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A study on the convergence conditions of generalized differential transform method. (English) Zbl 1354.26013

Summary: This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor’s series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order.

MSC:

26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Bagley, On the fractional calculus models of viscoelastic behaviour, Journal of Rheology 30 pp 133– (1986) · Zbl 0613.73034 · doi:10.1122/1.549887
[2] Gaul, Damping description involving fractional operators, Mechanical Systems and Signal Processing 5 pp 81– (1991) · doi:10.1016/0888-3270(91)90016-X
[3] Miller, An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[4] Samko, Fractional Integrals and Derivatives: Theory and Applications (1993)
[5] Mainardi, Waves and Stability in Continuous Media (Bologna pp 246– (1994)
[6] Metzler, Relaxation in filled polymers: a fractional calculus approach, Journal of Chemical Physics 103 pp 7180– (1995) · doi:10.1063/1.470346
[7] Rossikhin, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews 50 pp 15– (1997) · doi:10.1115/1.3101682
[8] Mainardi, Fractals and Fractional Calculus in Continuum Mechanics pp 291– (1997) · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6_7
[9] Podlubny, Fractional Differential Equations (1999)
[10] Metzler, The random walk’s guide to anomalous diffusion: a fractional dynamic approach, Physics Reports 339 (1) pp 1– (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[11] Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company (2000) · Zbl 0998.26002 · doi:10.1142/3779
[12] Matsuzaki, A chaos neuron model with fractional differential equation, Journal of the Physical Society of Japan 72 pp 2678– (2003) · doi:10.1143/JPSJ.72.2678
[13] Magin, Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering 32 pp 1– (2004) · doi:10.1615/CritRevBiomedEng.v32.10
[14] Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (2005)
[15] Kilbas, Theory and Applications of Fractional Differential Equations (2006) · Zbl 1138.26300
[16] Bonilla, Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation 187 (1) pp 79– (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[17] Guo, On the fractional mean-value theorem, International Journal of Bifurcation and Chaos 22 (5) (2010)
[18] Li, Numerical Method for Fractional Calculus (2015)
[19] Shawagfeh, The decomposition method for fractional differential equations, Journal of Fractional Calculus 16 pp 27– (1999) · Zbl 0956.34004
[20] Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Applied Mathematics and Computation 131 (2-3) pp 517– (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[21] Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos, Solitons and Fractals 28 (4) pp 930– (2006) · Zbl 1099.35118 · doi:10.1016/j.chaos.2005.09.002
[22] Momani, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Applied Mathematics and Computation 177 pp 488– (2006) · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025
[23] Odibat, Approximate solutions for boundary value problems of time-fractional wave equation, Applied Mathematics and Computation 181 pp 1351– (2006) · Zbl 1148.65100 · doi:10.1016/j.amc.2006.02.004
[24] Odibat, Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation 7 (1) pp 27– (2006) · Zbl 1401.65087 · doi:10.1515/IJNSNS.2006.7.1.27
[25] Momani, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons and Fractals 31 (5) pp 1248– (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[26] Momani, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics 207 (1) pp 96– (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[27] Odibat, Numerical methods for solving nonlinear partial differential equations of fractional order, Applied Mathematical Modelling 32 (1) pp 28– (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[28] Odibat, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications 58 (11-12) pp 2199– (2009) · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[29] Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling 51 (9-10) pp 1181– (2010) · Zbl 1198.65147 · doi:10.1016/j.mcm.2009.12.034
[30] Cang, Series solutions of non-linear Riccati differential equations with fractional order, Chaos, Solitons and Fractals 40 (1) pp 1– (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[31] Hashim, Homotopy analysis method for fractional IVPs, Communications in Nonlinear Science and Numerical Simulation 14 (3) pp 674– (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[32] Zurigat, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling 34 (1) pp 24– (2010) · Zbl 1185.65140 · doi:10.1016/j.apm.2009.03.024
[33] Odibat, A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Applied Mathematical Modelling 34 (3) pp 593– (2010) · Zbl 1185.65139 · doi:10.1016/j.apm.2009.06.025
[34] Erturk, Application of generalized differential transform method to multi-order fractional differential equations, Communications in Nonlinear Science and Numerical Simulation 13 (8) pp 1642– (2008) · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[35] Odibat, Generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters 21 (2) pp 194– (2008) · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[36] Momani, Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Physics Letters A 370 (5-6) pp 379– (2007) · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[37] Momani, A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula, Journal of Computational and Applied Mathematics 220 (1-2) pp 85– (2008) · Zbl 1148.65099 · doi:10.1016/j.cam.2007.07.033
[38] Odibat, Generalized differential transform method: application to differential equations of fractional order, Applied Mathematics and Computation 197 (2) pp 467– (2008) · Zbl 1141.65092 · doi:10.1016/j.amc.2007.07.068
[39] Odibat, Analytic study on linear systems of fractional differential equations, Computers & Mathematics with Applications 59 (3) pp 1171– (2010) · Zbl 1189.34017 · doi:10.1016/j.camwa.2009.06.035
[40] Odibat, Generalized Talyor’s formula, Applied Mathematics and Computation 186 pp 286– (2007) · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[41] Carpinteri, Fractals and Fractional Calculus in Continuum Mechanics (1997) · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6
[42] Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, Journal of the Royal Statistical Society 13 pp 529– (1967) · Zbl 1210.65130
[43] Zhou, Differential Transformation and Its Applications for Electrical Circuits (1986)
[44] Fatma, Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation 147 pp 547– (2004) · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4
[45] Bildik, Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Applied Mathematics and Computation 127 pp 551– (2006) · Zbl 1088.65085 · doi:10.1016/j.amc.2005.02.037
[46] Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons & Fractals 36 (1) pp 53– (2008) · Zbl 1152.65474 · doi:10.1016/j.chaos.2006.06.040
[47] El-Shahed, Application of differential transform method to non-linear oscillatory systems, Communications in Nonlinear Science and Numerical Simulation 13 (8) pp 1714– (2008) · doi:10.1016/j.cnsns.2007.03.005
[48] Odibat, A multi-step differential transform method and application to non-chaotic or chaotic systems, Computers & Mathematics with Applications 59 (4) pp 1462– (2010) · Zbl 1189.65170 · doi:10.1016/j.camwa.2009.11.005
[49] Kilbas, {\(\alpha\)}-Analytic solutions of some linear fractional differential equations with variable coefficients, Applied Mathematics and Computation 178 (1) pp 239– (2007) · Zbl 1121.34008 · doi:10.1016/j.amc.2006.08.121
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