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Status of the differential transformation method. (English) Zbl 1246.65107
Summary: Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the “traditional”-Taylor-method users (notably in the elaboration of software packages – numerical routines – for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the “traditional”-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the “misunderstandings” which have caused the controversy, the preceding topics are concretely illustrated. It is concluded that, for the sake of clarity, when the DTM is applied to ODEs, it should be mentioned that the DTM exactly coincides with the traditional Taylor method, contrary to what is currently written.
MSC:
65L05Initial value problems for ODE (numerical methods)
Software:
Taylor center
References:
[1]Pukhov, G. E.: Computational structure for solving differential equations by Taylor transformations, Cybern. syst. Anal. 14, 383 (1978)
[2]Pukhov, G. E.: Expansion formulas for differential transforms, Cybern. syst. Anal. 17, 460 (1981)
[3]Pukhov, G. E.: Differential transforms and circuit theory, Int. J. Circ. theory appl. 10 (1982) · Zbl 0491.94026 · doi:10.1002/cta.4490100307
[4]Pukhov, G. E.: Differential transforms of functions and equations, (1980) · Zbl 0453.44006
[5]Pukhov, G. E.: Differential transformations and mathematical modeling of physical processes, (1986)
[6]Zhou, J. K.: Differential transformation and its applications for electrical circuits, (1986)
[7]Bender, C. M.; Orszag, S. A.: Advanced mathematical methods for scientists engineers. 1. Asymptotic methods perturbation theory, (1999)
[8]Fernández, F. M.: Comment on solution of the Duffing – van der Pol oscillator equation by a differential transform method, Phys. scripta 84, 037002 (2011)
[9]Mukherjee, S.: Reply to comment on solution of the Duffing – van der Pol oscillator equation by the differential transform method, Phys. scripta 84, 037003 (2011)
[10]Chiou, J. S.; Tzeng, J. R.: Application of the Taylor transform to nonlinear vibration problems, J. vib. Acoust. 118, 83 (1996)
[11]Chen, C. -J.; Wu, W. -J.: Application of the Taylor differential transformation method to viscous damped vibration of hard and soft spring system, Comput. struct. 59, 631 (1996) · Zbl 0925.70011 · doi:10.1016/0045-7949(95)00304-5
[12]Chen, C. -L.; Lin, S. -H.; Chen, C. -K.: Application of Taylor transformation to nonlinear predictive control problem, Appl. math. Model. 20, 699 (1996) · Zbl 0860.93008 · doi:10.1016/0307-904X(96)00050-9
[13]Chen, C. -K.; Ho, S. -H.: Application of differential transformation to eigenvalue problems, Appl. math. Comput. 79, 173 (1996) · Zbl 0879.34077 · doi:10.1016/0096-3003(95)00253-7
[14]Jang, M. -J.; Chen, C. -L.: Analysis of the response of a strongly nonlinear damped system using a differential transformation technique, Appl. math. Comput. 88, 137 (1997) · Zbl 0911.65067 · doi:10.1016/S0096-3003(96)00308-6
[15]Chen, C. -L.; Liu, Y. C.: Solution of two-point boundary-value problems using the differential transformation method, J. optimiz. Theory appl. 99, 23 (1998) · Zbl 0935.65079 · doi:10.1023/A:1021791909142
[16]Chen, C. -L.; Liu, Y. -C.: Differential transformation technique for steady nonlinear heat conduction problems, Appl. math. Comput. 95, 155 (1998) · Zbl 0943.65082 · doi:10.1016/S0096-3003(97)10096-0
[17]H. Finkel, An iterated, multipoint differential transform method for numerically evolving PDE IVPs, unpublished, 2011, lt;arXiv:1102.3671gt;.
