scholarly journals Approximate analytic solution of the fractal Klein-Gordon equation

2021 ◽  
pp. 51-51
Author(s):  
Jian-She Sun
Keyword(s):  
Gordon Equation ◽  
Analytic Solution ◽  
Cantor Sets ◽  
Approximate Analytic Solution ◽  
Transform Method ◽  
Continuous Space ◽  
Fractional Complex Transform ◽  
Differential Transform ◽  
Klein Gordon ◽  
Linear And Nonlinear

The linear and nonlinear Klein-Gordon equations(KGE) are considered. The fractional complex transform is used to convert the equations on a continuous space/time to fractals ones on Cantor sets, the resultant equations are solved by local fractional reduced differential transform method (LFRDTM). Three examples are given to show the effectiveness of the technology.

ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850094
Author(s):  
JIANSHE SUN
Keyword(s):  
Approximate Solutions ◽  
Travelling Wave ◽  
Cantor Sets ◽  
Travelling Wave Solutions ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractal Models ◽  
Differential Transform ◽  
Harry Dym Equation

The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

Analytical and numerical validation for solving the fractional Klein-Gordon equation using the fractional complex transform and variational iteration methods

2016 ◽  
Vol 5 (3) ◽  
Author(s):  
M. M. Khader ◽  
M. Adel
Keyword(s):  
Fractional Derivatives ◽  
Gordon Equation ◽  
Variational Iteration Method ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Variational Iteration ◽  
Klein Gordon ◽  
Iteration Methods ◽  
Order Four ◽  
Klein Gordon Equation

AbstractIn this paper, we implement the fractional complex transform method to convert the nonlinear fractional Klein-Gordon equation (FKGE) to an ordinary differential equation. We use the variational iteration method (VIM) to solve the resulting ODE. The fractional derivatives are presented in terms of the Caputo sense. Some numerical examples are presented to validate the proposed techniques. Finally, a comparison with the numerical solution using Runge-Kutta of order four is given.

scholarly journals Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Seyyedeh Roodabeh Moosavi Noori ◽  
Nasir Taghizadeh
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Differential Transform Method ◽  
Hybrid Technique ◽  
Transform Method ◽  
Proportional Delays ◽  
Differential Transform ◽  
Computational Work ◽  
Linear And Nonlinear ◽  
First Time

AbstractIn this study, a hybrid technique for improving the differential transform method (DTM), namely the modified differential transform method (MDTM) expressed as a combination of the differential transform method, Laplace transforms, and the Padé approximant (LPDTM) is employed for the first time to ascertain exact solutions of linear and nonlinear pantograph type of differential and Volterra integro-differential equations (DEs and VIDEs) with proportional delays. The advantage of this method is its simple and trusty procedure, it solves the equations straightforward and directly without requiring large computational work, perturbations or linearization, and enlarges the domain of convergence, and leads to the exact solution. Also, to validate the reliability and efficiency of the method, some examples and numerical results are provided.

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 Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

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