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.

scholarly journals DEVELOPMENT AND ANALYSIS OF NEW APPROXIMATION OF EXTENDED CUBIC B-SPLINE TO THE NONLINEAR TIME FRACTIONAL KLEIN–GORDON EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040039 ◽  
Author(s):  
TAYYABA AKRAM ◽  
MUHAMMAD ABBAS ◽  
MUHAMMAD BILAL RIAZ ◽  
AHMAD IZANI ISMAIL ◽  
NORHASHIDAH MOHD. ALI
Keyword(s):  
Fractional Derivative ◽  
Numerical Solution ◽  
Gordon Equation ◽  
Basis Functions ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Caputo’S Fractional Derivative ◽  
Klein Gordon ◽  
Klein Gordon Equation

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

scholarly journals A FAMILY OF THE SPIRAL SOLUTIONS OF THE NONLINEAR KLEIN‐GORDON EQUATION

1998 ◽  
Vol 3 (1) ◽  
pp. 98-103
Author(s):  
V. V. Gudkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon ◽  
Klein Gordon Equation

A family of the functions, intended for a construction the exact travelling wave solutions of nonlinear partial differential equations, is given. Exact solutions of the Klein‐Gordon equation with a special potential are obtained. The behavior of complex and hypercomplex solutions of the second order is presented.

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