Current issue
Archive
About the Journal
Editorial Policies
For Authors
2/2020 vol. 25
Stats
Get citation
ORIGINAL PAPER
A Review: Differential Transform Method for Semi-Analytical Solution of Differential Equations
M.M. Rashidi 1
,
 
F. Rabiei 2
,
 
S. Naseri Nia 2
,
 
S. Abbasbandy 3
 
 
 
More details
Hide details
1
Shanghai Automotive Wind Tunnel Center, Tongji University, Shanghai 201804, China
 
2
School of Engineering, Monash University Malaysia, 47500, Selangor, Malaysia
 
3
Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran
 
 
Online publication date: 2020-06-05
 
 
Publication date: 2020-06-01
 
 
International Journal of Applied Mechanics and Engineering 2020;25(2):122-129
DOI: https://doi.org/10.2478/ijame-2020-0024
 
 
Article (PDF)
References (40)
 
KEYWORDS
differential transform method
differential transform-Pad é approximation
semi-analytical method
differential equations
fuzzy differential transformation method
gS-differentiability
 
ABSTRACT
In this article, the semi-analytical method known as the Differential Transform Method (DTM) for solving different types of differential equations is reviewed. First, basic definitions and formulas of DTM and Differential Transform-Padé approximation (DTM-Padé), which are used to increase the convergence and accuracy of DTM approximations, are discussed. Then both techniques of DTM and DTM-Padé, which have been successfully applied to partial differential equations, as well as the application of these methods in fluid mechanic and heat transfer are presented. In addition, the extension of DTM for integral differential equations and the fuzzy differential transformation method (FDTM) for fuzzy problems are discussed.
 
REFERENCES (40)
1.
Allahviranloo T., Kiani N.A. and Motamedi N. (2009): Solving fuzzy differential equations by differential transformation method.−Information Sciences, vol.179, No.7, pp.956-966.
Google Scholar
 
 
2.
Anwar Bég O., Tasveer A. Bég, Rashidi M.M. and Asadi M. (2013): DTM- Padé semi-numerical simulation of nanofluid transport in porous media.− International Journal of Applied Mathematics and Mechanics, vol.9, No.1, pp.80-107.
Google Scholar
 
 
3.
Arikoglu A. and OzkolI (2005): Solution of boundary value problems for integro-differential equations by using differential transform method.− Appl. Math. Comput., vol.168, pp.1145-1158.
Google Scholar
 
 
4.
Ayaz F. (2004): Application of differential transform method to differential-algebraic equations.− Appl. Math. Comput., vol.152, pp.649-657.
Google Scholar
 
 
5.
Ayaz F. (2004): Solutions of the system of differential equations by differential transform method.− Appl. Math. Comput., vol.147, pp.547-567.
Google Scholar
 
 
6.
Barkhordari Ahmadi M. and Kiani N.A. (2013): Differential transformation method for solving fuzzy differential inclusions by fuzzy partitions.− Int. J. Industrial Mathematics, vol.5, No.3, pp.237-249.
Google Scholar
 
 
7.
Bede B. and Gal S.G. (2004): Almost periodic fuzzy-number valued functions. −Fuzzy Sets and Systems, vol.147, pp.385-403.
Google Scholar
 
 
8.
Bede B. and Gal S.G. (2005): Generalizations of differentiability of fuzzy number valued function with application to fuzzy differential equations.− Fuzzy Sets and Systems, vol.151, pp.581-599.
Google Scholar
 
 
9.
Bede B., Imre J., Rudas C. and Attila L. (2007): First order linear fuzzy differential equations under generalized differentiability.−Information Sciences, vol.177, pp.3627-3635.
Google Scholar
 
 
10.
Bencsik A., Bede B, Tar J. and Fodor J. (2006): Fuzzy Differential Equations in modeling hydraulic differential servo cylinders.− In: Third Romanian–Hungarian Join Symposium on Applied Computational Intelligence (SACI), Timisoara, Romania.
Google Scholar
 
