A review: differential transform method for semi-analytical solution of differential equations

• Autorzy: Rashidi M. M. , Rabiei F. , Naser Nia S. , Abbasbandy S.

Identyfikatory

DOI

10.2478/ijame-2020-0024

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

differential transform method; differential transform - Pad'e approximation; semi-analytical method; differential equations; fuzzy differential transformation method; gS-differentiability

PL

metoda transformacji różniczkowej; równania różniczkowe; metoda półanalityczna

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mechanics and Engineering
• Rocznik: 2020
• Tom: Vol. 25, no. 2
• Strony: 122--129
• Opis fizyczny: Bibliogr. 40 poz., tab.

Twórcy

autor

Rashidi M. M.

• Shanghai Automotive Wind Tunnel Center, Tongji University Shanghai 201804, CHINA

autor

Rabiei F.

• School of Engineering, Monash University Malaysia 47500, Selangor, MALAYSIA

autor

Naser Nia S.

• School of Engineering, Monash University Malaysia 47500, Selangor, MALAYSIA

autor

Abbasbandy S.

• Department of Applied Mathematics Imam Khomeini International University Qazvin, IRAN

Bibliografia

• [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.
• [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.
• [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.
• [4] Ayaz F. (2004): Application of differential transform method to differential-algebraic equations.− Appl. Math. Comput., vol.152, pp.649-657.
• [5] Ayaz F. (2004): Solutions of the system of differential equations by differential transform method.− Appl. Math. Comput., vol.147, pp.547-567.
• [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.
• [7] Bede B. and Gal S.G. (2004): Almost periodic fuzzy-number valued functions. −Fuzzy Sets and Systems, vol.147, pp.385-403.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [18] Kaleva O. (1987): Fuzzy differential equations.− Fuzzy Sets and Systems, vol.24, pp.301-317.
• [19] Kaleva O. (1990): The Cauchy problem for fuzzy differential equations.− Fuzzy Sets and Systems, vol.35, pp.389-396.
• [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.
• [21] Liao S.J. (2003): Beyond perturbation: introduction to the homotopy analysis method. − CRC Press, Chapman Hall, Boca Raton.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [32] Seikkala S. (1987): On the fuzzy initial value problem.− Fuzzy Sets and Systems, vol.24, pp.319-330.
• [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.powtec.2013.12.042.
• [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.
• [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.
• [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.
• [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.
• [38] Zadeh L. (2005): Toward a generalized theory of uncertainty (GTU) an outline.− Information Sciences, vol.172, 140.
• [39] Zhou J.K. (1986): Differential transformation and its applications for electrical circuits.− Huazhong University Press, Wuhan, China.
• [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.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020)