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Home»Articles»Study of Properties of Differential Transform Method for Solving the Linear Differential Equation

Study of Properties of Differential Transform Method for Solving the Linear Differential Equation

Author : Nandita Das
Volume 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

Full Text:

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.

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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.

Editor-in-Chief
Dr. Seshadri Ramkumar
Department of Environmental Toxicology, Texas Tech University, Texas
[email protected]
Editorial Advisory Board
Dr. Kamarul Ariffin Bin Noordin
Department of Electrical Engineering, University of Malaya, Malaysia
[email protected]
Dr. Benjamin T.F. Chung
Department of Mechanical Engineering, University of Akron, Akron, USA
[email protected]
Dr. Mohd Faiz Bin Mohd Salleh
Department of Electrical Engineering, University of Malaya, Malaysia
[email protected]
Dr. Suhana Binti Mohd Said
Department of Electrical Engineering, University of Malaya, Malaysia
[email protected]
Dr. Norrima Binti Mokhtar
Department of Electrical Engineering, University of Malaya, Malaysia
[email protected]
Dr. Mohamadariff
Department of Electrical Engineering, University of Malaya, Malaysia
[email protected]

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    Editorial Note

    Editorial Dr. Seshadri Ramkumar

    Editor-in-Chief
    Dr. Seshadri Ramkumar
    Department of Environmental Toxicology, Texas Tech University, Texas
    [email protected]
    Editorial Advisory Board
    Dr. Kamarul Ariffin Bin Noordin
    Department of Electrical Engineering, University of Malaya, Malaysia
    [email protected]
    Dr. Benjamin T.F. Chung
    Department of Mechanical Engineering, University of Akron, Akron, USA
    [email protected]
    Dr. Mohd Faiz Bin Mohd Salleh
    Department of Electrical Engineering, University of Malaya, Malaysia
    [email protected]
    Dr. Suhana Binti Mohd Said
    Department of Electrical Engineering, University of Malaya, Malaysia
    [email protected]
    Dr. Norrima Binti Mokhtar
    Department of Electrical Engineering, University of Malaya, Malaysia
    [email protected]
    Dr. Mohamadariff
    Department of Electrical Engineering, University of Malaya, Malaysia
    [email protected]

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