 
11.
Biswas S. and Roy T.K. (2018): Generalization of Seikkala derivative and differential transform method for fuzzy Volterra integro-differential equations. − Journal of Intelligent and Fuzzy Systems, vol.34, No.4, pp.2795-2806.
Google Scholar
 
 
12.
Ebrahimi F., Ghadiri M., Salari E., Hoseini S. and Shaghaghi G. (2015): Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams.− Journal of Mechanical Science and Technology, vol.29, No.3, pp.1207-1215.
Google Scholar
 
 
13.
Ghazanfari B. and Ebrahimi P. (2015): Differential transformation method for solving fuzzy fractional heat equations. − Int. J. Mathematical Modelling and Computations, vol.5, No.1, pp.81-89.
Google Scholar
 
 
14.
Ghazanfari B. and Ebrahimi P. (2016): Differential transformation method for solving hybrid fuzzy differential equations.−Journal of Hyperstructures, vol.5, No.1, pp.69-83.
Google Scholar
 
 
15.
Gökdoğan A., Merdan M. and Yildirim A. (2012): Adaptive multi-Adaptive multi-step differential transformation method to solving nonlinear differential equations.−Mathematical and Computer Modelling, vol.55, No.3-4, pp.761-769.
Google Scholar
 
 
16.
Hajilou E., Paripour M. and Heidari H. (2015): Application of differential transform method to solve hybrid fuzzy differential equations.− Int. J. Mathematical Modelling and Computations, vol.5, No.3, pp.203-217.
Google Scholar
 
 
17.
Kadkhoda N., Sadeghi Roushan S. and Jafari H. (2018): Differential transform method: A tool for solving fuzzy differential equations.− Int. J. Applied and Computational Mathematics, 4:33.
Google Scholar
 
 
18.
Kaleva O. (1987): Fuzzy differential equations.− Fuzzy Sets and Systems, vol.24, pp.301-317.
Google Scholar
 
 
19.
Kaleva O. (1990): The Cauchy problem for fuzzy differential equations.− Fuzzy Sets and Systems, vol.35, pp.389-396.
Google Scholar
 
 
20.
Keimanesh M., Rashidi M.M., Ali J. Chamkha and Jafaric R. (2011): Study of a third grade non-Newtonian fluid flow between two parallel plates using the multi-step differential transform method.− Computers and Mathematics with Applications, vol.62, No.8, pp.2871-2891.
Google Scholar
 
 
21.
Liao S.J. (2003): Beyond perturbation: introduction to the homotopy analysis method. − CRC Press, Chapman Hall, Boca Raton.
Google Scholar
 
 
22.
Mikaeilvand N. and Khakrangin S. (2012): Solving fuzzy partial differential equations by fuzzy two-dimensional differential transform method.− Neural Computing and Applications, vol.21, No.1, pp.307-312.
Google Scholar
 
 
23.
Mohyud-Din, Usman Wei M. and Hamid W.M. (2018): A study of heat transfer analysis for squeezing flow of a Casson fluid via differential transform method.− Neural Computing and Applications, vol.30, No.10, pp.3253-3264.
Google Scholar
 
 
24.
Mohammed O.H. and Khaleel O.I. (2016): Fractional differential transform method for solving fuzzy integro-differential equations of fractional order. −Basrah Journal of Science (A), vol.4, No.2, pp.31-40.
Google Scholar
 
 
25.
Odibat Z. (2008): The differential transform method for solving Volterra integral equation with separable kernels.− Mathematical and Computer Modelling, vol.48, No.7-8, pp.1144-1149.
Google Scholar
 
 
26.
Paripour M., Karimi L. and Abbasbandy S. (2017): Differential transform method for Volterra’s population growth model.− Nonlinear Studies, vol.42, No.1, pp.227-234.
Google Scholar
 
 
27.
Rashidi M.M. and Erfani E. (2009): A Novel Analytical Solution of the Thermal Boundary-Layer over a Flat Plate with a Convective Surface Boundary Condition Using DTM-Padé.− International Conference on Applied Physics and Mathematics (ICAPM 2009) Singapore.
Google Scholar
 