[18]Gibbons, A.: A program for the automatic integration of differential equations using the method of Taylor series, Comput. J. 3, 108 (1960) · Zbl 0093.31604 · doi:10.1093/comjnl/3.2.108
[19]Barton, D.; Willers, I. M.; Zahar, R. V. M.: The automatic solution of systems of ordinary differential equations by the method of Taylor series, Comput. J. 14, 243 (1971) · Zbl 0221.65132 · doi:10.1093/comjnl/14.3.243
[20]Leavitt, J. A.: Methods and applications of power series, Math. comput. 20, 46 (1966) · Zbl 0134.33005 · doi:10.2307/2004267
[21]Norman, A. C.: Expanding the solutions of implicit sets of ordinary differential equations in power series, Comput. J. 19, 63 (1976) · Zbl 0321.65042 · doi:10.1093/comjnl/19.1.63
[22]Corliss, G.; Chang, Y. F.: Solving ordinary differential equations using Taylor series, ACM trans. Math. software 8, 114 (1982) · Zbl 0503.65046 · doi:10.1145/355993.355995
[23]Chang, Y. F.; Corliss, G.: ATOMFT: solving odes and daes using Taylor series, Comput. math. Appl. 28, 209 (1994) · Zbl 0810.65072 · doi:10.1016/0898-1221(94)00193-6
[24]Lara, M.; Elipe, A.; Palacios, M.: Automatic programming of recurrent power series, Math. comput. Simul. 49, 351 (1999)
[25]Gofen, A.: The Taylor center for pcs: exploring graphing and integrating odes with the ultimate accuracy, Lect. notes comput. Sci. 2329, 562 (2002) · Zbl 1054.65507 · doi:http://link.springer.de/link/service/series/0558/bibs/2329/23290562.htm
[26]Barrio, R.; Blesa, F.; Lara, M.: VSVO formulation of the Taylor method for the numerical solution of odes, Comput. math. Appl. 50, 93 (2005) · Zbl 1085.65056 · doi:10.1016/j.camwa.2005.02.010
[27]Makino, K.; Berz, M.: COSY INFINITY version 9, Nucl. instrum. Meth. phys. Res. A 558, 346 (2006)
[28]V. Satek, J. Kunovsky, J. Petrek, Multi-rate integration modern Taylor series method, in: Tenth International Conference on Computer Modeling and Simulation, 2008, p. 386.
[29]Nedialkov, N. S.; Pryce, J. D.: Solving differential-algebraic equations by Taylor series (III): the DAETS code, J. numer. Anal. ind. Appl. math. 3, 61 (2008) · Zbl 1188.65111
[30]J. Kunovsky, M. Drozdova, J. Kopriva, M. Pindryc, Methodology of the Taylor series based computations, in: Proceedings of the third Asia International Conference on Modelling and Simulation, AMS ’09, 2009, p. 206.
[31]Abad, A.; Barrio, R.; Blesa, F.; Rodriguez, M.: TIDES tutorial: integrating odes by using the Taylor series method, Monogr. acad. Ci. exact. Fís.-quím. Nat. Zaragoza 36 (2011)
[32]Barrio, R.: Performance of the Taylor series method for odes/daes, Appl. math. Comput. 163, 525 (2005) · Zbl 1067.65063 · doi:10.1016/j.amc.2004.02.015
[33]Jorba, A.; Zou, M.: A software package for the numerical integration of odes by means of high-order Taylor methods, Exp. math. 14, 99 (2005) · Zbl 1108.65072 · doi:10.1080/10586458.2005.10128904
[34]Abbasbandy, S.; Bervillier, C.: Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations, Appl. math. Comput. 218, 2178 (2011)
[35]Yu, L. -T.; Chen, C. -K.: The solution of the Blasius equation by the differential transformation method, Math. comput. Model. 28, 101 (1998) · Zbl 1076.34501 · doi:10.1016/S0895-7177(98)00085-5
[36]Chen, C. -K.; Chen, S. -S.: Application of the differential transformation method to a non-linear conservative system, Appl. math. Comput. 154, 431 (2004) · Zbl 1134.65353 · doi:10.1016/S0096-3003(03)00723-9
[37]Hassan, I. H. A.-H.: Differential transformation technique for solving higher-order initial value problems, Appl. math. Comput. 154, 299 (2004) · Zbl 1054.65069 · doi:10.1016/S0096-3003(03)00708-2
[38]Yaghoobi, H.; Torabi, M.: The application of differential transformation method to nonlinear equations arising in heat transfer, Int. commun. Heat mass 38, 815 (2011)
