scholarly journals The Improved Moving Least-Square Ritz Method for the One-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Qi Wei ◽  
Rongjun Cheng
Keyword(s):  
Gordon Equation ◽  
Ritz Method ◽  
Least Square ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Moving Least Square ◽  
Discrete Equations ◽  
Moving Least Square Approximation ◽  
Minimization Procedure ◽  
The One

Analysis of the one-dimensional sine-Gordon equation is performed using the improved moving least-square Ritz method (IMLS-Ritz method). The improved moving least-square approximation is employed to approximate the 1D displacement field. A system of discrete equations is obtained by application of the Ritz minimization procedure. The effectiveness and accuracy of the IMLS-Ritz method for the sine-Gordon equation are investigated by numerical examples in this paper.

scholarly journals Parametrically Driven One-Dimensional Sine-Gordon Equation and Chaos

1988 ◽  
Vol 43 (8-9) ◽  
pp. 727-733
Author(s):  
B. M. Herbst ◽  
W.-H. Steeb
Keyword(s):  
Driving Force ◽  
Gordon Equation ◽  
Zero Mode ◽  
Space Structure ◽  
Chaotic Behaviour ◽  
Force Frequency ◽  
Sine Gordon Equation ◽  
Mode Structure ◽  
One Dimensional ◽  
The One

AbstractThe chaotic behaviour of the parametrically driven one-dimensional sine-Gordon equation with periodic boundary conditions is studied. The initial condition is u(x, 0) = ƒ(x), ut (x, 0) = 0 where ƒ is the breather solution of the one-dimensional sine-Gordon equation at t = 0. We vary the amplitude of the driving force, the frequency of the driving force and the damping constant. For appropriate values of the driving force, frequency and damping constant chaotic behaviour with respect to the time-evolution of w(x = fixed, t) can be found. The space structure u(t = fixed, x) changes with increasing driving force from a zero mode structure to a breather-like structure consisting of a few modes.

scholarly journals Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse
Keyword(s):  
Iterative Method ◽  
Gordon Equation ◽  
Approximate Analytical Solution ◽  
Test Problems ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Nonlinear Part ◽  
Sumudu Transform ◽  
The One ◽  
The Given

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

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