The Properties of Certain Linear and Nonlinear Differential Equations

2019 ◽  
pp. 193-200 ◽  
Author(s):  
Galina Filipuk ◽  
Alexander Chichurin
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear And Nonlinear

Linear and Nonlinear Differential Equations

1984 ◽  
Vol 52 (7) ◽  
pp. 670-670 ◽  
Author(s):  
I. D. Huntley ◽  
R. M. Johnson ◽  
Fred Brauer
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear And Nonlinear

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

scholarly journals Principal Component Analysis in the Nonlinear Dynamics of Beams: Purification of the Signal from Noise Induced by the Nonlinearity of Beam Vibrations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
A. V. Krysko ◽  
Jan Awrejcewicz ◽  
Irina V. Papkova ◽  
Olga Szymanowska ◽  
V. A. Krysko
Keyword(s):  
Nonlinear Dynamics ◽  
Principal Component Analysis ◽  
Differential Equations ◽  
Geometric Nonlinearity ◽  
Nonlinear Differential Equations ◽  
Principal Component ◽  
Component Analysis ◽  
Harmonic Load ◽  
Linear And Nonlinear ◽  
The Impact

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).

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