scholarly journals A Note on the Generalized Camassa-Holm Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yun Wu ◽  
Ping Zhao
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Weak Solutions ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Blow Up ◽  
Holm Equation ◽  
Special Cases ◽  
Novikov Equation

We study the generalized Camassa-Holm equation which contains the Camassa-Holm (CH) equation and Novikov equation as special cases with the periodic boundary condition. We get a blow-up scenario and obtain the global existence of strong and weak solutions under suitable assumptions, respectively. Then, we construct the periodic peaked solutions and apply them to prove the ill-posedness inHswiths<3/2.

scholarly journals Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

2014 ◽  
Vol 12 ◽  
pp. 10-20
Author(s):  
Jiang Bo Zhou ◽  
Jun De Chen ◽  
Wen Bing Zhang
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Holm Equation ◽  
Special Cases ◽  
Long Time

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

scholarly journals Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Chunyu Shen
Keyword(s):  
Boundary Condition ◽  
Periodic Solution ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Holm Equation ◽  
Time Periodic ◽  
Time Periodic Solution

This paper is concerned with time-periodic solution of the weakly dissipative Camassa-Holm equation with a periodic boundary condition. The existence and uniqueness of a time periodic solution is presented.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals A new time-delayed periodic boundary condition for discrete element modelling of railway track under moving wheel loads

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Huiqi Li ◽  
Glenn McDowell ◽  
John de Bono
Keyword(s):  
Boundary Condition ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Discrete Element ◽  
Contact Forces ◽  
Railway Track ◽  
Discrete Element Modelling ◽  
Fixed Boundary ◽  
Element Modelling ◽  
New Time

Abstract A new time-delayed periodic boundary condition (PBC) has been proposed for discrete element modelling (DEM) of periodic structures subject to moving loads such as railway track based on a box test which is normally used as an element testing model. The new proposed time-delayed PBC is approached by predicting forces acting on ghost particles with the consideration of different loading phases for adjacent sleepers whereas a normal PBC simply gives the ghost particles the same contact forces as the original particles. By comparing the sleeper in a single sleeper test with a fixed boundary, a normal periodic boundary and the newly proposed time-delayed PBC (TDPBC), the new TDPBC was found to produce the closest settlement to that of the middle sleeper in a three-sleeper test which was assumed to be free of boundary effects. It appears that the new TDPBC can eliminate the boundary effect more effectively than either a fixed boundary or a normal periodic cell. Graphic abstract

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

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