×

Application of generalized differential transform method to multi-order fractional differential equations. (English) Zbl 1221.34022

Summary: In a recent paper [Appl. Math. Comput. 197, No. 2, 467–477 (2008; Zbl 1141.65092)] the authors presented a new generalization of the differential transform method that extend the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form \(y^{(\mu )}(t)=f(t,y(t),y^{(\beta_{1})}(t),y^{(\beta_{2})}(t),\ldots ,y^{(\beta_{n})}(t))\) with \(\mu >\beta _{n}>\beta _{n-1}>\ldots >\beta _{1}>0\), combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Citations:

Zbl 1141.65092
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bagley, R. L.; Torvik, P. J., A theoretical basis for the application of fractional calculus to viscoelasticity, J Rheol, 27, 3, 201-210 (1983) · Zbl 0515.76012
[2] Bagley, R. L.; Torvik, P. J., Fractional calculus-a different approach to the analysis of viscoelastically damped structures, AIAA J, 21, 5, 741-748 (1983) · Zbl 0514.73048
[3] Bagley, R. L.; Torvik, P. J., Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J, 23, 6, 918-925 (1985) · Zbl 0562.73071
[4] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J Electron Chem Interfacial Electrochem, 33, 253-265 (1971)
[5] Sun, H. H.; Onaral, B.; Tsao, Y., Application of positive reality principle to metal electrode linear polarization phenomena, IEEE Trans Biomed Eng, BME-31, 10, 664-674 (1984)
[6] Sun, H. H.; Abdelwahab, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans Automat Contr, AC-29, 5, 441-444 (1984) · Zbl 0532.93025
[7] Mandelbrot, B., Some noises with \(1/f\) spectrum, a bridge between direct current and white noise, IEEE Trans Inform Theor, 13, 2, 289-298 (1967) · Zbl 0148.40507
[8] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of viscoelastic structures, J Guid Contr Dynam, 14, 2, 304-311 (1991)
[9] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J Appl Mech, 51, 299-307 (1984) · Zbl 0544.73052
[10] Koeller, R. C., Polynomial operators. Stieltjes convolution and fractional calculus in hereditary mechanics, Acta Mech, 58, 251-264 (1986) · Zbl 0578.73040
[11] Skaar, S. B.; Michel, A. N.; Miller, R. K., Stability of viscoelastic control systems, IEEE Trans Automat Contr, AC-33, 4, 348-357 (1988) · Zbl 0641.93051
[12] Hartley, T. T.; Lorenzo, C. F.; Qammar, H. K., Chaos in a fractional order chua system, IEEE Trans Circ Syst I, 42, 8, 485-490 (1995)
[13] Mainardi, F., Fractional calculus: some basic problem in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and fractional calculus in continuum mechanics (1997), Springer: Springer Wein, New York), 291-348 · Zbl 0917.73004
[14] Rossikhin, Y. A.; Shitikova, M. V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl Mech Rev, 50, 15-67 (1997)
[15] Magin, R. L., Fractional calculus in bioengineering, Crit Rev Biomed Eng, 32, 1, 1-104 (2004)
[16] Magin, R. L., Fractional calculus in bioengineering – part 2, Crit Rev Biomed Eng, 32, 2, 105-193 (2004)
[17] Magin, R. L., Fractional calculus in bioengineering – part 3, Crit Rev Biomed Eng, 32, 3/4, 194-377 (2004)
[18] Oldham, K. B.; Spanier, J., The fractional calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[19] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives – theory and applications (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Longhorne, PA · Zbl 0818.26003
[20] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[21] Baker CTH, Derakhshan MS, Stability barriers to the costruction of Ρ;; Baker CTH, Derakhshan MS, Stability barriers to the costruction of Ρ;
[22] Blank L, Numerical treatment of differential equations of fractional order; 1996.; Blank L, Numerical treatment of differential equations of fractional order; 1996.
[23] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, 5, 1-6 (1997) · Zbl 0890.65071
[24] Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math Comp, 45, 463-469 (1985) · Zbl 0584.65090
[25] Diethelm, K.; Ford, N. J., Numerical solution of the Bagley-Torvik equation, BIT, 42, 490-507 (2002) · Zbl 1035.65067
[26] Diethelm, K.; Luchko, Y., Numerical solution of linear multi-term differential equations of fractional order, J Comput Anal Appl, 6, 243-263 (2004) · Zbl 1083.65064
