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 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 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 The absence of residual property for strong exponents of oscillation of linear systems

2021 ◽  
Vol 31 (1) ◽  
pp. 59-69
Author(s):  
A.G. Stash
Keyword(s):  
Linear Systems ◽  
Differential System ◽  
Finite Interval ◽  
Autonomous Systems ◽  
Residual Property ◽  
Differential Systems ◽  
Sign Changes

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. It is proved that all strong exponents of oscillations (unlike Sergeev's frequencies of sign changes, zeros and roots, as well as all the weak exponents of oscillations) considered as functions on the set of solutions to linear homogeneous $n$-dimensional differential systems with continuous coefficients on the semi-line are not residual (i.e. can be changed when changing solution on a finite interval). Besides, at any beforehand given natural $n\ge2$ we give the example of $n$-dimensional differential system, for some solution of which all strong oscillation exponents differ from corresponding weak exponents. In this case, all weak and all strong exponents on the chosen solution coincide with each other, respectively. When proving the results of this work, the case of parity and odd $n$ are considered separately.

Transformation of non-Gaussian random processes, signals and noise in the nonlinear differential systems by the method of cumulative functions

2018 ◽  
Vol 15 (1) ◽  
pp. 84-93
Author(s):  
V. I. Volovach ◽  
V. M. Artyushenko
Keyword(s):  
Spectral Characteristics ◽  
Random Processes ◽  
Spectral Densities ◽  
Differential Systems ◽  
Non Gaussian ◽  
Nonlinear Differential Systems

Reviewed and analyzed the issues linked with the torque and naguszewski cumulant description of random processes. It is shown that if non-Gaussian random processes are given by both instantaneous and cumulative functions, it is assumed that such processes are fully specified. Spectral characteristics of non-Gaussian random processes are considered. It is shown that higher spectral densities exist only for non-Gaussian random processes.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

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