scholarly journals On the Solutions of Nonlinear Higher-Order Boundary Value Problems by Using Differential Transformation Method and Adomian Decomposition Method

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Che Haziqah Che Hussin ◽  
Adem Kiliçman
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Transformation Method ◽  
Higher Order ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Adomian Decomposition

We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

Comparison of Adomian Decomposition Method and Differential Transformation Method for Vibration Problems of Euler-Bernoulli Beam

2012 ◽  
Vol 157-158 ◽  
pp. 476-483
Author(s):  
Zhi Feng Liu ◽  
Chun Hua Guo ◽  
Li Gang Cai ◽  
Wen Tong Yang ◽  
Zhi Min Zhang
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Differential Transformation ◽  
Adomian Decomposition ◽  
Vibration Problems ◽  
Euler Bernoulli Beam

In this paper, we compare the Differential transformation method and Adomian decomposition method to solve Euler-Bernoulli Beam vibration problems. The natural frequencies and mode shapes of the clamped-free uniform Euler-Bernoulli equation are calculated using the two methods. The Adomian decomposition method avoids the difficulties and massive computational work inherent in Differential transformation method by determining the very rapidly convergent analytic solutions directly. We found the solution between the two methods to be quite close. According to calculation of eigenvalues, natural frequencies and mode shapes, we compare the convergence of Differential transformation method and Adomian decomposition method. The two methods can be alternative ways to solve linear and nonlinear higher-order initial value problems.

scholarly journals A Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems

2010 ◽  
Vol 65 (12) ◽  
pp. 1093-1100 ◽  
Author(s):  
Weonbae Kim ◽  
Changbum Chun
Keyword(s):  
Boundary Value Problems ◽  
Decomposition Method ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Higher Order ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Adomian Polynomials ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition

In this paper, we present a reliable modification of the Adomian decomposition method for solving higher-order singular boundary value problems. He’s polynomials are also used to overcome the complex and difficult calculation of Adomian polynomials occurring in the application of the Adomian decomposition method. Numerical examples are given to illustrate the accuracy and efficiency of the presented method, revealing its reliability and applicability in handling the problems with singular nature.

scholarly journals DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 78 (6-4) ◽  
Author(s):  
Che Haziqah Che Hussin ◽  
Arif Mandangan ◽  
Adem Kilicman ◽  
Muhamad Azlan Daud ◽  
Nurliyana Juhan
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Relative Error ◽  
Decomposition Method ◽  
Boundary Value ◽  
Transformation Method ◽  
Sixth Order ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Absolute Relative Error

In this study, sixth-order boundary value problems for linear and nonlinear differential equations have been solved by using Differential Transformation Method (DTM). The numerical solutions are given in several examples. For each example, the solution given by DTM is compared with the exact solution. Absolute relative error (ARE) for each iteration can be computed. Therefore, the maximum absolute relative error (MARE) of the DTM can be obtained. To show that the solution given by the DTM has higher level of accuracy, the absolute relative error of the DTM has been compared with the other methods such as Adomian decomposition method with Green’s function, modified decomposition method (MDM), homotopy perturbation method (HPM), Variational Iteration Method (VIM) and Quintic B-Spline Collocation Method. Comparison graphs are given at the end of this paper. The obtained result shows that the proposed method is able to provide better approximation in term of accuracy.

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