The method of boundary integral equations for two-dimensional singularly perturbed nonstationary heat problems with nonlinear boundary conditions

2000 ◽  
Vol 36 (9) ◽  
pp. 1284-1295
Author(s):  
G. A. Nesenenko
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Nonlinear Boundary ◽  
Boundary Integral Equations ◽  
Nonlinear Boundary Conditions ◽  
Boundary Integral ◽  
Singularly Perturbed ◽  
Nonstationary Heat ◽  
Two Dimensional

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

2018 ◽  
Vol 24 (6) ◽  
pp. 1821-1848 ◽  
Author(s):  
Yuan Li ◽  
CuiYing Fan ◽  
Qing-Hua Qin ◽  
MingHao Zhao
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Elliptical Crack ◽  
Two Dimensional ◽  
Crack Face ◽  
Penny Shaped Crack ◽  
Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

Vibration of a Simply Supported L-Shaped Plate

1997 ◽  
Vol 119 (3) ◽  
pp. 464-467 ◽  
Author(s):  
R. Solecki
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Natural Vibration ◽  
Boundary Integral Equations ◽  
Rectangular Domain ◽  
Boundary Integral ◽  
Theoretical Development ◽  
Natural Vibrations ◽  
Simply Supported ◽  
Discontinuous Functions

Recently Solecki (1996) has shown that a differential equation for vibration of a rectangular plate with a cutout can be reduced to boundary integral equations. This was accomplished by filling the cutout with a “patch” made of the same material as the rest of the plate and separated from it by an infinitesimal gap. Thanks to this procedure it was possible to apply finite Fourier transformation of discontinuous functions in a rectangular domain. Subsequent application of the available boundary conditions led to a system of boundary integral equations. A plate simply supported along the perimeter, and fixed along the cutout (an L-shaped plate), was analyzed as an example. The general solution obtained by Solecki (1996) serves here to determine the frequencies of natural vibration of a L-shaped plate simply supported all around its perimeter. This problem is, however, more complicated than the previous example: to satisfy the boundary conditions an infinite series depending on discontinuous functions must be differentiated. The theoretical development is illustrated by numerical values of the frequencies of the natural vibrations of a square plate with a square cutout. The results are compared with the results obtained using finite elements method.

Sign in / Sign up

Export Citation Format

Share Document