Accuracy of the difference scheme of solving the eigenvalue problem for the Laplacian

2011 ◽  
Vol 47 (5) ◽  
pp. 783-790 ◽  
Author(s):  
N. V. Maiko ◽  
V. G. Prikazchikov ◽  
V. L. Ryabichev
Keyword(s):  
Eigenvalue Problem ◽  
Difference Scheme ◽  
The Difference

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

Eigenanalysis of the Rigid Body Motion Effect on Elastic Systems

1983 ◽  
Vol 105 (4) ◽  
pp. 461-466 ◽  
Author(s):  
A. Maher ◽  
A. L. Schlack
Keyword(s):  
Eigenvalue Problem ◽  
Rigid Body ◽  
Free Vibration Analysis ◽  
Plane Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Computer Calculations ◽  
Motion Effect ◽  
Ritz Procedure ◽  
The Difference

In this paper, the influence of rigid body motion on the behavior of a vibrating elastic system is treated by the development of a difference eigenvalue problem. The maximum possible changes in eigenfrequencies due to removal of constraints are obtained by the employment of the bound approach [1, 2]. As an application to a structural system the Rayleigh-Ritz procedure is employed for constructing the difference eigenvalue problem. Discussion of the use of the method for various types of engineering problems is outlined. An example of a free vibration analysis of a simply supported beam in plane motion with a nonuniform mass and elasticity distribution is solved. A comparison between computer calculations and previously published results is presented.

scholarly journals On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

2021 ◽  
Vol 26 (4) ◽  
pp. 738-758
Author(s):  
Regimantas Čiupaila ◽  
Kristina Pupalaigė ◽  
Mifodijus Sapagovas
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Nonlinear Elliptic Equations ◽  
Integral Boundary ◽  
Nonlinear Elliptic ◽  
Variable Weight ◽  
The Difference ◽  
Convergence Of Iterative Method

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Boundary Value Problem ◽  
Difference Scheme ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
The Difference

This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.

Propagator Method for Numerical Solution of Convective Fisher Equation

2011 ◽  
Vol 222 ◽  
pp. 387-390
Author(s):  
Daiga Zaime ◽  
Janis S. Rimshans ◽  
Sharif E. Guseynov
Keyword(s):  
Difference Scheme ◽  
Iteration Process ◽  
Fisher Equation ◽  
Implicit Difference Scheme ◽  
Nonlinear Convection ◽  
First Order ◽  
Truncation Errors ◽  
Propagator Method ◽  
Left And Right ◽  
The Difference

Propagator numerical method was developed as an effective tool for modeling of linear advective dispersive reactive (ADR) processes [1]. In this work implicit propagator difference scheme for Fisher equation with nonlinear convection (convective Fisher equation) is elaborated. Our difference scheme has truncation errors of the second order in space and of the first order in time. Iteration process for implicit difference scheme is proposed by introducing forcing terms in the left and right sides of the difference equation. Convergence and stability criterions for the elaborated implicit propagator difference scheme are obtained.

A CG-Type Method for Inverse Quadratic Eigenvalue Problems in Model Updating of Structural Dynamics

2011 ◽  
Vol 3 (1) ◽  
pp. 65-86
Author(s):  
Jiaofen Li ◽  
Xiyan Hu
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Model Updating ◽  
Inverse Eigenvalue Problem ◽  
Computationally Efficient ◽  
Type Method ◽  
Quadratic Pencil ◽  
Model Coefficient ◽  
Inverse Eigenvalue ◽  
The Difference

AbstractIn this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matricesM, DandKfor the quadratic pencilQ(λ) =λ2M+ λD+K, so thatQ(λ) has a prescribed subset of eigenvalues and eigenvectors. This method can determine the solvability of the inverse eigenvalue problem automatically. We then consider the least squares model for updating a quadratic pencilQ(λ). More precisely, we update the model coefficient matrices M, C and K so that (i) the updated model reproduces the measured data, (ii) the symmetry of the original model is preserved, and (iii) the difference between the analytical triplet (M, D, K) and the updated triplet (Mnew,Dnew,Knew) is minimized. In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.

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