Study of Properties of Differential Transform Method for Solving the Linear Differential Equation
DOI:
https://doi.org/10.51983/ajeat-2019.8.2.1138Keywords:
Differential Transformation Method, DTM, Taylor Series, Linear Properties, Differential EquationsAbstract
The differential transformation method (DTM) is an alternative procedure for obtaining an analytic Taylor series solution of differential linear and non-linear equations. However, the proofs of the properties of equation have been long ignored in the DTM literature. In this paper, we present an analytical solution for linear properties of differential equations by using the differential transformation method. This method has been discussed showing the proof of the equation which are presented to show the ability of the method for linear systems of differential equations. Most authors assume the knowledge of these properties, so they do not bother to prove the properties. The properties are therefore proved to serve as a reference for any work that would want to use the properties without proofs. This work argues that we can obtain the solution of differential equation through these proofs by using the DTM. The result also show that the technique introduced here is accurate and easy to apply.
References
C. Chen and S. Ho, "Application of differential transformation to eigenvalue problems," Applied Mathematics and Computation, vol. 79, no. 2-3, pp. 173-188, 1996. [Online]. Available: 10.1016/0096-3003(95)00253-7.
M. Hatami, D. Ganji, and M. Sheikholeslami, "Differential transformation method for mechanical engineering problems," Elsevier, 2003.
J. Zhou, "Differential Transformation and Its Applications for Electrical Circuits" (in Chinese), Huazhong University Press, 1986.
I. Abdel-Halim Hassan, "Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems," Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 53-65, 2008. [Online]. Available: 10.1016/j.chaos.2006.06.040.
J. Biazar and H. Ghazvini, "He’s variational iteration method for solving linear and non-linear systems of ordinary differential equations," Applied Mathematics and Computation, vol. 191, no. 1, pp. 287-297, 2007. [Online]. Available: 10.1016/j.amc.2007.02.153.
C. Kuang Chen and S. Huei Ho, "Solving partial differential equations by two-dimensional differential transform method," Applied Mathematics and Computation, vol. 106, no. 2-3, pp. 171-179, 1999. [Online]. Available: 10.1016/s0096-3003(98)10115-7.
F. Ayaz, "Solutions of the system of differential equations by differential transform method," Applied Mathematics and Computation, vol. 147, no. 2, pp. 547-567, 2004. [Online]. Available: 10.1016/s0096-3003(02)00794-4.
Z. Odibat, "Differential transform method for solving Volterra integral equation with separable kernels," Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1144-1149, 2008. [Online]. Available: 10.1016/j.mcm.2007.12.022.
F. Kangalgil and F. Ayaz, "Solitary wave solutions for the KdV and mKdV equations by differential transform method," Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 464-472, 2009. [Online]. Available: 10.1016/j.chaos.2008.02.009.
M. Jang, C. Chen, and Y. Liu, "Two-dimensional differential transform for partial differential equations," Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261-270, 2001. [Online]. Available: 10.1016/s0096-3003(99)00293-3.
I. Abdel-Halim Hassan, "Different applications for the differential transformation in the differential equations," Applied Mathematics and Computation, vol. 129, no. 2-3, pp. 183-201, 2002. [Online]. Available: 10.1016/s0096-3003(01)00037-6.
F. Ayaz, "On the two-dimensional differential transform method," Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 361-374, 2003. [Online]. Available: 10.1016/s0096-3003(02)00368-5.
A. Arikoglu and I. Ozkol, "Solution of boundary value problems for integro-differential equations by using differential transform method," Applied Mathematics and Computation, vol. 168, no. 2, pp. 1145-1158, 2005. [Online]. Available: 10.1016/j.amc.2004.10.009.
A. Kurnaz, G. Oturanç, and M. Kiris, "n-Dimensional differential transformation method for solving PDEs," International Journal of Computer Mathematics, vol. 82, no. 3, pp. 369-380, 2005. [Online]. Available: 10.1080/0020716042000301725.
A. Ravi Kanth and K. Aruna, "Differential transform method for solving linear and non-linear systems of partial differential equations," Physics Letters A, vol. 372, no. 46, pp. 6896-6898, 2008. [Online]. Available: 10.1016/j.physleta.2008.10.008.
A. Ravi Kanth and K. Aruna, "Differential transform method for solving the linear and nonlinear Klein–Gordon equation," Computer Physics Communications, vol. 180, no. 5, pp. 708-711, 2009. [Online]. Available: 10.1016/j.cpc.2008.11.012.
F. Mirzaee, "Differential transform method for solving linear and nonlinear systems of ordinary differential equations," Applied Mathematical Sciences, vol. 05, no. 70, pp. 3465 – 3472, 2011.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2019 The Research Publication
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
