Modified scattering for the mixed initial-boundary problem for the nonlinear Klein–Gordon equation

Nonlinearity ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 276-324
Author(s):  
Ivan Naumkin
Keyword(s):  
Boundary Problem ◽  
Gordon Equation ◽  
Initial Boundary ◽  
Initial Boundary Problem ◽  
Klein Gordon ◽  
Klein Gordon Equation

scholarly journals Longitudinal waves in an elastic rod caused by sudden damage to the foundation

2021 ◽  
pp. 1-1
Author(s):  
Ivan Shatskyi ◽  
Vasyl Perepichka ◽  
Maksym Vaskovskyi
Keyword(s):  
Gordon Equation ◽  
Integral Transformation ◽  
Initial Boundary ◽  
Transformation Method ◽  
Elastic Rod ◽  
Longitudinal Waves ◽  
Initial Boundary Problem ◽  
Klein Gordon ◽  
Integral Transformation Method ◽  
Thin Elastic Layer

We study the problem of propagating longitudinal waves in an elastic rod connected to a locally damaged foundation through a thin elastic layer. The motion of the rigid foundation blocks is considered predetermined. We formulated the initial-boundary problem for the Klein-Gordon equation with a discontinuous right-hand side. The nonstationary fields of displacements, velocities, and deformations were investigated by the Laplace integral transformation method. Examples of sudden divergence of fragments of the foundation by a given value and their mutual separation at a constant speed are considered.

scholarly journals Trigonometric Cubic B-spline Collocation Method for Solitons of the Klein-Gordon Equation

2016 ◽  
Author(s):  
Alper Korkmaz ◽  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Coupled System ◽  
Initial Boundary ◽  
Spline Collocation ◽  
B Spline ◽  
Klein Gordon ◽  
Initial Boundary Value ◽  
Klein Gordon Equation

In the present study, we derive a new B-spline technique namely trigonometric B-spline collocation algorithm to solve some initial boundary value problems for the nonlinear Klein-Gordon equation. In order to carry out the time integration with Crank-Nicolson implicit method, the order of the equation is reduced to give a coupled system of nonlinear partial differential equations. The collocation approximation based on trigonometric cubic B-splines for spatial discretization is followed by the linearization of the nonlinear term. The efficiency and accuracy of the present method are validated by measuring the error between the numerical and analytical solutions when exist. The conservation laws representing momentum and energy are also computed for all problems.

scholarly journals Sharp Condition for Global Existence and Blow-Up on Klein-Gordon Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Zhao Junsheng ◽  
Li Shufeng
Keyword(s):  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Potential Wells ◽  
Sharp Condition ◽  
Klein Gordon ◽  
Initial Boundary Value ◽  
Klein Gordon Equation

We study the initial boundary value problem of the nonlinear Klein-Gordon equation. First we introduce a family of potential wells. By using them, we obtain a new existence theorem of global solutions and show the blow-up in finite time of solutions. Especially the relation between the above two phenomena is derived as a sharp condition.

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