scholarly journals Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

2021 ◽  
Vol 54 (1) ◽  
pp. 377-409
Author(s):  
Appanah Rao Appadu ◽  
Abey Sherif Kelil
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Experiment ◽  
Difference Method ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Kdv Equations ◽  
Adomian Decomposition

Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

A Reliable Treatment for Solving Nonlinear Two-Point Boundary Value Problems

2007 ◽  
Vol 62 (9) ◽  
pp. 483-489
Author(s):  
Mustafa Inc
Keyword(s):  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Shooting Method ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Point Boundary ◽  
Adomian Decomposition

In this paper, we study the modified decomposition method (MDM) for solving nonlinear twopoint boundary value problems (BVPs) and show numerical experiments. The modified form of the Adomian decomposition method which is more fast and accurate than the standard decomposition method (SDM) was introduced by Wazwaz. In addition, we will compare the performance of the MDM and the new nonlinear shooting method applied to the solutions of nonlinear two-point BVPs. The comparison shows that the MDM is reliable, efficient and easy for solving the nonlinear twopoint BVPs.

A finite-difference method for the numerical solution of the Sal’nikov thermokinetic oscillator problem

1995 ◽  
Vol 449 (1936) ◽  
pp. 255-271
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Stationary Point ◽  
Difference Method ◽  
Numerical Experiments ◽  
Euler Method ◽  
Initial Value ◽  
First Order ◽  
Highly Nonlinear ◽  
Implicit Finite Difference

A finite-difference method is developed for solving two coupled, ordinary differential equations that model a sequence of chemical reactions. The initial-value problem is highly nonlinear and involves three parameters. Various types of theoretical solution of this problem (the Sal’nikov thermokinetic oscillator problem) may be found, depending on these parameters; this is because the stationary point is surrounded by up to two limit cycles. The well-known, first-order, explicit Euler method and an implicit finite difference method of the same order are used to compute the solution. It is shown that this implicit method may, in fact, be used explicitly and extensive numerical experiments are made to confirm the superior stability properties of the alternative method.

scholarly journals Numerical solution of Drinfeld–Sokolov–Wilso system by using modified Adomian decomposition method

2021 ◽  
Vol 23 (1) ◽  
pp. 590
Author(s):  
Badran Jasim Salim ◽  
Oday Ahmed Jasim ◽  
Zeiad Yahya Ali
Keyword(s):  
Genetic Algorithm ◽  
Exact Solution ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Linear Partial Differential Equations ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Optimal Value ◽  
Mean Square Errors

<p class="Char">In this paper, the modified Adomian decomposition method (MADM) is usedto solve different types of differential equations, one of the numerical analysis methods for solving non linear partial differential equations (Drinfeld–Sokolov–Wilson system) and short (DSWS) that occur in shallow water flows. A Genetic Algorithm was used to find the optimal value for the parameter (a). We numerically solved the system (DSWS) and compared the result to the exact solution. When the value of it is low and close to zero, the MADM provides an excellent approximation to the exact solution. As well as the lower value of leads to the numerical algorithm of (MADM) approaching the real solution.  Finally, found the optimal value when a=-10 by using the Genetic Algorithm (G-MADM). All the computations were carried out with the aid of Maple 18 and Matlab to find the parameter value (a) by using the genetic algorithm as well as to figures drawing. The errors in this paper resulted from cut errors and mean square errors.</p>

scholarly journals Exact Solution of Space-Time Fractional Partial Differential Equations by Adomian Decomposition Method

2021 ◽  
pp. 75-87
Author(s):  
Vidya N. Bhadgaonkar ◽  
Bhausaheb R. Sontakke
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Decomposition Method ◽  
Graphical Representation ◽  
Adomian Decomposition Method ◽  
Physical Models ◽  
Partial Differential ◽  
Conduction Model ◽  
Adomian Decomposition ◽  
Present Technique

The intention behind this paper is to achieve exact solution of one dimensional nonlinear fractional partial differential equation(NFPDE) by using Adomian decomposition method(ADM) with suitable initial value. These equations arise in gas dynamic model and heat conduction model. The results show that ADM is powerful, straightforward and relevant to solve NFPDE. To represent usefulness of present technique, solutions of some differential equations in physical models and their graphical representation are done by MATLAB software.

Spiral Tip Identification via Deterministic Learning

2019 ◽  
Vol 29 (03) ◽  
pp. 1950040 ◽  
Author(s):  
Xunde Dong ◽  
Chen Song ◽  
Cong Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Numerical Experiments ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Wave Source ◽  
Deterministic Learning ◽  
New Perspective

A spiral tip can be considered as a wave source, i.e. a wave is sent out after the tip rotates one circle. Therefore, the dynamics of the spiral tip is vital to understand the behavior of spiral waves. In this paper, we study the spiral tip dynamics from a new perspective by using deterministic learning. A Barkley model described by partial differential equations (PDEs) is employed to illustrate the method. It is first transformed into a set of ordinary differential equations (ODEs) by using finite difference method. Then, the position states of spiral tip are extracted from the spiral wave generated by the transformed Barkley model by using an isocontour method. Finally, with the recurrent trajectory of spiral tip, its dynamics is accurately identified by using the deterministic learning theory. It is shown that the dynamics underlying the periodic or recurrent trajectory of spiral tips can be accurately identified by using the proposed approach. Numerical experiments are included to demonstrate the effectiveness and feasibility of the proposed method.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

On Solution of Singular Integral Equations by Operator Method

2019 ◽  
Vol 14 ◽  
pp. 41-48
Author(s):  
Najem A. Mohammad ◽  
Mohammad Shami Hasso
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Singular Integral ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Operator Method ◽  
Singular Integral Equations ◽  
Transform Method ◽  
Suggested Technique ◽  
Adomian Decomposition

In this paper, we study the exact solution of singular integral equations using two methods, including Adomian Decomposition Method and Elzaki Transform Method. We propose an analytical method for solving singular integral equations and system of singular integral equations, and have some goals in our paper related to suggested technique for solving singular integral equations. The primary goal is for giving analytical solutions of such equations with simple steps, another goal is to compare the suggested method with other methods used in this study.

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