scholarly journals Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

2022 ◽  
Vol 7 (4) ◽  
pp. 5275-5290
Author(s):  
Sayed Saifullah ◽  
◽  
Amir Ali ◽  
Zareen A. Khan ◽  
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Double Laplace Transform ◽  
General Series ◽  
Klein Gordon ◽  
Localized Mode ◽  
Klein Gordon Equation

<abstract><p>We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.</p></abstract>

scholarly journals Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Sayed Saifullah ◽  
Amir Ali ◽  
Muhammad Irfan ◽  
Kamal Shah
Keyword(s):  
Laplace Transform ◽  
Decomposition Method ◽  
Wave Amplitude ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Double Laplace Transform ◽  
Proposed Model ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

scholarly journals Conformable fractional derivative and its application to fractional Klein-Gordon equation

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3745-3749
Author(s):  
Kangle Wang ◽  
Shaowen Yao
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Gordon Equation ◽  
Variational Iteration Method ◽  
Conformable Fractional Derivative ◽  
Fractional Complex Transform ◽  
Klein Gordon ◽  
Fractional Differential ◽  
Fractional Equation ◽  
Klein Gordon Equation

This paper adopts conformable fractional derivative to describe the fractional Klein-Gordon equations. The conformable fractional derivative is a new simple well-behaved definition. The fractional complex transform and variational iteration method are used to solve the fractional equation. The result shows that the proposed technology is very powerful and efficient for fractional differential equations.

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.

Numerical Solutions of Linear Time-fractional Klein-Gordon Equation by Using Power Series Approach

2018 ◽  
Author(s):  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Omar Abu Arqub ◽  
Rokiah Rozita Ahmad ◽  
Shaher Momani
Keyword(s):  
Power Series ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Linear Time ◽  
Klein Gordon ◽  
Klein Gordon Equation

scholarly journals Novel Recursive Approximation for Fractional Nonlinear Equations within Caputo-Fabrizio Operator

2018 ◽  
Vol 22 ◽  
pp. 01008 ◽  
Author(s):  
Mehmet Yavuz
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Solution Method ◽  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Rapid Convergence ◽  
Derivative Operator ◽  
Klein Gordon ◽  
Novel Method ◽  
Klein Gordon Equation

This study displays a novel method for solving time-fractional nonlinear partial differential equations. The suggested method namely Laplace homotopy method (LHM) is considered with Caputo-Fabrizio fractional derivative operator. In order to show the efficiency and accuracy of the mentioned method, we have applied it to time-fractional nonlinear Klein-Gordon equation. Comparisons between obtained solutions and the exact solutions have been made and the analysis shows that recommended solution method presents a rapid convergence to the exact solutions of the problems.

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