[39]Jang, M. -J.; Chen, C. -L.; Liu, Y. -C.: Two-dimensional differential transform for partial differential equations, Appl. math. Comput. 121, 261 (2001) · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3
[40]Jang, M. -J.; Chen, C. -L.; Liy, Y. -C.: On solving the initial-value problems using the differential transformation method, Appl. math. Comput. 115, 145 (2000) · Zbl 1023.65065 · doi:10.1016/S0096-3003(99)00137-X
[41]Hairer, E.; Norsett, S. P.; Wanner, G.: Solving ordinary differential equations I. Nonstiff problems, (1993)
[42]Davis, H. T.: Introduction to nonlinear differential and integral equations, (1962) · Zbl 0106.28904
[43]Henrici: Elements of numerical analysis, (1964) · Zbl 0149.10901
[44]Hulme, B. L.: Piecewise polynomial Taylor methods for initial value problems, Numer. math. 17, 367 (1971) · Zbl 0209.47003 · doi:10.1007/BF01436086
[45]Asaithambi, A.: Solution of the Falkner – Skan equation by recursive evaluation of Taylor coefficients, J. comput. Appl. math. 176, 203 (2005) · Zbl 1063.65065 · doi:10.1016/j.cam.2004.07.013
[46]Rashidi, M. M.: The modified differential transform method for solving MHD boundary-layer equations, Comput. phys. Commun. 180, 2210 (2009) · Zbl 1197.76156 · doi:10.1016/j.cpc.2009.06.029
[47]Erfani, E.; Rashidi, M. M.; Parsa, A. B.: The modified differential transform method for solving off-centered stagnation flow toward a rotating disc, Int. J. Comput. meth. 7, 655 (2010)
[48]Rashidi, M. M.; Keimanesh, M.: Using differential transform method and Padé approximant for solving MHD flow in a laminar liquid film from a horizontal stretching surface, Math. probl. Eng. 2010, 491319 (2010) · Zbl 1191.76108 · doi:10.1155/2010/491319
[49]Zou, L.; Zong, Z.; Wang, Z.; Tian, S.: Differential transform method for solving solitary wave with discontinuity, Phys. lett. A 374, 3451 (2010)
[50]Gökdogan, A.; Merdan, M.; Yildirim, A.: The modified algorithm for the differential transform method to solution of Genesio systems, Commun. nonlinear sci. 17, 45 (2012)
[51]Zou, L.; Wang, Z.; Zong, Z.: Generalized differential transform method to differential-difference equation, Phys. lett. A 373, 4142 (2009)
[52]Willers, I. M.: A new integration algorithm for ordinary differential equations based on continued fraction approximations, Commun. ACM 17, 504 (1974) · Zbl 0285.65045 · doi:10.1145/361147.361150
[53]Odibat, Z. M.; Bertelle, C.; Aziz-Alaoui, M. A.; Duchamp, G. H. E.: A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. math. Appl. 59, 1462 (2010) · Zbl 1189.65170 · doi:10.1016/j.camwa.2009.11.005
[54]Keimanesh, M.; Rashidi, M. M.; Chamkha, A. J.; Jafari, R.: Study of a third grade non-Newtonian fluid flow between two parallel plates using the multi-step differential transform method, Comput. math. Appl. 62, 2871 (2011) · Zbl 1232.76003 · doi:10.1016/j.camwa.2011.07.054
[55]Rashidi, M. M.; Chamkha, A. J.; Keimanesh, M.: Application of multi-step differential transform method on flow of a second-grade fluid over a stretching or shrinking sheet, Am. J. Comput. math. 1, 119 (2011)
[56]Alomari, A. K.: A new analytic solution for fractional chaotic dynamical systems using the differential transform method, Comput. math. Appl. 61, 2528 (2011) · Zbl 1221.65191 · doi:10.1016/j.camwa.2011.02.043
[57]Gökdogan, A.; Merdan, M.; Yildirim, A.: A multistage differential transformation method for approximate solution of hantavirus infection model, Commun. nonlinear sci. 17, 1 (2012)
[58]Ertürk, V. S.; Odibat, Z. M.; Momani, S.: An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of T-cells, Comput. math. Appl. 62, 996 (2011) · Zbl 1228.92064 · doi:10.1016/j.camwa.2011.03.091
[59]Arikoglu, A.; Ozkol, I.: Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. math. Comput. 168, 1145 (2005) · Zbl 1090.65145 · doi:10.1016/j.amc.2004.10.009