[27] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (Kiel, F.; Mackens, W.; Voß, H.; Werther, J., Scientific computing in chemical engineering II - computional fluid dynamics. Scientific computing in chemical engineering II - computional fluid dynamics, Reaction engineering, and molecular properties (1999), Springer: Springer Heidlberg), 217-224
[28] Momani, S.; Al-Khaled, K., Numerical solutions for systems of fractional differential equations by the decomposition method, Appl Math Comput, 162, 3, 1351-1365 (2005) · Zbl 1063.65055
[29] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl Math Comput, 131, 517-529 (2002) · Zbl 1029.34003
[30] Momani S, Odibat Z, Numerical solution of fractional differential equations: a selection of semi-analytical techniques, Arab J Math Math Sci, accepted for publication.; Momani S, Odibat Z, Numerical solution of fractional differential equations: a selection of semi-analytical techniques, Arab J Math Math Sci, accepted for publication. · Zbl 1133.65116
[31] Edwards, J. T.; Ford, N. J.; Simpson, A. C., The numerical solution of linear multi-order fractional differential equations: systems of equations, J Comput Math, 148, 401-418 (2002) · Zbl 1019.65048
[32] El-Mesiry, A. E.M.; El-Sayed, A. M.A.; El-Saka, H. A.A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl Math Comput, 160, 3, 683-699 (2005) · Zbl 1062.65073
[33] Momani, S., A numerical scheme for the solution of multi-order fractional differential equations, Appl Math Comput, 182, 1, 761-770 (2006) · Zbl 1107.65119
[34] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), Academic Press: Academic Press New York · Zbl 0924.34008
[35] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J Roy Aust Soc, 13, 529-539 (1967)
[36] Zhou, J. K., Differential transformation and its applications for electrical circuits (1986), Huazhong University Press: Huazhong University Press Wuhan, China, [in Chinese]
[37] Ayaz, Fatma, Solutions of the system of differential equations by differential transform method, Appl Math Comput, 147, 547-567 (2004) · Zbl 1032.35011
[38] 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
[39] 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, 172, 551-567 (2006) · Zbl 1088.65085
[40] Abdel-Halim Hassan IH. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons and Fractals, in press. doi:10.1016/j.chaos.2006.06.040; Abdel-Halim Hassan IH. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons and Fractals, in press. doi:10.1016/j.chaos.2006.06.040
[41] Liu, H.; Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl Math Comput, 184, 748-753 (2007) · Zbl 1115.65089
[42] Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method, Chaos, Solitons and Fractals 2006. doi:10.1016/j.chaos.2006.09.004; Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method, Chaos, Solitons and Fractals 2006. doi:10.1016/j.chaos.2006.09.004 · Zbl 1148.65310
[43] Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order. Appl Math Comput, submitted for publication.; Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order. Appl Math Comput, submitted for publication. · Zbl 1141.65092
[44] Odibat Z, Shawagfeh N. Generalized Taylor’s formula. Appl Math Comput, in press. doi:10.1016/j.amc.2006.07.102; Odibat Z, Shawagfeh N. Generalized Taylor’s formula. Appl Math Comput, in press. doi:10.1016/j.amc.2006.07.102 · Zbl 1122.26006
[45] Narahari Achar, B. N.; Hanneken, J. W.; Enck, T.; Clarke, T., Dynamics of the fractional oscillator, Physica A, 297, 361-367 (2001) · Zbl 0969.70511
[46] Bagley, R. L.; Torvik, P. J., On the appearance of the fractional derivative in the behavior of real materials, J Appl Mech, 51, 275 (1984), 294-298 · Zbl 1203.74022
[47] Canat, S.; Faucher, J., Modeling, identification and simulation of induction machine with fractional derivative, (Le Mahaute, A.; Tenreiro Machado, J. A.; Trigeassou, J. C.; Sabatier, J., Fractional differentiation and its applications (2006), Ubooks Verlag: Ubooks Verlag Neusäb), 459-471
[48] Riu, D.; Retiére, N., Implicit half-order systems utilisation for diffusion phenomenon modelling, (Le Mahaute, A.; Tenreiro Machado, J. A.; Trigeassou, J. C.; Sabatier, J., Fractional differentiation and its applications (2006), Ubooks Verlag: Ubooks Verlag Neusäb), 447-459
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.