Differential transform method for solving the linear and nonlinear Klein–Gordon equation

2009 ◽  
Vol 180 (5) ◽  
pp. 708-711 ◽  
Author(s):  
A.S.V. Ravi Kanth ◽  
K. Aruna
Keyword(s):  
Gordon Equation ◽  
Differential Transform Method ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Linear And Nonlinear ◽  
Klein Gordon Equation

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.

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 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.

scholarly journals Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 314-320 ◽  
Author(s):  
Muhammet Yazici ◽  
Süleyman Şengül
Keyword(s):  
Gordon Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Initial Value ◽  
Transform Method ◽  
Sitter Spacetime ◽  
Klein Gordon ◽  
Klein Gordon Equation

AbstractWe consider initial value problems for the nonlinear Klein-Gordon equation in de Sitter spacetime. We use the differential transform method for the solution of the initial value problem. In order to show the accuracy of results for the solutions, we use the variational iteration method with Adomian’s polynomials for the nonlinearity. We show that the methods are effective and useful.

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