalue and the resonance problem for several potentials show that this new method is better than other similar methods.

Optimal Operation of Multireservoir Systems by Enhanced Water Cycle Algorithm

2019 ◽  
Vol 7 (1) ◽  
pp. 27-43
Author(s):  
Yanjun Kong ◽  
Yadong Mei ◽  
Weinan Li ◽  
Ben Yue ◽  
Xianxun Wang
Keyword(s):  
Water Cycle ◽  
Water System ◽  
System Size ◽  
Optimal Operation ◽  
Test Functions ◽  
Water Cycle Algorithm ◽  
Adaptive Parameter ◽  
Fitness Value ◽  
Better Than

In this article, an enhanced water cycle algorithm (EWCA) is proposed and applied to optimize the operation of multireservoir systems. Three improvements have been made to the water cycle algorithm (WCA). They refer to high-quality initial solutions obtained by the chaos-based method, balancing of exploration of streams using a dynamic adaptive parameter, and dynamic variation of sub-water system size using the fitness value of rivers. For the purpose of verifying the improvements, three typical benchmark functions were selected as test functions. It has shown that EWCA performs better than WCA and water cycle algorithm with evaporation rate (ER-WCA). And then these three algorithms were also applied to optimize the operation of a multireservoir system with complex constrains as the case study. By comparing the results, it is found that the EWCA has higher ability to find a feasible solution in a narrow searching space. The effectiveness of the improvements is confirmed.

scholarly journals Solving Nonstiff Higher-Order Ordinary Differential Equations Using 2-Point Block Method Directly

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Hazizah Mohd Ijam ◽  
Mohamed Suleiman ◽  
Ahmad Fadly Nurullah Rasedee ◽  
Norazak Senu ◽  
Ali Ahmadian ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Divided Difference ◽  
Approximate Solutions ◽  
Higher Order ◽  
Numerical Results ◽  
The Other ◽  
Block Method ◽  
Better Than

We describe the development of a 2-point block backward difference method (2PBBD) for solving system of nonstiff higher-order ordinary differential equations (ODEs) directly. The method computes the approximate solutions at two points simultaneously within an equidistant block. The integration coefficients that are used in the method are obtained only once at the start of the integration. Numerical results are presented to compare the performances of the method developed with 1-point backward difference method (1PBD) and 2-point block divided difference method (2PBDD). The result indicated that, for finer step sizes, this method performs better than the other two methods, that is, 1PBD and 2PBDD.

scholarly journals Modified Preconditioned GAOR Methods for Systems of Linear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xue-Feng Zhang ◽  
Qun-Fa Cui ◽  
Shi-Liang Wu
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Least Squares ◽  
Linear Equations ◽  
Generalized Least Squares ◽  
Numerical Results ◽  
Least Squares Problem ◽  
Comparison Results ◽  
Systems Of Linear Equations ◽  
Better Than

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.

scholarly journals Adaptive Opposition Slime Mould Algorithm

2021 ◽  
Author(s):  
Manoj Kumar Naik ◽  
Rutuparna Panda ◽  
Ajith Abraham
Keyword(s):  
Random Search ◽  
Optimal Solution ◽  
Optimization Methods ◽  
Function Optimization ◽  
Rank Test ◽  
Test Functions ◽  
Slime Mould ◽  
Opposition Based Learning ◽  
Suggested Technique ◽  
Better Than

Abstract Recently, the slime mould algorithm (SMA) has become popular in function optimization, because it effectively uses exploration and exploitation to reach an optimal solution or near-optimal solution. However, the SMA uses two random search agents from the whole population to decide the future displacement and direction from the best search agents, which limits its exploitation and exploration. To solve this problem, we investigate an adaptive approach to decide whether opposition based learning (OBL) will be used or not. Sometimes the OBL is used to further increase the exploration. In addition, it maximizes the exploitation by replacing one random search agent with the best one in the position updating. The suggested technique is called an adaptive opposition slime mould algorithm (AOSMA). The qualitative and quantitative analysis of AOSMA is reported using 29 test functions consisting of 23 classical test functions and 6 recently used composition functions from the IEEE CEC 2014 test suite. The results are compared with state-of-the-art optimization methods. Results presented in this paper show that AOSMA’s performance is better than other optimization algorithms. The AOSMA is evaluated using Wilcoxon’s rank-sum test. It also ranked one in Friedman’s mean rank test. The proposed AOSMA algorithm would be useful for function optimization to solve real-world engineering problems.

scholarly journals Approximation by modified Jain–Baskakov operators

2020 ◽  
Vol 27 (3) ◽  
pp. 403-412
Author(s):  
Vishnu Narayan Mishra ◽  
Preeti Sharma ◽  
Marius Mihai Birou
Keyword(s):  
Weighted Approximation ◽  
Approximation Order ◽  
Continuous Functions ◽  
Basis Functions ◽  
Test Functions ◽  
Approximation Properties ◽  
Baskakov Operators ◽  
Closed Intervals ◽  
Direct Results ◽  
Modified Forms

AbstractIn the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions {e_{0}} and {e_{1}}. It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

A sequence of positive linear operators related to powered Baskakov basis

2019 ◽  
Vol 35 (1) ◽  
pp. 51-58
Author(s):  
ADRIAN HOLHOS ◽  
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Properties ◽  
Type Result ◽  
Baskakov Operators ◽  
Better Than

In this paper we study some approximation properties of a sequence of positive linear operators defined by means of the powered Baskakov basis. We prove that in the particular case of squared Baskakov basis the operators behave better than the classical Baskakov operators. For this particular case we give also a quantitative Voronovskaya type result.

scholarly journals NUCLEON AND PION FORM FACTORS IN DIFFERENT FORMS OF RELATIVISTIC KINEMATICS

2005 ◽  
Vol 20 (08n09) ◽  
pp. 1601-1606 ◽  
Author(s):  
B. DESPLANQUES
Keyword(s):  
Single Particle ◽  
Form Factors ◽  
Numerical Results ◽  
Relativistic Kinematics ◽  
High Q ◽  
Particle Current ◽  
Better Than

Calculations of form factors in different forms of relativistic kinematics are presented. They involve the instant, front and point forms. In the two first cases, different kinematical conditions are considered while in the latter case, both a Dirac-inspired approach and a hyperplane-based one are incorporated in our study. Numerical results are presented for the pion form factors with emphasis on both the low and high Q2 range. A new argument is presented, explaining why some approaches do considerably much better than other ones when only a single-particle current is considered.

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