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 A Unified KBM Method for Obtaining the Second Approximate Solution of a Third Order Weakly Non-linear Differential System with Strong Damping and Slowly Varying Coefficients

2011 ◽  
Vol 35 (1) ◽  
pp. 77-89
Author(s):  
M Alhaz Uddin ◽  
MAM Talukder ◽  
M Hasanuzzaman ◽  
MST Mumtahinah
Keyword(s):  
Approximate Solution ◽  
Differential System ◽  
Initial Conditions ◽  
Numerical Procedure ◽  
Oscillatory Process ◽  
Third Order ◽  
Strong Damping ◽  
Varying Coefficients ◽  
Kbm Method ◽  
Linear Differential

To obtain the second order approximate solution of a third order weakly nonlinear ordinary differential system with strong damping and slowly varying coefficients modeling a damped oscillatory process is considered based on the extension of a unified Krylov-Bogoliubov- Mitropolskii (KBM) method. The asymptotic solution for different initial conditions shows a good coincidence with those obtained by the numerical procedure for obtaining the transient’s response. The method is illustrated by an example.DOI: http://dx.doi.org/10.3329/jbas.v35i1.7973Journal of Bangladesh Academy of Sciences, Vol.35, No.1, 77-89, 2011

scholarly journals APPROXIMATE SOLUTIONS OF DAMPED NON LINEAR SYSTEM WITH VARYING PARAMETER AND DAMPING FORCE

2019 ◽  
pp. 13-18
Author(s):  
REZAUL KARIM ◽  
PINAKEE DEY ◽  
SOMI AKTER ◽  
MOHAMMAD ASIF AREFIN ◽  
SAIKH SHAHJAHAN MIAH
Keyword(s):  
Approximate Solution ◽  
Dynamical Systems ◽  
Linear System ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Damping Force ◽  
Kbm Method ◽  
Non Linear System

The study of second-order damped nonlinear differential equations is important in the development of the theory of dynamical systems and the behavior of the solutions of the over-damped process depends on the behavior of damping forces. We aim to develop and represent a new approximate solution of a nonlinear differential system with damping force and an approximate solution of the damped nonlinear vibrating system with a varying parameter which is based on Krylov–Bogoliubov and Mitropolskii (KBM) Method and Harmonic Balance (HB) Method. By applying these methods we solve and also analyze the finding result of an example. Moreover, the solutions are obtained for different initial conditions, and figures are plotted accordingly where MATHEMATICA and C++ are used as a programming language.

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 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

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

Signature change in p-adic and noncommutative FRW cosmology

2014 ◽  
Vol 29 (27) ◽  
pp. 1450155 ◽  
Author(s):  
Goran S. Djordjevic ◽  
Ljubisa Nesic ◽  
Darko Radovancevic
Keyword(s):  
Loop Quantum Gravity ◽  
Initial Conditions ◽  
Planck Scale ◽  
The Other ◽  
Frw Cosmology ◽  
Signature Change ◽  
Discrete Nature ◽  
Frw Metric ◽  
The Universe ◽  
Friedmann Robertson Walker

The significant matter for the construction of the so-called no-boundary proposal is the assumption of signature transition, which has been a way to deal with the problem of initial conditions of the universe. On the other hand, results of Loop Quantum Gravity indicate that the signature change is related to the discrete nature of space at the Planck scale. Motivated by possibility of non-Archimedean and/or noncommutative structure of space–time at the Planck scale, in this work we consider the classical, p-adic and (spatial) noncommutative form of a cosmological model with Friedmann–Robertson–Walker (FRW) metric coupled with a self-interacting scalar field.

scholarly journals Interaction of buoyant plumes in open-channel flow

1992 ◽  
Vol 33 (4) ◽  
pp. 451-473
Author(s):  
P. J. Wicks
Keyword(s):  
Diffusion Equation ◽  
Channel Flow ◽  
Open Channel ◽  
Nonlinear Term ◽  
Initial Conditions ◽  
Hermite Polynomials ◽  
Open Channel Flow ◽  
The Other ◽  
Buoyant Plumes ◽  
Rich Variety

AbstractIn this paper, a model for lateral dispersion in open-channel flow is studied involving a diffusion equation which has a nonlinear term describing the effect of buoyancy. The model is used to investigate the interaction of two buoyant pollutant plumes. An approximate analytic technique involving Hermite polynomials is applied to the resulting PDEs to reduce them to a system of ODEs for the centroids and widths of the two plumes. The ODEs are then solved numerically. A rich variety of behaviour occurs depending on the relative positions, widths and strengths of the initial discharges. It is found that for two plumes of equal strength and width discharged side-by-side, the plumes move apart and the rate of spreading is inhibited by their interaction, whereas when one plume is initially much wider than the other, both plumes tend to drift to the side of the narrower plume. Finally, the PDEs are solved numerically for two sets of initial conditions and a comparison is made with the ODE solutions. Agreement is found to be good.

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