scholarly journals A New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Method

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed Elbadri ◽  
Shams A. Ahmed ◽  
Yahya T. Abdalla ◽  
Walid Hdidi
Keyword(s):  
Decomposition Method ◽  
Kdv Equation ◽  
Adomian Decomposition Method ◽  
Convergent Series ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Coupled Kdv Equation ◽  
A New Technique ◽  
Natural Decomposition ◽  
Made In

In this article, we applied a new technique for solving the time-fractional coupled Korteweg-de Vries (KdV) equation. This method is a combination of the natural transform method with the Adomian decomposition method called the natural decomposition method (NDM). The solutions have been made in a convergent series form. To demonstrate the performances of the technique, two examples are provided.

scholarly journals Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

2019 ◽  
Vol 11 (1) ◽  
pp. 99-116 ◽  
Author(s):  
Ali Khalouta ◽  
Abdelouahab Kadem
Keyword(s):  
Approximate Method ◽  
Decomposition Method ◽  
Nonlinear Term ◽  
Adomian Decomposition Method ◽  
Variable Coefficients ◽  
Transform Method ◽  
Adomian Polynomials ◽  
Clear Advantage ◽  
Adomian Decomposition ◽  
Natural Decomposition

Abstract In this paper, we propose a new approximate method, namely fractional natural decomposition method (FNDM) in order to solve a certain class of nonlinear time-fractional wave-like equations with variable coefficients. The fractional natural decomposition method is a combined form of the natural transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a clear advantage of this technique over the decomposition method. Some examples are given to illustrate the applicability and the easiness of this approach.

scholarly journals Application of double natural decomposition method for solving singular one dimensional Boussinesq equation

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4389-4401 ◽  
Author(s):  
Hassan Gadain
Keyword(s):  
Decomposition Method ◽  
Boussinesq Equation ◽  
Adomian Decomposition Method ◽  
Boussinesq Equations ◽  
Science And Engineering ◽  
Transform Method ◽  
One Dimensional ◽  
Adomian Decomposition ◽  
Linear And Nonlinear ◽  
Natural Decomposition

In this work, a combined form of the double Natural transform method with the Adomian decomposition method is developed for analytic treatment of the linear and nonlinear singular one dimensional Boussinesq equations. Two examples are provided to illustrate the simplicity and reliability of this method. Moreover, the results show that the proposed method is effective and is easy to implement for certain problems in science and engineering.

scholarly journals Natural Decomposition Approximation Solution for First Order Nonlinear Differential Equations

2018 ◽  
Vol 7 (4.5) ◽  
pp. 442
Author(s):  
A. Patra ◽  
T. T. Shone ◽  
B. B. Mishra
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Approximation Solution ◽  
Transform Method ◽  
First Order ◽  
Adomian Decomposition ◽  
Natural Decomposition ◽  
First Order Differential Equations

In this research paper, we propose the Natural decomposition method (NDM) to solve nonlinear first order differential equations. We compare the results obtained by NDM with the exact solutions. This method is a combination of the natural transform method and adomian decomposition method. By showing the less error one can be concluded that the NDM is easy to use and efficient.  

scholarly journals On Semi-Analytical Solutions for Linearized Dispersive KdV Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1769
Author(s):  
Appanah Rao Appadu ◽  
Abey Sherif Kelil
Keyword(s):  
Decomposition Method ◽  
Kdv Equation ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Test Problems ◽  
Transform Method ◽  
Kdv Equations ◽  
Adomian Decomposition ◽  
Adomian Method ◽  
Relative Errors

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are applied to solve linearized dispersive KdV equations with homogeneous and inhomogeneous source terms. These methods are the Laplace-Adomian decomposition method (LADM), Homotopy perturbation method (HPM), Bernstein-Laplace-Adomian Method (BALDM), and Reduced Differential Transform Method (RDTM). Three numerical experiments are considered. As the main contribution, we proposed a new scheme, known as BALDM, which involves Bernstein polynomials, Laplace transform and Adomian decomposition method to solve inhomogeneous linearized dispersive KdV equations. Besides, some modifications of HPM are also considered to solve certain inhomogeneous KdV equations by first constructing a newly modified homotopy on the source term and secondly by modifying Laplace’s transform with HPM to build HPTM. Both modifications of HPM numerically confirm the efficiency and validity of the methods for some test problems of dispersive KdV-like equations. We also applied LADM and RDTM to both homogeneous as well as inhomogeneous KdV equations to compare the obtained results and extended to higher dimensions. As a result, RDTM is applied to a 3D-dispersive KdV equation. The proposed iterative schemes determined the approximate solution without any discretization, linearization, or restrictive assumptions. The performance of the four methods is gauged over short and long propagation times and we compute absolute and relative errors at a given time for some spatial nodes.

scholarly journals Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Manar A. Alqudah ◽  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad
Keyword(s):  
Integral Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Singular Integral Equations ◽  
Fractional Integrals ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Weakly Singular ◽  
Adomian Decomposition ◽  
A New Technique

In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian decomposition method, Laplace transform method, and Ψ-Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

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