Study of Properties of Differential Transform Method for Solving the Linear Differential Equation
Author : Nandita DasVolume 8 No.2 April-June 2019 pp 50-56
Abstract
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.
Keywords
Differential Transformation Method, DTM, Taylor Series, Linear Properties, Differential Equations
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References
[1] 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 at: 10.1016/0096-3003(95)00253-7.
[2] M. Hatami, D. Ganji and M. Sheikholeslami, “Differential transformation method for mechanical engineering problems”. Elsevier, 2003.
[3] J. Zhou, “Differential Transformation and Its Applications for Electrical Circuits” (in Chinese), Huazhong University Press, 1986.
[4] 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 at: 10.1016/j.chaos.2006.06.040.
[5] 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 at: 10.1016/j.amc.2007.02.153.
[6] 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 at: 10.1016/s0096-3003(98)10115-7.
[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 at: 10.1016/s0096-3003(02)00794-4.
[8] 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 at: 10.1016/j.mcm.2007.12.022.
[9] 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 at: 10.1016/j.chaos.2008.02.009.
[10] 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 at: 10.1016/s0096-3003(99)00293-3.
[11] 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 at: 10.1016/s0096-3003(01)00037-6.
[12] F. Ayaz, “On the two-dimensional differential transform method”, Applied Mathematics and Computation, Vol. 143, No. 2-3, pp. 361-374, 2003. [Online] Available at: 10.1016/s0096-3003(02)00368-5.
[13] 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 at: 10.1016/j.amc.2004.10.009.
[14] 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 at: 10.1080/0020716042000301725.
[15] 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 at: 10.1016/j.physleta.2008.10.008.
[16] 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.
[17] 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.