scholarly journals Analysis of Numerical Measure and Numerical Integration Based on Measure

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yinkun Wang ◽  
Jianshu Luo ◽  
Xiangling Chen
Keyword(s):  
Numerical Method ◽  
Convergence Analysis ◽  
Numerical Integration ◽  
Error Bound ◽  
Lagrange Interpolation ◽  
Singular Integrals ◽  
Specific Method ◽  
Numerical Examples ◽  
Measure Function ◽  
Theoretical Results

We present a convergence analysis for a general numerical method to estimate measure function. By combining Lagrange interpolation, we propose a specific method for approximating the measure function and analyze the convergence order. Further, we analyze the error bound of numerical measure integration and prove that the numerical measure integration can decrease the singularity for singular integrals. Numerical examples are presented to confirm the theoretical results.

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

2015 ◽  
Vol 5 (4) ◽  
pp. 301-311 ◽  
Author(s):  
Lijun Yi
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Error Bound ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Time Stepping ◽  
Theoretical Results

AbstractThe h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

scholarly journals Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 99 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Yurii Shunkin
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Difference Method ◽  
Divided Difference ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Theoretical Results

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.

scholarly journals Convergence analysis for a fast class of multi-step Chebyshev-Halley-type methods under weak conditions

2021 ◽  
Vol 4 (1) ◽  
pp. 34-43
Author(s):  
Samundra Regmi ◽  
◽  
Ioannis K. Argyros ◽  
Santhosh George ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Computational Cost ◽  
Convergence Domain ◽  
Convergence Region ◽  
Type Method ◽  
Precise Information ◽  
Numerical Examples ◽  
Lipschitz Constants ◽  
Theoretical Results

In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results.

Numerical quadrature for computing of singular integrals

2015 ◽  
Vol 23 (2) ◽  
Author(s):  
Petr Stašek ◽  
Josef Kofron ◽  
Karel Najzar
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Second Order ◽  
Third Order ◽  
Rule Based ◽  
Numerical Examples ◽  
The Third ◽  
Hadamard Finite Part ◽  
Work First ◽  
Theoretical Results

AbstractThe paper is concerned with the superconvergence of numerical evaluation of Hadamard finite-part integral. Following the works [6-9], we studied the second-order and the third-order quadrature formulae of Newton-Cotes type and introduced new rules. The rule for the second-order gives the same convergence rate as the rule [6] but in more general cases, the rule for the third-order gives better results than the rule in [9] In this work, first we mention the main results on the superconvergence of the Newton-Cotes rules, we mention trapezoidal and Simpson’s rules and then we introduce a rule based on the cubic approximation. In the second part we describe important error estimates and in the last section we demonstrate theoretical results by numerical examples.

scholarly journals Convergence analysis of the scaled boundary finite element method for the Laplace equation

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Fleurianne Bertrand ◽  
Daniele Boffi ◽  
Gonzalo G. de Diego
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Element Method ◽  
Convergence Analysis ◽  
Interpolation Operator ◽  
Numerical Examples ◽  
Theoretical Results ◽  
Discrete Functions ◽  
Scaled Boundary Finite Element ◽  
Element Method

AbstractThe scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to partial differential equations (PDEs) without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation operator onto this space. We prove error estimates for this interpolation operator and show that optimal convergence to the solution can be obtained in SBFEM. These theoretical results are backed by two numerical examples.

scholarly journals Improved error estimate and applications of the complete quartic spline

2019 ◽  
Vol 27 (2) ◽  
pp. 71-83
Author(s):  
Alexandru Mihai Bica ◽  
Diana Curilă ◽  
Zoltan Satmari
Keyword(s):  
Error Estimate ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Numerical Integration ◽  
Error Bound ◽  
Fourth Order ◽  
Boundary Value ◽  
Continuous Functions ◽  
Point Boundary ◽  
Lipschitzian Functions

AbstractIn this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

scholarly journals Design and analysis of a faster King-Werner-type derivative free method

2022 ◽  
Vol 40 ◽  
pp. 1-18
Author(s):  
J. R. Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Examples ◽  
Weak Center ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
Methods Numerical

We introduce a new faster  King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local  convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.

scholarly journals On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 286
Author(s):  
Fairouz Tchier ◽  
Ioannis Dassios ◽  
Ferdous Tawfiq ◽  
Lakhdar Ragoub
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Efficient Method ◽  
Pseudospectral Method ◽  
Numerical Examples ◽  
Expansion Coefficients ◽  
Cardinal Functions ◽  
Theoretical Results ◽  
Chebyshev Functions

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

scholarly journals Bipartite Synchronization of Heterogeneous Signed Networks with Distributed Impulsive Control

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Guizhen Feng ◽  
Jian Ding ◽  
Jinde Cao ◽  
Qingqing Cao
Keyword(s):  
Error Bound ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Mathematical Expression ◽  
Synchronization Error ◽  
Error Level ◽  
Numerical Examples ◽  
Signed Networks ◽  
Average Impulsive Interval ◽  
Theoretical Results

This study investigates the bipartite synchronization of heterogeneous signed networks with distributed impulsive control. Leader-follower bipartite synchronization within a nonzero error bound is analyzed when the average impulsive interval is T a < ∞ or T a = ∞ . Some sufficient conditions to achieve the bipartite quasi-synchronization are presented, and the synchronization error level is estimated by the specific mathematical expression. The correctness of the theoretical results is verified by numerical examples.

scholarly journals Multiple orthogonality in the space of trigonometric polynomials of semi-integer degree

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2227-2237
Author(s):  
Marija Stanic ◽  
Tatjana Tomovic
Keyword(s):  
Numerical Method ◽  
Recurrence Relations ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Optimal Set ◽  
Numer Math ◽  
Theoretical Results

In this paper we consider multiple orthogonal trigonometric polynomials of semi-integer degree, which are necessary for constructing of an optimal set of quadrature rules with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994) 271-288]. We prove that such multiple orthogonal trigonometric polynomials satisfy certain recurrence relations and present numerical method for their construction, as well as for construction of mentioned optimal set of quadrature rules. Theoretical results are illustrated by some numerical examples.

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