Accurate numerical solutions of conservative nonlinear oscillators

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Khan Nasir Uddin ◽  
Khan Nadeem Alam
Keyword(s):  
Differential Equation ◽  
Plasma Physics ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Arbitrary Power

AbstractThe objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.

scholarly journals New Properties of Modified Second Kind Chebyshev Polynomials

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Semaa Hassan Aziz ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Digital Computer ◽  
Higher Order ◽  
Algebraic Equations ◽  
Equation Problem ◽  
Higher Order Differential Equations

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

scholarly journals Numerical Algorithm to Solve a Class of Variable Order Fractional Integral-Differential Equation Based on Chebyshev Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Kangwen Sun ◽  
Ming Zhu
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Numerical Algorithm ◽  
Chebyshev Polynomials ◽  
Fractional Integral ◽  
Numerical Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Integral Differential Equation

The purpose of this paper is to study the Chebyshev polynomials for the solution of a class of variable order fractional integral-differential equation. The properties of Chebyshev polynomials together with the four kinds of operational matrixes of Chebyshev polynomials are used to reduce the problem to the solution of a system of algebraic equations. By solving the algebraic equations, the numerical solutions are acquired. Further some numerical examples are shown to illustrate the accuracy and reliability of the proposed approach and the results have been compared with the exact solution.

The Rapid Series Method for Solving Differential Equation of the Odd Power Oscillator

2012 ◽  
Vol 226-228 ◽  
pp. 138-141
Author(s):  
Song Lin He ◽  
Yan Huang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Series Method ◽  
Approximate Methods ◽  
Harmonic Term ◽  
Power Oscillator

The new rapid series method to solve the differential equation of the periodic vibration of the strongly odd power nonlinear oscillator has been put forward in this paper. By adding the exponentially decaying factor to each harmonic term of the Fourier series of the periodic solution, the high accurate solution can be obtained with a few harmonic terms. The number of truncated terms is determined by the requirement of accuracy. Comparing with other approximate methods, the calculation of rapid series method is very easy and the accurate degrees of solution can be control. By comparing the analytical approximate solutions obtained by this method with numerical solutions of the cubic and fifth power oscillators, it is proven that this method is valid.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

scholarly journals New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Main Idea ◽  
Initial Condition ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Expansion Coefficients

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented.

scholarly journals Parameters Approach Applied on Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Nadeem Alam Khan
Keyword(s):  
Natural Frequency ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Power Exponent ◽  
Restoring Force ◽  
Arbitrary Power ◽  
Collocation Points ◽  
Approximate Frequency

We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.

A Second-Order Scheme for Nonlinear Fractional Oscillators Based on Newmark-β Algorithm

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Q. X. Liu ◽  
J. K. Liu ◽  
Y. M. Chen
Keyword(s):  
Fractional Derivatives ◽  
Solution Method ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Second Order ◽  
Algebraic Equations ◽  
Order Accuracy ◽  
Hybrid Solution ◽  
Nonlinear Algebraic Equations ◽  
Order Scheme

This paper presents an accurate and efficient hybrid solution method, based on Newmark-β algorithm, for solving nonlinear oscillators containing fractional derivatives (FDs) of arbitrary order. Basically, this method employs a quadrature method and the Newmark-β algorithm to handle FDs and integer derivatives, respectively. To reduce the computational burden, the proposed approach provides a strategy to avoid nonlinear algebraic equations arising routinely in the Newmark-β algorithm. Numerical results show that the presented method has second-order accuracy. Importantly, it can be applied to both linear and nonlinear oscillators with FDs of arbitrary order, without losing any precision and efficiency.

scholarly journals An Optimal Iteration Method for Strongly Nonlinear Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Convergence Of Approximate Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

scholarly journals NUMERICAL CALCULATION OF NONSTATIONARY FRACTIONAL DIFFERENTIAL EQUATION IN PROBLEMS OF MODELING TOXIC SUBSTANCES DISTRIBUTION IN GROUND WATERS

2019 ◽  
pp. 70-80
Author(s):  
Alexandra Alekseyevna Afanasyeva ◽  
Tatyana Nikolayevna Shvetsova-Shilovskaya ◽  
Dmitriy Evgenevich Ivanov ◽  
Denis Igorevich Nazarenko ◽  
Elena Victorovna Kazarezova
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Differential Equations ◽  
Difference Scheme ◽  
Numerical Method ◽  
Fractional Differential Equation ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

Sign in / Sign up

Export Citation Format

Share Document