[60]Darania, P.; Ebadian, A.: A method for the numerical solution of the integro-differential equations, Appl. math. Comput. 188, 657 (2007) · Zbl 1121.65127 · doi:10.1016/j.amc.2006.10.046
[61]Ho, S. -H.; Chen, C. -K.: Analysis of general elastically end restrained non-uniform beams using differential transform, Appl. math. Model. 22, 219 (1998)
[62]Chen, C. -K.; Ho, S. -H.: Solving partial differential equations by two-dimensional differential transform method, Appl. math. Comput. 106, 171 (1999) · Zbl 1028.35008 · doi:10.1016/S0096-3003(98)10115-7
[63]Ayaz, F.: On the two-dimensional differential transform method, Appl. math. Comput. 143, 361 (2003) · Zbl 1023.35005 · doi:10.1016/S0096-3003(02)00368-5
[64]Ayaz, F.: Solutions of the system of differential equations by differential transform method, Appl. math. Comput. 147, 547 (2004) · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4
[65]Kurnaz, A.; Oturanç, G.; Kiris, M. E.: N-dimensional differential transformation method for solving pdes, Int. J. Comput. math. 82, 369 (2005) · Zbl 1065.35011 · doi:10.1080/0020716042000301725
[66]Bildik, N.; Konuralp, A.; Bek, F. O.; Küçükarslan, S.: Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. math. Comput. 172, 551 (2006) · Zbl 1088.65085 · doi:10.1016/j.amc.2005.02.037
[67]Kangalgil, F.; Ayaz, F.: Solution of linear and nonlinear heat equations by differential transform method, Selçuk J. Appl. math. 8, 75 (2007) · Zbl 1229.65185
[68]Hassan, I. H. A.-H.: Comparison differential transformation technique with Adomian decomposition method for linear nonlinear initial value problems, Chaos solitons fractals 36, 53 (2008) · Zbl 1152.65474 · doi:10.1016/j.chaos.2006.06.040
[69]Kanth, A. S. V. Ravi; Aruna, K.: Differential transform method for solving linear and non-linear systems of partial differential equations, Phys. lett. A 372, 6896 (2008)
[70]Kanth, A. S. V. Ravi; Aruna, K.: Differential transform method for solving the linear and nonlinear Klein – Gordon equation, Comput. phys. Commun. 180, 708 (2009) · Zbl 1198.81038 · doi:10.1016/j.cpc.2008.11.012
[71]Arikoglu, A.; Ozkol, I.: Solution of difference equations by using differential transform method, Appl. math. Comput. 174, 1216 (2006) · Zbl 1138.65309 · doi:10.1016/j.amc.2005.06.013
[72]Arikoglu, A.; Ozkol, I.: Solution of differential-difference equations by using differential transform method, Appl. math. Comput. 181, 153 (2006) · Zbl 1148.65310 · doi:10.1016/j.amc.2006.01.022
[73]Ayaz, F.: Applications of differential transform method to differential-algebraic equations, Appl. math. Comput. 152, 649 (2004) · Zbl 1077.65088 · doi:10.1016/S0096-3003(03)00581-2
[74]Liu, H.; Song, Y.: Differential transform method applied to high index differential-algebraic equations, Appl. math. Comput. 184, 748 (2007) · Zbl 1115.65089 · doi:10.1016/j.amc.2006.05.173
[75]Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos solitons fractals 34, 1473 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[76]Momani, S.; Odibat, Z. M.; Ertürk, V. S.: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. lett. A 370, 379 (2007) · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[77]Odibat, Z. M.; Momani, S.; Ertürk, V. S.: Generalized differential transform method: application to differential equations of fractional order, Appl. math. Comput. 197, 467 (2008) · Zbl 1141.65092 · doi:10.1016/j.amc.2007.07.068
[78]Odibat, Z. M.; Momani, S.: A generalized differential transform method for linear partial differential equations of fractional order, Appl. math. Lett. 21, 194 (2008) · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[79]Ertürk, V. S.; Momani, S.: Solving systems of fractional differential equations using differential transform method, J. comput. Appl. math. 215, 142 (2008) · Zbl 1141.65088 · doi:10.1016/j.cam.2007.03.029
