# zbMATH — the first resource for mathematics

A multi-step differential transform method and application to non-chaotic or chaotic systems. (English) Zbl 1189.65170
Summary: The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

##### MSC:
 65L99 Numerical methods for ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text:
##### References:
 [1] Zhou, J.K., Differential transformation and its applications for electrical circuits, (1986), Huazhong University Press Wuhan, China, (in Chinese) [2] Ayaz, Fatma, Solutions of the system of differential equations by differential transform method, Appl. math. comput., 147, 547-567, (2004) · Zbl 1032.35011 [3] Ayaz, Fatma, Application of differential transform method to differential-algebraic equations, Appl. math. comput., 152, 649-657, (2004) · Zbl 1077.65088 [4] Arikoglu, A.; Ozkol, I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. math. comput., 168, 1145-1158, (2005) · Zbl 1090.65145 [5] Bildik, N.; Konuralp, A.; Bek, F.; Kucukarslan, S., Solution of different type of the partial differential equation by differential transform method and adomian’s decomposition method, Appl. math. comput., 127, 551-567, (2006) · Zbl 1088.65085 [6] Arikoglu, A.; Ozkol, I., Solution of difference equations by using differential transform method, Appl. math. comput., 173, 1, 126-136, (2006) [7] Arikoglu, A.; Ozkol, I., Solution of differential difference equations by using differential transform method, Appl. math. comput., 181, 1, 153-162, (2006) · Zbl 1148.65310 [8] Liu, H.; Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl. math. comput., 184, 2, 748-753, (2007) · Zbl 1115.65089 [9] Momani, S.; Noor, M., Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. math. lett., 191, 1, 218-224, (2007) · Zbl 1193.65135 [10] Hassan, I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos solitons fractals, 36, 1, 53-65, (2008) · Zbl 1152.65474 [11] Hassan, I., Application to differential transformation method for solving systems of differential equations, Appl. math. model., 32, 12, 2552-2559, (2008) · Zbl 1167.65417 [12] El-Shahed, M., Application of differential transform method to non-linear oscillatory systems, Commun. nonlinear sci. numer. simul., 13, 8, 1714-1720, (2008) [13] Odibat, Z., Differential transform method for solving Volterra integral equation with separable kernels, Math. comput. modelling, 48, 7-8, 144-1149, (2008) · Zbl 1187.45003 [14] Momani, S.; Odibat, Z.; Erturk, V., Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. lett. A, 370, 5-6, 379-387, (2007) · Zbl 1209.35066 [15] Erturk, V.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Commun. nonlinear sci. numer. simul., 13, 8, 1642-1654, (2008) · Zbl 1221.34022 [16] Odibat, Z.; Momani, S., Generalized differential transform method for linear partial differential equations of fractional order, Appl. math. lett., 21, 2, 194-199, (2008) · Zbl 1132.35302 [17] Momani, S.; Odibat, Z., A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized taylor’s formula, J. comput. appl. math., 220, 1-2, 85-95, (2008) · Zbl 1148.65099 [18] Odibat, Z.; Momani, S.; Erturk, V., Generalized differential transform method: application to differential equations of fractional order, Appl. math. comput., 197, 2, 467-477, (2008) · Zbl 1141.65092 [19] Kuo, B.; Lo, C., Application of the differential transformation method to the solution of a damped system with high nonlinearity, Nonlinear anal. TMA, 70, 4, 1732-1737, (2009) · Zbl 1168.34301 [20] Al-Sawalha, M.; Noorani, M., Application of the differential transformation method for the solution of the hyperchaotic Rössler system, Commun. nonlinear sci. numer. simul., 14, 4, 1509-1514, (2009) [21] Chen, S.; Chen, C., Application of the differential transformation method to the free vibrations of strongly non-linear oscillators, Nonlinear anal. RWA, 10, 2, 881-888, (2009) · Zbl 1167.70328 [22] Kanth, A.; Aruna, K., Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations, Chaos solitons fractals, 41, 5, 2277-2281, (2009) · Zbl 1198.81089 [23] Kanth, A.; Aruna, K., Differential transform method for solving the linear and nonlinear klein – gordon equation, Comput. phys. commun., 180, 5, 708-711, (2009) · Zbl 1198.81038 [24] Chen, C.; Ueta, T., Yet another chaotic attractor, Internat. J. bifur. chaos, 9, 7, 1465-1466, (1999) · Zbl 0962.37013 [25] Ueta, T.; Chen, C., Bifurcation analysis of chen’s equation, Internat. J. bifur. chaos, 10, 8, 1917-1931, (2000) · Zbl 1090.37531 [26] Lu, J.; Zhou, T.; Chen, G.; Zhang, S., Local bifurcations of the Chen system, Internat. J. bifur. chaos, 12, 10, 2257-2270, (2002) · Zbl 1047.34044 [27] Lorenz, E., Deterministic nonperiodic flow, J. atmospheric sci., 20, 130-141, (1963) · Zbl 1417.37129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.