Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods

2004 ◽  
Vol 57 (1) ◽  
pp. B6-B7 ◽  
Author(s):  
J Awrejcewicz, ◽  
VA Krys’ko, ◽  
MV Shitikova,
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Finite Difference ◽  
Thermoelastic Problems

Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Muhad H. Abregov ◽  
Vladimir Z. Kanchukoev ◽  
Maryana A. Shardanova
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Deviating Argument ◽  
Second Order Differential Equation

AbstractThis work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

scholarly journals Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

2021 ◽  
pp. 66-70
Author(s):  
S. Homeniuk ◽  
S. Grebenyuk ◽  
D. Gristchak
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Time Dependent ◽  
Forced Oscillations

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

scholarly journals Reflection of Solar Activity Dynamics in Radionuclide Data

Radiocarbon ◽  
1992 ◽  
Vol 34 (2) ◽  
pp. 207-212 ◽  
Author(s):  
A. V. Blinov ◽  
M. N. Kremliovskij
Keyword(s):  
Nonlinear Dynamics ◽  
Solar Activity ◽  
Numerical Methods ◽  
Chaotic Behavior ◽  
Magnetic Activity ◽  
Data Sets ◽  
Cosmogenic Nuclide ◽  
Proxy Records ◽  
Solar Magnetic Activity ◽  
Activity Dynamics

Variability of solar magnetic activity manifested within sunspot cycles demonstrates features of chaotic behavior. We have analyzed cosmogenic nuclide proxy records for the presence of the solar activity signals. We have applied numerical methods of nonlinear dynamics to the data showing the contribution of the chaotic component. We have also formulated what kind of cosmogenic nuclide data sets are needed for investigations on solar activity.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

2016 ◽  
Vol 27 (09) ◽  
pp. 1650097 ◽  
Author(s):  
A. H. Encinas ◽  
V. Gayoso-Martínez ◽  
A. Martín del Rey ◽  
J. Martín-Vaquero ◽  
A. Queiruga-Dios
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Nonlinear Phenomena ◽  
Klein Gordon ◽  
Implicit Finite Difference ◽  
The Stability ◽  
Stability And Consistency ◽  
New Algorithms

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

scholarly journals Analytical and Numerical Methods for the CMKdV-II Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Ömer Akin ◽  
Ersin Özuğurlu
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Iterative Method ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Difference Method ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Form ◽  
De Vries

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.

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