Closed form solutions to the matrix sine-Gordon equation

2012 ◽  
Vol 77 (3) ◽  
pp. 308-315 ◽  
Author(s):  
F. Demontis ◽  
C. van der Mee
Keyword(s):  
Closed Form ◽  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Closed Form Solutions ◽  
The Matrix

Modal Analysis of Multi-Stepped Distributed-Parameter Rotor-Bearing Systems Using Exact Dynamic Elements

2001 ◽  
Vol 123 (3) ◽  
pp. 401-403 ◽  
Author(s):  
Seong-Wook Hong ◽  
Jong-Heuck Park
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Complete Analysis ◽  
Distributed Parameter ◽  
Analysis Method ◽  
Closed Form Solutions ◽  
Analysis Scheme ◽  
Rotor Bearing ◽  
The Matrix ◽  
Transcendental Functions

Although the exact dynamic elements have been suggested by the authors [1] and proved to be useful for the dynamic analysis of distributed-parameter rotor-bearing systems, difficulty remains in computation because of the presence of transcendental functions in the matrix. This paper proposes a complete analysis scheme for the exact dynamic elements, a generalized modal analysis method, to obtain exact and closed form solutions of time and frequency domain responses for multi-stepped distributed-parameter rotor-bearing systems. A numerical example is provided for validating the proposed method.

scholarly journals Riccati equations

1987 ◽  
Vol 10 (1) ◽  
pp. 205-207
Author(s):  
Lloyd K. Williams
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Closed Form ◽  
Riccati Equations ◽  
Second Order ◽  
Closed Form Solutions ◽  
The Matrix ◽  
Matrix Case

In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.

Optical solitons and closed form solutions to the (3+1)-dimensional resonant Schrödinger dynamical wave equation

2020 ◽  
Vol 34 (30) ◽  
pp. 2050291 ◽  
Author(s):  
Usman Younas ◽  
Aly R. Seadawy ◽  
M. Younis ◽  
S. T. R. Rizvi
Keyword(s):  
Closed Form ◽  
Gordon Equation ◽  
Velocity Dispersion ◽  
Group Velocity Dispersion ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Closed Form Solutions ◽  
Contour Plots ◽  
Complex Phenomena ◽  
Selection Of

This paper investigates the new solitons and closed form solutions to [Formula: see text] dimensional resonant nonlinear Schrödinger equation (RNLSE) that explains the behavior of waves with the effect of group velocity dispersion and resonant nonlinearities in the optical fiber. The soliton solutions in single and combined forms like dark, singular, and dark-singular in mixed form are extracted by means of two innovative integration norms namely extended sinh-Gordon equation expansion and [Formula: see text]-expansion function methods. Moreover, kink and closed form solutions are also observed under different constraint conditions. By choosing the suitable selection of the parameters, three dimensional, two dimensional, and contour plots are sketched. The obtained outcomes show that the applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations.

Micromechanical Property Prediction for Flexible Matrix Composite Materials

1987 ◽  
Vol 109 (1) ◽  
pp. 29-33
Author(s):  
B. P. Gupta
Keyword(s):  
Closed Form ◽  
Elastic Constants ◽  
Matrix Composites ◽  
Finite Element Analyses ◽  
Property Prediction ◽  
Micromechanical Property ◽  
Closed Form Solutions ◽  
The Matrix ◽  
Composite Models ◽  
Flexible Matrix Composite

Several closed-form solutions exit to predict elastic constants of a composite material. Most of these methods give comparable results for epoxy matrix composites, but not for flexible matrix composites, where the matrix is much softer than the fiber. We have devised a method that uses energy values given by finite element analyses of composite models, subjected to various independent displacement conditions. Results for flexible matrix composites thus obtained are compared with those determined by some of the existing methods. Closed-form solutions are recommended for approximate prediction of the different elastic constants by this comparison.

scholarly journals Applicability of the interface spring model for micromechanical analyses with interfacial imperfections to predict the modified exterior Eshelby tensor and effective modulus

2019 ◽  
Vol 24 (9) ◽  
pp. 2944-2960 ◽  
Author(s):  
Sangryun Lee ◽  
Youngsoo Kim ◽  
Jinyeop Lee ◽  
Seunghwa Ryu
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Eshelby Tensor ◽  
Effective Moduli ◽  
Spring Model ◽  
Interfacial Damage ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
The Matrix ◽  
Linear Spring

Closed-form solutions for the modified exterior Eshelby tensor, strain concentration tensor, and effective moduli of particle-reinforced composites are presented when the interfacial damage is modeled as a linear-spring layer of vanishing thickness; the solutions are validated against finite element analyses. Based on the closed-form solutions, the applicability of the interface spring model is tested by calculating those quantities using finite element analysis augmented with a matrix–inhomogeneity non-overlapping condition. The results indicate that the interface spring model reasonably captures the characteristics of the stress distribution and effective moduli of composites, despite its well-known problem of unphysical overlapping between the matrix and inhomogeneity.

Digging into the Elusive Localised Solutions of (2+1) Dimensional sine-Gordon Equation

2018 ◽  
Vol 73 (5) ◽  
pp. 415-423
Author(s):  
R. Radha ◽  
C. Senthil Kumar
Keyword(s):  
Closed Form ◽  
Gordon Equation ◽  
Rogue Waves ◽  
Sine Gordon Equation ◽  
Space And Time ◽  
Collisional Dynamics ◽  
Lower Dimensional

AbstractIn this paper, we revisit the (2+1) dimensional sine-Gordon equation analysed earlier [R. Radha and M. Lakshmanan, J. Phys. A Math. Gen.29, 1551 (1996)] employing the Truncated Painlevé Approach. We then generate the solutions in terms of lower dimensional arbitrary functions of space and time. By suitably harnessing the arbitrary functions present in the closed form of the solution, we have constructed dromion solutions and studied their collisional dynamics. We have also constructed dromion pairs and shown that the dynamics of the dromion pairs can be turned ON or OFF desirably. In addition, we have also shown that the orientation of the dromion pairs can be changed. Apart from the above classes of solutions, we have also generated compactons, rogue waves and lumps and studied their dynamics.

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