scholarly journals Extended High Order Algorithms for Equations under the Same Set of Conditions

Algorithms ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 207
Author(s):  
Ioannis K. Argyros ◽  
Debasis Sharma ◽  
Christopher I. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda ◽  
...  
Keyword(s):  
Higher Order ◽  
High Order ◽  
Sixth Order ◽  
Convergence Order ◽  
First Derivative ◽  
Solving Equations

A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

scholarly journals Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Sixth Order ◽  
Convergence Order ◽  
Not Given ◽  
Numerical Examples ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Order Scheme ◽  
Solving Equations

Abstract The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

scholarly journals Extending the Convergence of Two Similar Sixth Order Schemes for Solving Equations under Generalized Conditions

2021 ◽  
pp. 246-257
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Sixth Order ◽  
Convergence Order ◽  
Not Given ◽  
Numerical Examples ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Solving Equations

The applicability of two competing efficient sixth convergence order schemes is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the schemes. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided not given in the earlier works based on ω-continuity conditions. Our technique extends other schemes analogously, since it is so general. Numerical examples complete this work.

A comparison between two competing sixth convergence order algorithms under the same set of conditions

2021 ◽  
Vol 30 (1) ◽  
pp. 19-28
Author(s):  
GUS ARGYROS ◽  
MICHAEL ARGYROS ◽  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Banach Space ◽  
Error Estimates ◽  
Taylor Series ◽  
High Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Convergence Conditions ◽  
First Derivative ◽  
Banach Space Operators

There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines.

scholarly journals Convergence and Dynamics of a Higher-Order Method

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 420 ◽  
Author(s):  
Alejandro Moysi ◽  
Ioannis K. Argyros ◽  
Samundra Regmi ◽  
Daniel González ◽  
Á. Alberto Magreñán ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Higher Order ◽  
High Order ◽  
Local Form ◽  
Order Method ◽  
First Derivative ◽  
The Family ◽  
Higher Order Method

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

VERY HIGH-ORDER SPATIALLY IMPLICIT SCHEMES FOR COMPUTATIONAL ACOUSTICS ON CURVILINEAR MESHES

2001 ◽  
Vol 09 (04) ◽  
pp. 1259-1286 ◽  
Author(s):  
MIGUEL R. VISBAL ◽  
DATTA V. GAITONDE
Keyword(s):  
Three Dimensional ◽  
Radiation Condition ◽  
Higher Order ◽  
High Order ◽  
High Frequencies ◽  
Decomposition Strategies ◽  
The Impact ◽  
Spurious Modes ◽  
Very High ◽  
Computational Strategy

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. For three-dimensional computations, special metric relations are employed to assure the fidelity of the scheme in highly curvilinear meshes. A variety of problems, including several benchmark computations, demonstrate the success of the overall computational strategy.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Towards High-Order Large Eddy Simulation of Aero-Thermal Flows for Turbomachinery Applications

2017 ◽  
Author(s):  
R. Bhaskaran ◽  
Feilin Jia ◽  
Gregory M. Laskowski ◽  
Z. J. Wang ◽  
Umesh Paliath
Keyword(s):  
Heat Transfer ◽  
Large Eddy Simulation ◽  
Numerical Algorithms ◽  
High Order ◽  
Sixth Order ◽  
Computational Grids ◽  
Variable Order ◽  
Free Stream Turbulence ◽  
Eddy Simulation ◽  
Large Eddy

The solution accuracy and computational efficiency of high order Large Eddy Simulation (LES) solvers are evaluated on two benchmark open literature blade cascade problems. The first problem concerns wake development in the T106A low pressure turbine cascade [1]. The second problem examines the effect of free-stream turbulence on heat transfer from the VKI first stage high pressure turbine vane [2]. The calculations are performed with two independently developed high order LES solvers using completely different numerical algorithms. The first solver FDL3Di [3] was originally developed at the Airforce Research Laboratory (AFRL) and employs structured overset grids. It uses a sixth order compact finite difference scheme in space along with an implicit Beam-Warming scheme for time marching. The second solver, hpMusic, is developed at the University of Kansas [4]. This is a variable order (up to sixth order) unstructured grid solver employing a discontinuous formulation known as flux reconstruction (FR) / correction procedure via reconstruction (CPR) [5]. The computational grids used are independently tuned for each application. The solvers are benchmarked against experimental data for wake development and blade heat transfer coefficient. Further physical insights in to the test cases are also obtained, filling gaps in experimental results, especially for the VKI problem.

scholarly journals Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

2019 ◽  
Vol 4 (2) ◽  
pp. 34
Author(s):  
Deasy Wahyuni ◽  
Elisawati Elisawati
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Convergence Order ◽  
Matlab Program ◽  
Analytical Results

Newton method is one of the most frequently used methods to find solutions to the roots of nonlinear equations. Along with the development of science, Newton's method has undergone various modifications. One of them is the hasanov method and the newton method variant (vmn), with a higher order of convergence. In this journal focuses on the three-step iteration method in which the order of convergence is higher than the three methods. To find the convergence order of the three-step iteration method requires a program that can support the analytical results of both methods. One of them using the help of the matlab program. Which will then be compared with numerical simulations also using the matlab program.  Keywords : newton method, newton method variant, Hasanov Method and order of convergence

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