[80]Momani, S.; Odibat, Z. M.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J. comput. Appl. math. 220, 85 (2008) · Zbl 1148.65099 · doi:10.1016/j.cam.2007.07.033
[81]Ertürk, V. S.; Momani, S.; Odibat, Z. M.: Application of generalized differential transform method to multi-order fractional differential equations, Commun. nonlinear sci. 13, 1642 (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[82]Arikoglu, A.; Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method, Chaos solitons fractals 40, 521 (2009) · Zbl 1197.45001 · doi:10.1016/j.chaos.2007.08.001
[83]Liu, J. -C.; Hou, G. -L.: New approximate solution for time-fractional coupled KdV equations by generalised differential transform method, Chin. phys. B 19, 110203 (2010)
[84]Kurulay, M.; Bayram, M.: Approximate analytical solution for the fractional modified KdV by differential transform method, Commun. nonlinear sci. 15, 1777 (2010) · Zbl 1222.35172 · doi:10.1016/j.cnsns.2009.07.014
[85]Al-Rabtah, A.; Ertürk, V. S.; Momani, S.: Solutions of a fractional oscillator by using differential transform method, Comput. math. Appl. 59, 1356 (2010) · Zbl 1189.34068 · doi:10.1016/j.camwa.2009.06.036
[86]Nazari, D.; Shahmorad, S.: Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. comput. Appl. math. 234, 883 (2010) · Zbl 1188.65174 · doi:10.1016/j.cam.2010.01.053
[87]Gökdogan, A.; Yildirim, A.; Merdan, M.: Solving a fractional order model of HIV infection of CD4+T cells, Math. comput. Model. 54, 2132 (2011)
[88]X. Chen, L. Wei, J. Sui, X. Zhang, L. Zheng, Solving fractional partial differential equations in fluid mechanics by generalized differential transform method, in: Proceedings of the International Conference on Multimedia Technology (ICMT), 2011, p. 2573.
[89]Liu, J. -C.; Hou, G. -L.: Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Appl. math. Comput. 217, 7001 (2011) · Zbl 1213.65131 · doi:10.1016/j.amc.2011.01.111
[90]Das, S.; Kumar, R.: Approximate analytical solutions of fractional gas dynamic equations, Appl. math. Comput. 217, 9905 (2011)
[91]Allahviranloo, T.; Kiani, N. A.; Motamedi, N.: Solving fuzzy differential equations by differential transformation method, Inform. sci. 179, 956 (2009) · Zbl 1160.65322 · doi:10.1016/j.ins.2008.11.016
[92]El-Shahed, M.; Gaber, M.: Two-dimensional q-differential transformation and its application, Appl. math. Comput. 217, 9165 (2011) · Zbl 1223.35109 · doi:10.1016/j.amc.2011.03.152
[93]Kanwal, R. P.; Liu, K. C.: A Taylor expansion approach for solving integral equations, Int. J. Math. ed. Sci. tech. 20, 411 (1989) · Zbl 0683.45001 · doi:10.1080/0020739890200310
[94]Sezer, M.: Taylor polynomial solutions of Volterra integral equations, Int. J. Math. ed. Sci. tech. 25, 625 (1994) · Zbl 0823.45005 · doi:10.1080/0020739940250501
[95]Yalcinbas, S.: Taylor polynomial solutions of nonlinear Volterra Fredholm integral equations, Appl. math. Comput. 127, 195 (2002) · Zbl 1025.45003 · doi:10.1016/S0096-3003(00)00165-X
[96]Zitoun, F. B.; Cherruault, Y.: A Taylor expansion approach using faádi bruno’s formula for solving nonlinear integral equations of the second and third kind, Kybernetes 38, 800 (2009) · Zbl 1198.65252 · doi:10.1108/03684920910962687
[97]Yalcinbas, S.; Sezer, M.: The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials, Appl. math. Comput. 112, 291 (2000) · Zbl 1023.65147 · doi:10.1016/S0096-3003(99)00059-4
[98]Maleknejad, K.; Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Appl. math. Comput. 145, 641 (2003) · Zbl 1032.65144 · doi:10.1016/S0096-3003(03)00152-8
