scholarly journals Approximate Solution of Fourth Order Near Critically Damped Nonlinear Systems with Special Conditions

2012 ◽  
Vol 36 (2) ◽  
pp. 187-197
Author(s):  
Md Asraful Alom ◽  
M Alhaz Uddin
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Numerical Method ◽  
Fourth Order ◽  
Perturbation Technique ◽  
Kbm Method ◽  
Academy Of Sciences

A perturbation technique has been developed based on the Krylov-Bogoliubov-Mitropolskii (KBM) method to investigate the solution of fourth order near critically damped nonlinear systems in the case of ?1??2, ?4 ? ?3 + 2?1 but ?4<2?3 among the eigenvalues ?1, ?2, ?3, ?4. The solutions obtained by this technique were compared with those obtained by numerical method. The method has been explained by an example. DOI: http://dx.doi.org/10.3329/jbas.v36i2.12962 Journal of Bangladesh Academy of Sciences, Vol. 36, No. 2, 187-197, 2012

scholarly journals Approximate Solution of a Fourth Order Weakly Non-Linear Differential System with Strong Damping and Slowly Varying Coefficients by Unified KBM Method

1970 ◽  
Vol 34 (1) ◽  
pp. 71-82
Author(s):  
M Alhaz Uddin ◽  
MA Sattar
Keyword(s):  
Approximate Solution ◽  
Differential System ◽  
Fourth Order ◽  
Initial Conditions ◽  
Numerical Procedure ◽  
The Other ◽  
Oscillatory Process ◽  
Strong Damping ◽  
Varying Coefficients ◽  
Kbm Method

The unified Krylov-Bogoliubov-Mitropolskii (KBM) method is used for determining theanalytical approximate solution of a fourth order weakly nonlinear differential system with strongdamping and slowly varying coefficients when a pair of eigen-values of the unperturbed equationis a multiple (approximately or perfectly) of the other pair or pairs. In a damped case, one of thenatural frequencies of the linearized equation may be a multiple of the other. The analytical firstorder approximate solution for different initial conditions shows a good coincidence with thoseobtained by the numerical procedure. The method is illustrated by an example.Key words: Perturbation method; Weak nonlinearity; Oscillatory process; Strong damping; Varying coefficientsDOI: 10.3329/jbas.v34i1.5493Journal of Bangladesh Academy of Sciences, Vol.34, No.1, 71-82, 2010

scholarly journals An Efficient Zero-Stable Numerical Method for Fourth-Order Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
S. J. Kayode
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Numerical Method ◽  
Fourth Order ◽  
Optimal Order ◽  
Direct Solution ◽  
Order Of Accuracy ◽  
First Order ◽  
Stable Numerical Method

The purpose of this paper is to produce an efficient zero-stable numerical method with the same order of accuracy as that of the main starting values (predictors) for direct solution of fourth-order differential equations without reducing it to a system of first-order equations. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is consistent, symmetric, and of optimal order . The main predictor for the method is also consistent, symmetric, zero-stable, and of optimal order .

scholarly journals Approximate solution of nonlinear differential system with time variation

2021 ◽  
Vol 44 (2) ◽  
pp. 121-130
Author(s):  
Rezaul Karim ◽  
Pinakee Dey ◽  
Saikh Shahjahan Miah
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Differential System ◽  
Time Variation ◽  
Autonomous Systems ◽  
Differential Systems ◽  
Matlab Software ◽  
Kbm Method ◽  
Academy Of Sciences ◽  
Nonlinear Differential Systems

this paper develops a reliable algorithm based on the general Struble’s technique and extended KBM method for solving nonlinear differential systems. Moreover, we find a solution based on the KBM and general Struble’s technique of nonlinear autonomous systems with time variation, which is more powerful than the existing perturbation method. Finally, results are discussed, primarily to enrich the physical prospects, and shown graphically by utilizing MATHEMATICA and MATLAB software. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 121-130, 2020

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

scholarly journals PEMAHAMAN METODE NUMERIK MENGGUNAKAN PEMPROGRMAN MATLAB (Studi Kasus : Metode Secant)

2017 ◽  
Vol 1 (1) ◽  
pp. 89
Author(s):  
Melda Panjaitan
Keyword(s):  
Approximate Solution ◽  
Problem Solving ◽  
Numerical Methods ◽  
Numerical Method ◽  
Mathematical Problem ◽  
Secant Method ◽  
Mathematical Problem Solving ◽  
Real Solution ◽  
True Solution ◽  
Solution Approach

Abstract - The numerical method is a powerful mathematical problem solving tool. With numerical methods, we get a solution that approaches or approaches a true solution so that a numerical solution is also called an approximate solution or solution approach, but almost the solution can be made as accurately as we want. The solution almost certainly isn't exactly the same as the real solution, so there is a difference between the two. This difference is called an error. the solution using numerical methods is always in the form of numbers. The secant method requires two initial estimates that must enclose the roots of the equation. Keywords - Numerical Method, Secant Method

scholarly journals Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1257 ◽  
Author(s):  
Denis Sidorov ◽  
Aleksandr Tynda ◽  
Ildar Muftahov ◽  
Aliona Dreglea ◽  
Fang Liu
Keyword(s):  
Nonlinear Systems ◽  
Power Systems ◽  
Numerical Method ◽  
Storage Systems ◽  
Piecewise Smooth ◽  
Volterra Equations ◽  
Dynamical Models ◽  
Smooth Kernels ◽  
Power Systems Operation ◽  
Real World Problems

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

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