scholarly journals On the Analytical Approximation of the Nonlinear Cubic Oscillator by an Iteration Method

2019 ◽  
pp. 1-9
Author(s):  
B. M. Ikramul Haque ◽  
A. K. M. Selim Reza ◽  
Md. Mominur Rahman
Keyword(s):  
Nonlinear Oscillator ◽  
Analytic Solution ◽  
Analytical Approximation ◽  
Approximate Analytic Solution ◽  
Rapid Convergence ◽  
Modified Technique ◽  
Validity Range ◽  
Approximate Frequency ◽  
Truncated Fourier Series ◽  
Good Agreement

A modified approximate analytic solution of the cubic nonlinear oscillator “ ” has been obtained based on an iteration procedure. Here we have used the truncated Fourier series in each iterative step. The approximate frequencies obtained by this technique show a good agreement with the exact frequency. The percentage of error between exact frequency and our fifth approximate frequency is as low as 0.009%. The calculation with this technique is very easy. This easily-calculated modified technique accelerates the rapid convergence, reduces the error and increases the validity range of the solution.

scholarly journals A Modified Solution of the Nonlinear Singular Oscillator by Extended Iteration Procedure

2019 ◽  
pp. 1-9
Author(s):  
B. M. Ikramul Haque ◽  
M. M. Ayub Hossain
Keyword(s):  
Fourier Series ◽  
Analytic Solution ◽  
Rate Of Change ◽  
Iteration Procedure ◽  
Approximate Analytic Solution ◽  
Modified Technique ◽  
The Third ◽  
Present Technique ◽  
Approximate Frequency ◽  
Good Agreement

A modified solution of the nonlinear singular oscillator has been obtained based on the extended iteration procedure. We have used an appropriate truncation of the obtained Fourier series in each step of iterations to determine the approximate analytic solution of the oscillator. The third approximate frequency of the nonlinear singular oscillator shows a good agreement with its exact values. Earlier different authors presented the analytic solution of the oscillator by using various types of methods. We have compared the results obtained by the modified technique with some of the existing results. We see that some of their techniques deviate from higher-order approximations and the present technique performs comparatively better.  The rate of change of percentage of error of the presented modified solution shows the validity of convergence.

scholarly journals Iterative Homotopy Harmonic Balance Approach for Determining the Periodic Solution of a Strongly Nonlinear Oscillator

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Huaxiong Chen ◽  
Mingkang Ni
Keyword(s):  
Periodic Solution ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Order Approximation ◽  
Analytical Approximation ◽  
Higher Order ◽  
Strongly Nonlinear ◽  
Novel Approach ◽  
Approximate Frequency ◽  
Strongly Nonlinear Oscillator

A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution can be obtained easily. Comparison of the obtained results with those of the exact solutions shows the high accuracy, simplicity, and efficiency of the approach. The approach can be extended to other nonlinear oscillators in engineering and physics.

scholarly journals On the analytical approximation of the quadratic non-linear oscillator by modified extended iteration method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
B. M. Ikramul Haque ◽  
Shajia Afrin Flora
Keyword(s):  
Analytical Solution ◽  
Iteration Method ◽  
Analytical Approximation ◽  
Linear Oscillator ◽  
Algebraic Complexity ◽  
Non Linear ◽  
Approximate Frequency ◽  
Non Linear Oscillator ◽  
Good Agreement

AbstractA modified analytical solution of the quadratic non-linear oscillator has been obtained based on an extended iteration method. In this study, truncated Fourier terms have been used in each step of iterations. The frequencies obtained by this technique show good agreement with the exact frequency. The percentage of error between the exact frequency and our third approximate frequency is as low as 0.001%. There is no algebraic complexity in our calculation, which is why this technique is very easy. The results have been compared with the exact and other existing results, which are both convergent and consistent.

A shallow-water sloshing model for wave breaking in rectangular tanks

2014 ◽  
Vol 746 ◽  
pp. 437-465 ◽  
Author(s):  
Matteo Antuono ◽  
Andrea Bardazzi ◽  
Claudio Lugni ◽  
Maurizio Brocchini
Keyword(s):  
Shallow Water ◽  
Wave Breaking ◽  
Analytic Solution ◽  
Turbulent Viscosity ◽  
Approximate Analytic Solution ◽  
Experimental Campaign ◽  
Proposed Model ◽  
Water Conditions ◽  
Reliability And Robustness ◽  
Good Agreement

AbstractWe propose a simple, robust and efficient sloshing model that accounts for breaking phenomena evolving in rectangular tanks and in shallow-water conditions. The model has been obtained by applying Fourier analysis to Boussinesq-type equations and using an approximate analytic solution for the vorticity generated by wave breaking. The toe of the breaker and the intensity of the vorticity injected at the free surface are computed on the basis of literature results for coastal-type breakers. A first experimental campaign has been used to calibrate the turbulent viscosity of the sloshing model, while a second campaign has been run as final validation. The overall good agreement between the numerical outputs and the experimental data confirms the reliability and robustness of the proposed model.

Theoretical Assessment of Compressibility Factor of Gases by Using Second Virial Coefficient

2018 ◽  
Vol 73 (2) ◽  
pp. 121-125
Author(s):  
Bahtiyar A. Mamedov ◽  
Elif Somuncu ◽  
Iskender M. Askerov
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Compressibility Factor ◽  
Analytical Approximation ◽  
Accurate Evaluation ◽  
Numerical Examples ◽  
Lennard Jones ◽  
Theoretical Assessment ◽  
Good Agreement

AbstractWe present a new analytical approximation for determining the compressibility factor of real gases at various temperature values. This algorithm is suitable for the accurate evaluation of the compressibility factor using the second virial coefficient with a Lennard–Jones (12-6) potential. Numerical examples are presented for the gases H2, N2, He, CO2, CH4 and air, and the results are compared with other studies in the literature. Our results showed good agreement with the data in the literature. The consistency of the results demonstrates the effectiveness of our analytical approximation for real gases.

scholarly journals An approximate analytic solution of a particular boundary value problem

2001 ◽  
Vol 27 (8) ◽  
pp. 513-520
Author(s):  
Ugur Tanriver ◽  
Aravinda Kar
Keyword(s):  
Boundary Value Problem ◽  
Heat Conduction Equation ◽  
Analytic Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Welding Process ◽  
Approximate Analytic Solution ◽  
Analytic Expression ◽  
Steady State Heat ◽  
Laser Welding Process

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the wholex′y′-plane and finite inz′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution.

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