[99]Gülsu, M.; Sezer, M.: The approximate solution of high-order linear difference equations with variable coefficients in terms of Taylor polynomials, Appl. math. Comput. 168, 76 (2005) · Zbl 1082.65592 · doi:10.1016/j.amc.2004.08.043
[100]Sezer, M.; Akyuz-Dascioglu, A.: Taylor polynomial solutions of general linear differential-difference equations with variable coefficients, Appl. math. Comput. 174, 1526 (2006) · Zbl 1090.65087 · doi:10.1016/j.amc.2005.07.002
[101]Abbasbandy, S.; Viranloo, T. A.: Numerical solutions of fuzzy differential equations by Taylor method, Comput. meth. Appl. math. 2, 113 (2002) · Zbl 1019.34061 · doi:emis:journals/CMAM/issues/v2/n2/paper1.html
[102]Gomashi, A.; Haghi, E.; Abbasi, M.: An approach for solving fuzzy Fredholm integro-difference equations with mixed argument, Int. J. Ind. math. 2, 143 (2010)
[103]Huang, L.; Li, X. -F.; Zhao, Y.; Duan, X. -Y.: Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. math. Appl. 62, 1127 (2011) · Zbl 1228.65133 · doi:10.1016/j.camwa.2011.03.037
[104]Pryce, J. D.: Solving high-index daes by Taylor series, Numer. algorithms 19, 195 (1998) · Zbl 0921.34014 · doi:10.1023/A:1019150322187
[105]Pryce, J. D.: A simple structural analysis method for daes, BIT numer. Math. 41, 364 (2001) · Zbl 0989.34005 · doi:10.1023/A:1021998624799
[106]Nedialkov, N. S.; Pryce, J. D.: Solving differential-algebraic equations by Taylor series (I): computing Taylor coefficients, BIT numer. Math. 45, 561 (2005) · Zbl 1084.65075 · doi:10.1007/s10543-005-0019-y
[107]Nedialkov, N. S.; Pryce, J. D.: Solving differential-algebraic equations by Taylor series (II): computing the system Jacobian, BIT numer. Math. 47, 121 (2007) · Zbl 1123.65080 · doi:10.1007/s10543-006-0106-8
[108]Goldfine, A.: Taylor series methods for the solution of Volterra integral and integro-differential equations, Math. comput. 31, 691 (1977) · Zbl 0372.65054 · doi:10.2307/2006001
[109]Odibat, Z. M.; Shawagfeh, N. T.: Generalized Taylor’s formula, Appl. math. Comput. 186, 286 (2007) · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[110]Barrio, R.; Rodriguez, M.; Abad, A.; Blesa, F.: Breaking the limits: the Taylor series method, Appl. math. Comput. 217, 7940 (2011) · Zbl 1219.65064 · doi:10.1016/j.amc.2011.02.080
[111]Chen, C. -K.; Ju, S. -P.: Application of differential transformation to transient advective-dispersive transport equation, Appl. math. Comput. 155, 25 (2004) · Zbl 1053.76055 · doi:10.1016/S0096-3003(03)00755-0
[112]I. Newton, Methodus fluxionum et serierum infinitorum, 1671, unpublished. Translated in English by John Colson in 1736 as ”The method of fluxions and infinite series; with its application to the geometry of curve-lines”.
[113]Miletics, E.; Molnarka, G.: Implicit extension of Taylor series method with numerical derivatives for initial value problems, Comput. math. Appl. 50, 1167 (2005) · Zbl 1092.65056 · doi:10.1016/j.camwa.2005.08.017
[114]Miller, J. C. P.: The Airy integral, (1946) · Zbl 0061.30506
[115]Steffensen, J. F.: On the restricted problem of three bodies, Kong danske videnskab. Selskab, Math.-phys. Medd. 30, No. 18 (1956) · Zbl 0071.08901
[116]Sconzo, P.: Contribution to the solution of the three-body problem in power series form, Astronomische nachrichten 290, 163 (1967) · Zbl 0173.51903 · doi:10.1002/asna.19672900405
[117]Knuth, D. E.: Seminumerical algorithms, The art of computer programming 2 (1997)
[118]Henrici, P.: Automatic computations with power series, J. ACM 3, 10 (1956)
[119]Norman, A. C.: Computing with formal power series, ACM trans. Math. software 1, 346 (1975) · Zbl 0315.65044 · doi:10.1145/355656.355660
[120]A.C. Norman, A system for the solution of initial and two-point boundary value problems, in: Proceedings of the ACM Annual Conference, vol. 2, 1972, p. 826.
