Journal of Numerical Analysis and Approximation Theory

Influence of control parameters on the stabilization of an Euler-Bernoulli flexible beam

Goh André-Pascal Abro — 2024-07-11

In this work, we numerically study the influence of control parameters on the stabilization of a flexible Euler-Bernoulli beam fixed at one end and subjected at the other end to a force control and a punctual moment control proportional respectively to velocity and rotation velocity. First, we analyze the displacement stabilization and the asymptotic behavior of the beam energy using a stable numerical scheme, resulting from the Crank-Nicholson algorithm for time discretization and the finite element method based on the approximation by Hermite's cubic polynomial functions, for discretization in space. Then, by means of the finite element method, we represent the spectrum of the operator associated with this beam problem and we carry out a qualitative study of the
locus of the eigenvalues according to the positive control parameters. From these studies we conclude that rotation velocity control has more effect on the stabilization of the beam compared to velocity control. Finally, this result is confirmed by a sensitivity study on the control parameters involved in the stabilization of the beam.

Convergence of \(\lambda\)-Bernstein - Kantorovich operators in the \(L_p\)- norm

Purshottam N. Agrawal — 2024-07-11

We show the convergence of \(\lambda\)-Bernstein - Kantorovich operators defined by Acu et al. [J. Ineq. Appl. 2018], for functions in \(L_p[0,1],\, p\geq 1\). We also determine the convergence rate via integral modulus of smoothness.

Adaptation of the composite finite element framework for semilinear parabolic problems

Anjaly Anand — 2024-04-16

In this article, we discuss one type of finite element method (FEM), known as the composite finite element method (CFE). Dimensionality reduction is the primary benefit of CFE as it helps to reduce the complexity of the domain space. The number of degrees of freedom is greater in standard FEM compared to CFE. We consider the semilinear parabolic problem in a 2D convex polygonal domain. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization is carried out only in space. Then, the fully discrete problem is taken into account, where both the spatial and time components get discretized.
In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework are adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

Local convergence of a two-step Gauss-Newton Werner-type method for solving least squares problems

Ioannis K. Argyros — 2024-07-11

The aim of this paper is to extend the applicability of a two-step Gauss-Newton-Werner-type method (TGNWTM) for solving nonlinear least squares problems. The radius of convergence, error bounds and the information on the location of the solution are improved under the same information as in earlier studies. Numerical examples further validate the theoretical results.

Preserving properties of some Szasz-Mirakyan type operators

Jorge Bustamante — 2024-07-11

For a family of Szasz-Mirakyan type operators we prove that they preserve convex-type functions and that a monotonicity property verified by Cheney and Sharma in the case Szasz-Mirakyan operators holds for the variation study here. We also verify that several modulus of continuity are preserved.

The second order modulus revisited: remarks, applications, problems

Heiner Gonska — 2024-07-11

Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.

A Stancu type extension of the Cheney-Sharma Chlodovsky operators

Eduard Ștefan Grigoriciuc — 2024-07-11

In this paper we introduce a Stancu type extension of the Cheney-Sharma Chlodovsky operators based on the ideas presented by Cătinaș and Buda, Bostanci and Bașcanbaz-Tunca, respectively Söylemez and Tașdelen. For this new operators we study some approximation and convexity properties and the preservation of the Lipschitz constant and order. Finally, we study approximation properties of the new operators with the help of Korovkin type theorems.

New accelerated modulus-based iteration method for solving large and sparse linear complementarity problem

Bharat Kumar — 2024-07-11

For the large and sparse linear complementarity problem, we provide a family of new accelerated modulus-based iteration methods in this article. We provide some sufficient criteria for the convergence analysis when the system matrix is a \(P\)-matrix or an \(H_+\)-matrix. In addition, we provide some numerical examples of the different parameters to illustrate the efficacy of our proposed methods. These methods help us reduce the number of iterations and the time required by the CPU, which improves convergence performance.

A comparative study of Filon-type rules for oscillatory integrals

Hassan Majidian — 2024-03-06

Our aim is to answer the following question: "Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).

Controlling numerical diffusion in solving advection-dominated transport problems

Nicolae Suciu — 2024-07-11

Numerical schemes for advection-dominated transport problems are are evaluated in a comparative study. Explicit and implicit finite difference methods are analyzed together with a global random walk algorithm in the frame of a splitting procedure. The efficiency of the methods with respect to the control of the numerical diffusion is investigated numerically on one-dimensional problems with constant coefficients and two-dimensional problems with variable coefficients consisting of realizations of space-random functions.

Iterative schemes for coupled flow and transport in porous media -- Convergence and truncation errors

Nicolae Suciu — 2024-07-11

Nonlinearities of coupled flow and transport problems for partially saturated porous media are solved with explicit iterative L-schemes. Their behavior is analyzed with the aid of the computational orders of convergence. This approach allows highlighting the influence of the truncation errors in the numerical schemes on the convergence of the iterations. Further, by using manufactured exact solutions, error-based orders of convergence of the iterative schemes are assessed and the convergence of the numerical solutions is demonstrated numerically through grid-convergence tests.