 
28.
Rashidi M.M. and Erfani E. (2009): New analytical method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM.− Computer Physics Communications, vol.180, pp.1539-1544.
Google Scholar
 
 
29.
Rashidi M.M., Hayat T., Keimanesh T. and Yousefian H. (2013): A study on heat transfer in a second grade fluid through a porous medium with the modified differential transform method.− Heat Transfer-Asian Research, vol.42, No.1, pp.31-45.
Google Scholar
 
 
30.
Rashidi M.M. (2009): The modified differential transform method for solving MHD boundary-layer equations.− Computer Physics Communications, vol.180, No.11, pp.2210-2217.
Google Scholar
 
 
31.
Salahshour S. and Allahviranloo T. (2013): Application of fuzzy differential transform method for solving fuzzy Volterra integral equations.− Applied Mathematical Modelling, vol.37, No.3, pp.1016-1027.
Google Scholar
 
 
32.
Seikkala S. (1987): On the fuzzy initial value problem.− Fuzzy Sets and Systems, vol.24, pp.319-330.
Google Scholar
 
 
33.
Sheikholeslami M. and Ganji D.D. (2014): Three dimensional heat and mass transfer in a rotating system using nanofluid.− Powder Technol., vol.253, pp.789-796, http://dx.doi.org/10.1016/j.po....
CrossRef
 
Google Scholar
 
 
34.
Sepasgozar S., Faraji M. and Valipour P. (2017): Application of differential transformation method (DTM) for heat and mass transfer in a porous channel.− Propulsion and Power Research, vol.6, No.1, pp.41-48.
Google Scholar
 
 
35.
Shukla1H.S., Tamsi M., Srivastava V.K. and Rashidi M.M. (2016): Modified cubic B-spline differential quadrature method for numerical solution of three dimensional coupled viscous burger equation.− Modern Physics Letters B, vol.30, No.11.
Google Scholar
 
 
36.
Usman M., Hamida M., Khanb U., Mohyud Dinc S., Iqbald M.A. and Wei Wang (2018): Differential transform method for unsteady nanofluid flow and heat transfer.− Alexandria Engineering Journal, vol.57, No.3, pp.1867-1875.
Google Scholar
 
 
37.
Yousif M., Mahmood B. and Rashidi M.M (2017): Using differential transform method and Padé approximant for solving MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet.− Journal of Mathematics and Computer Science, vol.17, pp.169-178.
Google Scholar
 
 
38.
Zadeh L. (2005): Toward a generalized theory of uncertainty (GTU) an outline.− Information Sciences, vol.172, 140.
Google Scholar
 
 
39.
Zhou J.K. (1986): Differential transformation and its applications for electrical circuits.− Huazhong University Press, Wuhan, China.
Google Scholar
 
 
40.
Zou L., Zong Z., Wang Z. and Tian S. (): Differential transform method for the degasperis-procesi equation. − Advances in Electric and Electronics, LNEE, vol.155, pp.197-203.
Google Scholar
 
 
Share
Send by email
RELATED ARTICLE
Free Vibration Analysis Of Isotropic Rectangular Plates On Winkler Foundation Using Differential Transform Method
A New Analytical Procedure for Solving the Non-Linear Differential Equation Arising in the Stretching Sheet Problem
Propagation of SH waves in an regular non homogeneous monoclinic crustal layer lying over a non-homogeneous semi-infinite medium
Indexes
 
Keywords index
Topics index
Authors index
eISSN:2353-9003
ISSN:1734-4492
Journals System - logo
In the years 2019-2020, the International Journal of Applied Mechanics and Engineering is financed under contract no. 634/P-DUN/2019 by the Minister of Science and Higher Education from funds for science dissemination activities. The financing amounts to 83.500 PLN.
Task: edition of a scientific journal, its internationalization and ensuring open access to the journal via the Internet
© 2006-2024 Journal hosting platform by Bentus
Scroll to top