Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

1998 ◽  
Vol 99 (1) ◽  
pp. 23-35 ◽  
Author(s):  
C. L. Chen ◽  
Y. C. Liu
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Point Boundary ◽  
Differential Transformation

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.

scholarly journals On the solutions of some boundary value problems by using differential transformation method with convolution terms

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 917-928 ◽  
Author(s):  
Adem Kılıçman ◽  
Ömer Altun
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Transformation Method ◽  
New Method ◽  
Differential Transformation Method ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transformation

In this study, we consider some boundary value problems by using the differential transformation method with convolutions term. Further, we also propose a new method to solve the differential equations having singularity by using the convolution. In this new method when the operator has some singularities then we multiply the partial differential operator with continuously differential functions by using the convolution in order to regularize the singularity. Then the differential transform method will be applied to the new partial differential equations that might also have some fractional order.

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.

Node Selection for Two-Point Boundary-Value Problems

1985 ◽  
Vol 107 (3) ◽  
pp. 364-369 ◽  
Author(s):  
C. M. Ablow ◽  
S. Schechter ◽  
W. H. Zwisler
Keyword(s):  
Boundary Value Problems ◽  
Boundary Layers ◽  
Boundary Value ◽  
System Size ◽  
Transformation Method ◽  
Point Boundary ◽  
First Order ◽  
Standard Problem ◽  
Tem Method ◽  
Irregular Spacing

The solutions of two-point boundary-value problems often have boundary layers, narrow regions of sharp variation, that can occur in any part of the interval between the points. A finite difference method of numerical solution will generally require more closely spaced nodes in the boundary layers than elsewhere. An automatic method is needed for achieving the irregular spacing when the location of the boundary layer is not known in advance. Several automatic node-insertion or node-movement methods have been proposed. A new node-movement method is presented that is optimal under the criterion of producing the least sum of squares of the truncation errors at the nodes. For the Keller box scheme applied to a system of N coupled first-order differential equations this truncation-error minimizing (TEM) method increases the system size to N+6 equations. The campylotropic coordinate transformation method and other published methods based on heuristically derived monitor functions are node-movement methods that involve systems of only N+1 or N+2 first order equations. A comparison is made of the accuracies of several such methods and the TEM method in the solution of a standard problem.

scholarly journals Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Nurettin Doğan ◽  
Vedat Suat Ertürk ◽  
Ömer Akın
Keyword(s):  
Boundary Value Problems ◽  
Present Method ◽  
Boundary Value ◽  
Transformation Method ◽  
High Accuracy ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Transform Method ◽  
Differential Transform ◽  
First Time

Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

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