scholarly journals STABILITY ANALYSIS OF RIGID SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
pp. 54-64
Author(s):  
S.G. Bulanov
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations

The Analysis of the Systems Stability of Linear Differential Equations Based on the Transformation of Difference Schemes

2019 ◽  
Vol 20 (9) ◽  
pp. 542-549 ◽  
Author(s):  
S. G. Bulanov
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computer Analysis ◽  
Linear Differential Equations ◽  
System Stability ◽  
The Difference ◽  
Linear Differential ◽  
The Stability

The approach to the analysis of Lyapunov systems stability of linear ordinary differential equations based on multiplicative transformations of difference schemes of numerical integration is presented. As a result of transformations, the stability criteria in the form of necessary and sufficient conditions are formed. The criteria are invariant with respect to the right side of the system and do not require its transformation with respect to the difference scheme, the length of the gap and the step of the solution. A distinctive feature of the criteria is that they do not use the methods of the qualitative theory of differential equations. In particular, for the case of systems with a constant matrix of the coefficients it is not necessary to construct a characteristic polynomial and estimate the values of the characteristic numbers. When analyzing the system stability with variable matrix coefficients, it is not necessary to calculate the characteristic indicators. The varieties of criteria in an additive form are obtained, the stability analysis based on them being equivalent to the stability assessment based on the criteria in a multiplicative form. Under the conditions of a linear system stability (asymptotic stability) of differential equations, the criteria of the systems stability (asymptotic stability) of linear differential equations with a nonlinear additive are obtained. For the systems of nonlinear ordinary differential equations the scheme of stability analysis based on linearization is presented, which is directly related to the solution under study. The scheme is constructed under the assumption that the solution stability of the system of a general form is equivalent to the stability of the linearized system in a sufficiently small neighborhood of the perturbation of the initial data. The matrix form of the criteria allows implementing them in the form of a cyclic program. The computer analysis is performed in real time and allows coming to an unambiguous conclusion about the nature of the system stability under study. On the basis of a numerical experiment, the acceptable range of the step variation of the difference method and the interval length of the difference solution within the boundaries of the reliability of the stability analysis is established. The approach based on the computer analysis of the systems stability of linear differential equations is rendered. Computer testing has shown the feasibility of using this approach in practice.

STABILITY ANALYSIS OF A PROPOSED SCHEME OF ORDER FIVE FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 898
Author(s):  
Sunday Emmanuel Fadugba ◽  
Roseline Bosede Ogunrinde ◽  
Rowland Rotimi Ogunrinde
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Step Length ◽  
Step Method ◽  
First Order ◽  
Absolute Relative Error ◽  
One Step ◽  
Test Equation ◽  
The Stability

This paper presents the stability analysis of a proposed scheme of order five (FCM) for first order Ordinary Differential Equations (ODEs). The proposed FCM is derived by means of an interpolating function of polynomial and exponential forms. The properties of FCM were discussed extensively. The linear stability of FCM in the context of the Third Order One-Step Method (TCM) and Second Order One-Step Method (SCM) for the solution of initial value problems of first order differential equations is presented. The stability region of FCM, TCM and SCM is investigated using the Dahlquist’s test equation. The numerical results obtained via FCM are compared with TCM and SCM. Moreover, by varying the step length, the accuracy and convergence of the methods in terms of the final absolute relative error are measured. The results show that FCM converges faster and more stable than its counterparts.

Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation

2016 ◽  
Vol 30 (11) ◽  
pp. 1650110 ◽  
Author(s):  
H. S. Shukla ◽  
Mohammad Tamsir ◽  
Vineet K. Srivastava ◽  
Mohammad Mehdi Rashidi
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Burger Equation ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Differential Quadrature ◽  
Solution Stability ◽  
Quadrature Method ◽  
B Spline

In this paper, we propose a modified cubic B-spline differential quadrature method (MCB-DQM) to solve three-dimensional (3D) coupled viscous Burger equation with appropriate initial and boundary conditions. In this method, modified cubic B-spline is treated as a basis function in the differential quadrature method (DQM) to compute the weighting coefficients. In this way, the Burger equation is reduced into a system of ordinary differential equations. An optimal strong stability-preserving Runge–Kutta (SSP-RK) method is employed to solve the resulting system of ordinary differential equations. In order to illustrate the accuracy and efficiency of the proposed method, a numerical problem is considered. From the numerical experiment, it is found that the computed result is in good agreement with the exact solution. Stability analysis of the method is also carried out using the matrix stability analysis method and found to be unconditionally stable.

scholarly journals A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1154
Author(s):  
Essam R. El-Zahar ◽  
José Tenreiro Machado ◽  
Abdelhalim Ebaid
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stability Property ◽  
Order Of Convergence ◽  
Numerical Solutions ◽  
Explicit Method ◽  
Stiff Ordinary Differential Equations ◽  
Integration Schemes ◽  
Special Cases

A new generalised Taylor-like explicit method for stiff ordinary differential equations (ODEs) is proposed. The algorithm is presented in its component and vector forms. The error and stability analysis of the method are developed showing that it has an arbitrary high order of convergence and the L-stability property. Moreover, it is verified that several integration schemes are special cases of the new general form. The method is applied on stiff problems and the numerical solutions are compared with those of the classical Taylor-like integration schemes. The results show that the proposed method is accurate and overcomes the shortcoming of the classical Taylor-like schemes in their component and vector forms.

scholarly journals Lyapunov stability for discontinuous systems

2020 ◽  
Vol 42 ◽  
pp. e17
Author(s):  
Iguer Santos
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Stability ◽  
Lyapunov Functions ◽  
Discontinuous Systems ◽  
Discontinuous Differential Equations ◽  
Nonautonomous Differential Equations ◽  
The Stability ◽  
Lyapunov Sense

The present work studies the stability analysis of equilibrium of ordinary differential equations with the discontinuous right side, also called discontinuous differential equations, using the notion of Carathéodory solution for differential equations. This way, it is studied the stability of equilibrium in the Lyapunov sense for discontinuous systems through nonsmooth Lyapunov functions. Then two existing Lyapunov theorems are obtained. The results established refer to systems determined by nonautonomous differential equations.

scholarly journals Stability Analysis and Numerical Solutions of a Competition Model with the Effects of Distribution Parameters

2019 ◽  
Vol 43 (1) ◽  
pp. 95-106
Author(s):  
Md Kamrujjaman ◽  
Sadia Akter Lima ◽  
Sonia Akter ◽  
Tanzila Eva
Keyword(s):  
Mathematical Biology ◽  
Stability Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Population Models ◽  
Distribution Parameters ◽  
Academy Of Sciences ◽  
Linearization Techniques

A system of two nonlinear differential equations in mathematical biology is considered. These models are originally stimulated by population models in biology when solutions are required to be non-negative, but the ordinary differential equations can be understood outside of this conventional scope of population models. The focus of this paper is on the use of linearization techniques, and Hartman Grobman theory to analyze nonlinear differential equations. We provide stability analysis and numerical solutions for these models that describe behaviors of solutions based only on the parameters used in the formulation of the systems. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 95-106, 2019

scholarly journals Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 185
Author(s):  
Angelamaria Cardone ◽  
Dajana Conte ◽  
Raffaele D’Ambrosio ◽  
Beatrice Paternoster
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Experiments ◽  
Collocation Methods ◽  
Spline Collocation ◽  
Fractional Differential ◽  
Convergence And Stability Analysis

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

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