[121]Kedem, G.: Automatic differentiation of computer programs, ACM trans. Math. software 6, 150 (1980) · Zbl 0441.68041 · doi:10.1145/355887.355890
[122]N. Saunderson, The Method of Fluxions, printed for A. Millar, London, 1756.
[123]Zeilberger, D.: The J.C.P. Miller recurrence for exponentiating a polynomial and its q- analog, J. differ. Equ. appl. 1, 57 (1995) · Zbl 0838.05006 · doi:10.1080/10236199508808006
[124]L. Euler, Introductio in analysin infinitorum, vol. 1, Section 76, 1748.
[125]Griewank, A.; Corliss, G.: Automatic differentiation of algorithms: theory, implementation and application, (1991) · Zbl 0747.00030
[126]Berz, M.; Bischof, C.; Corliss, G.; Griewank, A.: Computational differentiation: techniques, applications and tools, (1996) · Zbl 0857.00033
[127]Wengert, R. E.: A simple automatic derivative evaluation program, Commun. ACM 7, 463 (1964) · Zbl 0131.34602 · doi:10.1145/355586.364791
[128]Rall, L. B.: Automatic differentiation: techniques and applications, Lect. notes comput. Sci. 120 (1981) · Zbl 0473.68025
[129]Nedialkov, N. S.; Jackson, K. R.; Corliss, G.: Validated solutions of initial value problems for ordinary differential equations, Appl. math. Comput. 105, 21 (1999) · Zbl 0934.65073 · doi:10.1016/S0096-3003(98)10083-8
[130]Hoefkens, J.; Berz, M.; Makino, K.: Computing validated solutions of implicit differential equations, Adv. comput. Math. 19, 231 (2003) · Zbl 1028.34004 · doi:10.1023/A:1022858921155
[131]Parker, G. E.; Sochacki, J. S.: Implementing the Picard iteration neural, Parallel sci. Comput. 4, 97 (1997) · Zbl 1060.34501
[132]Berz, M.: Algorithms for higher order automatic differentiation in many variables with applications to beam physics, Automatic differentiation of algorithms: theory, implementation and application, 147 (1991) · Zbl 0782.65018
[133]Barrio, R.: Sensitivity analysis of odes/daes using the Taylor series method, SIAM J. Sci. comput. 27, 1929 (2006) · Zbl 1108.65077 · doi:10.1137/030601892
[134]Barrio, R.; Rodriguez, M.; Abad, A.; Serrano, S.: Uncertainty propagation or box propagation, Math. comput. Model. 54, 2602 (2011)
[135]Rentrop, P.: A Taylor series method for the numerical solution of two-point boundary value problems, Numer. math. 31, 359 (1978) · Zbl 0421.65051 · doi:10.1007/BF01404566
[136]Berz, M.; Makino, K.: Verified integration of odes and flows using differential algebraic methods on high-order Taylor models, Reliab. comput. 4, 361 (1998) · Zbl 0976.65061 · doi:10.1023/A:1024467732637
[137]Gofen, A.: Odes and redefining the concept of elementary functions, Lect. notes comput. Sci. 2329, 1000 (2002) · Zbl 1052.34500 · doi:http://link.springer.de/link/service/series/0558/bibs/2329/23291000.htm
[138]Jr., G. A. Baker: Essentials of Padé approximants, (1975) · Zbl 0315.41014
[139]Jr., G. A. Baker; Graves-Morris, P.: Padé approximants, part I, basic theory, Encyclopedia of mathematics 13 (1981) · Zbl 0603.30044
[140]Jr., G. A. Baker; Graves-Morris, P.: Padé approximants, part II, extensions and applications, Encyclopedia of mathematics 14 (1996) · Zbl 0468.30033
[141]Barton, D.: On Taylor series and stiff equations, ACM trans. Math. software 6, 280 (1980) · Zbl 0434.65045 · doi:10.1145/355900.355902
[142]Chang, Y. F.: Solving stiff systems by Taylor series, Appl. math. Comput. 31, 251 (1989) · Zbl 0688.65053
