Wyniki wyszukiwania - BazTech

1

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Sriwastav Nikhil, Barnwal Amit K., Wazwaz Abdul-Majid, Singh Mehakpreet

Opuscula Mathematica

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2023

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Vol. 43, no. 4

575--601

EN

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

2

A posteriori error estimates for beams with inexact flexural stiffness representation

Torii André J., Gracite Paula M.A., Miguel Leandro F.F., Lopez Rafael H.

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 2

62--74

EN

In this work, we present a posteriori error estimates for the Euler-Bernoulli beam theory with inexact flexural stiffness representation. This is an important subject in practice because beams with non-uniform flexural stiffness are frequently modeled using a mesh of elements with constant stiffness. The error estimates obtained in this work are validated by means of two numerical examples. The estimates presented here can be employed for adaptive mesh refinement.

3

Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition

Sekar Elango, Tamilselvan Ayyadurai

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

99--110

EN

A class of third order singularly perturbed delay differential equations of reaction diffusion type with an integral boundary condition is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The method suggested is of almost first order convergent. An error estimate is derived in the discrete norm. Numerical examples are presented, which validate the theoretical estimates.

4

Local accuracy and error bounds of the improved Runge-Kutta numerical methods

Qureshi S., Memon Z., Shaikh A. A.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 4

73--84

EN

In this paper, explicit Improved Runge-Kutta (IRK) methods with two, three and four stages have been analyzed in detail to derive the error estimates inherent in them whereas their convergence, order of local accuracy, stability and arithmetic complexity have been proved in the relevant literature. Using single and multivariate Taylor series expansion for a mathematical function of one and two variables respectively, slopes involved in the IRK methods have been expanded in order to obtain the general expression for the leading or principal term in the local truncation error of the methods. In addition to this, principal error functions of the methods have also been derived using the idea of Lotkin bounds which consequently gave rise to the error estimates for the IRK methods. Later, these error estimates were compared with error estimates of the two, three, and four-stage standard explicit Runge-Kutta (RK) methods to show the better performance of the IRK methods in terms of the error bounds on the constant step-size h used for solving the initial value problems in ordinary differential equations. Finally, a couple of initial value problems have been tested to determine the maximum absolute global errors, absolute errors at the final nodal point of the integration interval and the CPU times (seconds) for all the methods under consideration to get a better idea of how the methods behave in a particular situation especially when it comes to analyzing the error terms.

5

An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

Barboteu M., Bartosz K., Kalita P.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 2

263--276

EN

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

6

The concept of the variance estimation for the neural network approximator by jackknife subsampling

Pietraszek J., Gądek-Moszczak A.

Czasopismo Techniczne. Mechanika

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2013

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R. 110, z. 1-M

309--316

EN

The estimation of a variance for a semi-parametric neural network model variance for geometric properties of sintered metal will be done on the basis of jackknife subsampling method. Calculation results are of great practical significance because it will be possible to use proposed approach in similar microscale modelling. The proposed approach is simple and has many advantages if model identification procedure is computational expensive.

PL

W artykule przedstawiono estymację wariancji półparametrycznego modelu neuronowego cech geometrycznych spieku metali przeprowadzoną za pomocą metody podpróbkowania jackknife. Obliczone wyniki są cenne z uwagi na możliwość zastosowania proponowanego podejścia do analogicznych zagadnień modelowania w mikroskali.

7

A finite difference method for nonlinear parabolic-elliptic systems of second-order partial differential equations

Malec M., Sapa L.

Opuscula Mathematica

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2007

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Vol. 27, no. 2

259-289

EN

This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in R1+n. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.

8

A direct and accurate adaptive semi-Lagrangian scheme for the Vlasov-Poisson equation

Campos Pinto M. M.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

351-359

EN

This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)- dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter \epsilon which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L\infty towards the exact ones as \epsilon and \delta t tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.

9

Adaptive analysis of inelastic problems with Bodner-Partom constitutive model

Cecot W.

Computer Assisted Mechanics and Engineering Sciences

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2006

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Vol. 13, No. 4

513-521

EN

The Bodner-Partom elastic-visco-plastic constitutive eąuations [4] were used for numerical analysis of inelastic problems. This rate-dependent model makes it possible to describe elastic, plastic and viscous processes in metals, including temperaturę and continuum damage effects. The adaptive finite element method [9] was applied to approximate solution of the governing eąuations with two a posteriori error es-timates that control accuracy of time and space discretization of displacements and internal variables. The paper addresses a further development of the methodology proposed by the author in previous works [7, 8] and used in [6]. We present here certain additional theoretical background and propose a novel strategy of adaptation as well as verify the method of solution transfer.

10

Budżetowa metoda szacowania błędów konstruowanych urządzeń i systemów pomiarowych

Borek R.

Zeszyty Naukowe Politechniki Rzeszowskiej. Elektrotechnika

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2002

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z. 22 [192]

33-42

PL

W artykule omówiono budżetową metodę oszacowywania błędów projektowanego urządzenia pomiarowego na przykładzie toru przetwarzania analogowo-cyfrowego. Zwrócono uwagę na zalety tej metody w porównaniu do metod fenomenologicznych i strukturalnych. Przedstawiono przykład wykorzystania tej metody do oceny błędów wzmacniacza instrumentalnego. Zwrócono uwagę na możliwość rozszerzenia obszaru zastosowań tej metody na systemy o strukturze złożonej z modułowych wyspecjalizowanych podsystemów.

11

Numerical Analysis and Simulations of Quasistatic Frictionless Contact Problems

Fernandez Garcia J. R., Han W., Shillor M., Sofonea M.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 1

205-222

EN

A summary of recent results concerning the modelling as well as the variational and numerical analysis of frictionless contact problems for viscoplastic materials are presented. The contact is modelled with the Signorini or normal compliance conditions. Error estimates for the fully discrete numerical scheme are described, and numerical simulations based on these schemes are reported.