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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 (μ) (t)=f(t,y(t),y (β 1 ) (t),y (β 2 ) (t),...,y (β n ) (t)) with μ>β n >β n-1 >...>β 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:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A08Fractional differential equations
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
References:
[1]Bagley, R. L.; Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity, J rheol 27, No. 3, 201-210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724
[2]Bagley, R. L.; Torvik, P. J.: Fractional calculus-a different approach to the analysis of viscoelastically damped structures, Aiaa j 21, No. 5, 741-748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142
[3]Bagley, R. L.; Torvik, P. J.: Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j 23, No. 6, 918-925 (1985) · Zbl 0562.73071 · doi:10.2514/3.9007
[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 31, No. 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 29, No. 5, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[7]Mandelbrot, B.: Some noises with 1/f spectrum, a Bridge between direct current and white noise, IEEE trans inform theor 13, No. 2, 289-298 (1967) · Zbl 0148.40507 · doi:10.1109/TIT.1967.1053992
[8]Bagley, R. L.; Calico, R. A.: Fractional order state equations for the control of viscoelastic structures, J guid contr dynam 14, No. 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 · doi:10.1115/1.3167616
[10]Koeller, R. C.: Polynomial operators. Stieltjes convolution and fractional calculus in hereditary mechanics, Acta mech 58, 251-264 (1986) · Zbl 0578.73040 · doi:10.1007/BF01176603
[11]Skaar, S. B.; Michel, A. N.; Miller, R. K.: Stability of viscoelastic control systems, IEEE trans automat contr 33, No. 4, 348-357 (1988) · Zbl 0641.93051 · doi:10.1109/9.192189
[12]Hartley, T. T.; Lorenzo, C. F.; Qammar, H. K.: Chaos in a fractional order Chua system, IEEE trans circ syst I 42, No. 8, 485-490 (1995)
[13]Mainardi, F.: Fractional calculus: some basic problem in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[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, No. 1, 1-104 (2004)
[16]Magin, R. L.: Fractional calculus in bioengineering – part 2, Crit rev biomed eng 32, No. 2, 105-193 (2004)
[17]Magin, R. L.: Fractional calculus in bioengineering – part 3, Crit rev biomed eng 32, No. 3/4, 194-377 (2004)
[18]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[19]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives – theory and applications, (1993) · Zbl 0818.26003
[20]Podlubny, I.: Fractional differential equations, (1999)
[21]Baker CTH, Derakhshan MS, Stability barriers to the costruction of nbsp;,nbsp;-reducible and fractional quadrautre rules. in: BraB H, Hammerlin BG, editors. Numerical integration III, vol. 85; 1988, p. 1 – 15.
[22]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, An algorithm for the numerical solution of differential equations of fractional order 5, 1-6 (1997) · Zbl 0890.65071 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[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 · doi:10.2307/2008136
[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, Reaction engineering, and molecular properties, 217-224 (1999)
[28]Momani, S.; Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method, Appl math comput 162, No. 3, 1351-1365 (2005) · Zbl 1063.65055 · doi:10.1016/j.amc.2004.03.014
[29]Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations, Appl math comput 131, 517-529 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[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.
[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 · doi:10.1016/S0377-0427(02)00558-7
[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, No. 3, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[33]Momani, S.: A numerical scheme for the solution of multi-order fractional differential equations, Appl math comput 182, No. 1, 761-770 (2006) · Zbl 1107.65119 · doi:10.1016/j.amc.2006.04.037
[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)
[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)
[37]Ayaz, Fatma: Solutions of the system of differential equations by differential transform method, Appl math comput 147, 547-567 (2004) · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4
[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 · doi:10.1016/j.amc.2004.10.009
[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 · doi:10.1016/j.amc.2005.02.037
[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.
[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 · doi:10.1016/j.amc.2006.05.173
[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.
[43]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 · doi:10.1016/j.amc.2007.07.068
[44]Odibat Z, Shawagfeh N. Generalized Taylor’s formula. Appl Math Comput, in press. doi:10.1016/j.amc.2006.07.102.
[45]Achar, B. N. Narahari; Hanneken, J. W.; Enck, T.; Clarke, T.: Dynamics of the fractional oscillator, Physica A 297, 361-367 (2001) · Zbl 0969.70511 · doi:10.1016/S0378-4371(01)00200-X
[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) · Zbl 1203.74022 · doi:10.1115/1.3167615
[47]Canat, S.; Faucher, J.: Modeling, identification and simulation of induction machine with fractional derivative, Fractional differentiation and its applications, 459-471 (2006)
[48]Riu, D.; Retiére, N.: Implicit half-order systems utilisation for diffusion phenomenon modelling, Fractional differentiation and its applications, 447-459 (2006)