[143]G. Corliss, G. Kirlinger, On implicit Taylor series methods for stiff ODEs, in: Proceedings of SCAN 91: International Symposium on Computer Arithmetic and Scientific Computing, 1991. · Zbl 0838.65075
[144]Corliss, G.; Griewank, A.; Henneberger, P.; Kirlinger, G.; Potra, F.; Stetter, H.: High-order stiff ODE solvers via automatic differentiation and rational prediction, Lect. notes comput. Sci. 1196, 114 (1997)
[145]Wu, X. -Y.: A sixth-order A-stable explicit one-step method for stiff systems, Comput. math. Appl. 35, 59 (1998) · Zbl 0999.65066 · doi:10.1016/S0898-1221(98)00057-1
[146]Barrio, R.; Blesa, F.; Lara, M.: High-precision numerical solution of ODE with high-order Taylor methods in parallel, Monogr. acad. Ci. exact. Fís.-quím. Nat. Zaragoza 22, 67 (2003) · Zbl 1071.65096
[147]Nedialkov, N. S.; Jackson, K. R.; Pryce, J. D.: An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE, Reliab. comput. 7, 449 (2001) · Zbl 1003.65077 · doi:10.1023/A:1014798618404
[148]Jackson, K. R.; Nedialkov, N. S.: Some recent advances in validated methods for ivps for odes, Appl. numer. Math. 42, 269 (2002) · Zbl 0998.65068 · doi:10.1016/S0168-9274(01)00155-6
[149]Nedialkov, N. S.; Kreinovich, V.; Starks, S. A.: Interval arithmetic, affine arithmetic, Taylor series methods: why, what next?, Numer. algorithms 37, 325 (2004) · Zbl 1078.65035 · doi:10.1023/B:NUMA.0000049478.42605.cf
[150]N.S. Nedialkov, Interval tools for ODEs and DAEs, in: IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, 2006, p. 4.
[151]Neher, M.; Jackson, K. R.; Nedialkov, N. S.: On Taylor model based integration of odes, SIAM J. Numer. anal. 45, 236 (2007) · Zbl 1141.65056 · doi:10.1137/050638448
[152]Berz, M.; Makino, K.: New methods for high-dimensional verified quadrature, Reliab. comput. 5, 13 (1999) · Zbl 0947.65026 · doi:10.1023/A:1026437523641
[153]Makino, K.; Berz, M.: Higher order verified inclusions of multidimensional systems by Taylor models, Nonlinear anal. 47, 3503 (2001) · Zbl 1042.41501 · doi:10.1016/S0362-546X(01)00467-9
[154]Berz, M.; Hoefkens, J.: Verified high-order inversion of functional dependencies and interval Newton methods, Reliab. comput. 7, 379 (2001) · Zbl 1016.65030 · doi:10.1023/A:1011423909873
[155]Makino, K.; Berz, M.: Taylor models and other validated functional inclusion methods, Int. J. Pure appl. Math. 6, 239 (2003) · Zbl 1055.65095
[156]H. Finkel, The differential transformation method and Miller’s recurrence, unpublished, 2010, lt;arXiv:1007.2178gt;.
[157]Chang, S. -H.; Chang, I. -L.: A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. math. Comput. 195, 799 (2008) · Zbl 1132.65062 · doi:10.1016/j.amc.2007.05.026
[158]Chang, S. -H.; Chang, I. -L.: A new algorithm for calculating two-dimensional differential transform of nonlinear functions, Appl. math. Comput. 215, 2486 (2009) · Zbl 1179.65121 · doi:10.1016/j.amc.2009.08.046
[159]Momani, S.; Ertürk, V. S.: Solutions of non-linear oscillators by the modified differential transform method, Comput. math. Appl. 55, 833 (2008) · Zbl 1142.65058 · doi:10.1016/j.camwa.2007.05.009
[160]Abu-Gurra, S.; Ertürk, V. S.; Momani, S.: Application of the modified differential transform method to fractional oscillators, Kybernetes 40, 751 (2011)
[161]I.N. Efimov, E.D. Golovin, O.V. Stukach, Exactitude of the electronic devices analysis by the differential transformations method, in: Proceedings of the 4th Annual 2003 Siberian Russian Workshop on Electron Devices and Materials, 2003.