zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The solution of the Bagley-Torvik equation with the generalized Taylor collocation method. (English) Zbl 1188.65107
Summary: The Bagley-Torvik equation, which has an important role in fractional calculus, is solved by generalizing the Taylor collocation method [cf. R. L. Bagley and P. J. Torvik [J. Appl. Mech. 51, 294–298 (1984; Zbl 1203.74022)]. The proposed method has a new algorithm for solving fractional differential equations. This new method has many advantages over variety of numerical approximations for solving fractional differential equations. To assess the effectiveness and preciseness of the method, results are compared with other numerical approaches. Since the Bagley-Torvik equation represents a general form of the fractional problems, its solution can give many ideas about the solution of similar problems in fractional differential equations.
MSC:
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34A08Fractional differential equations
65L05Initial value problems for ODE (numerical methods)
References:
[1]Bagley, R. L.; Torvik, P. J.: On the appearance of the fractional derivative in the behavior of real materials, Applied mechanics 51, 294-298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615
[2]Podlubny, I.: Fractional differential equations, (1999)
[3]Trinks, C.; Ruge, P.: Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals, Computational mechanics 29, 471-476 (2002) · Zbl 1146.76634 · doi:10.1007/s00466-002-0356-5
[4]C. Leszczynski, M. Ciesielski, A numerical method for solution of ordinary differential equations of fractional order. Parallel Processing and Applied Mathematics (2002) 695–702. · Zbl 1057.65507 · doi:http://link.springer.de/link/service/series/0558/bibs/2328/23280695.htm
[5]Edwards, J. T.; Ford, N. J.; Simpson, A. C.: The numerical solution of linear multi-term fractional differential equations: systems of equations, Journal of computational and applied mathematics 148, 401-448 (2002) · Zbl 1019.65048 · doi:10.1016/S0377-0427(02)00558-7
[6]El-Sayed, A. M. A.; El-Mesiry, A. E. M.; El-Saka, H. A. A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations, Computational and applied mathematics 23, 33-54 (2004) · Zbl 1213.34025 · doi:10.1590/S0101-82052004000100002 · doi:http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022004000100002&lng=en&nrm=iso&tlng=en
[7]Ray, S. S.; Bera, R. K.: Analytical solution of the bagley–torvik equation by Adomian decomposition method, Applied mathematics and computation 168, 398-410 (2005) · Zbl 1109.65072 · doi:10.1016/j.amc.2004.09.006
[8]Hu, Y.; Luo, Y.; Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method, Journal of computational and applied mathematics 215, 220-229 (2008) · Zbl 1132.26313 · doi:10.1016/j.cam.2007.04.005
[9]Daftardar-Gejji, V.; Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations, Journal of mathematical analysis and applications 301, 508-518 (2005) · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039
[10]Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos, solitons and fractals 34, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[11]Y. Çenesiz, Bagley–Torvik denkleminin kesirli diferensiyel dönüşüm metodu ile çözümü ve digbreve;er yöntemlerle karşılaştırılması, Master Thesis, 2007 (in Turkish).
[12]Momani, S.; Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, solitons and fractals 31, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[13]Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations, Applied mathematics and computation 131, 517-529 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[14]Ray, S. S.; Bera, R. K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied mathematics and computation 167, 561-571 (2005) · Zbl 1082.65562 · doi:10.1016/j.amc.2004.07.020
[15]Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, Journal of computational and applied mathematics 207, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[16]Hu, Y.; Luo, Y.; Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method, Journal of computational and applied mathematics 215, 220-229 (2008) · Zbl 1132.26313 · doi:10.1016/j.cam.2007.04.005
[17]Abbasbandy, S.: An approximation solution of a nonlinear equation with Riemann–Liouville’s fractional derivatives by he’s variational iteration method, Journal of computational and applied mathematics 207, 53-58 (2007) · Zbl 1120.65133 · doi:10.1016/j.cam.2006.07.011
[18]Abdulaziz, O.; Hashim, I.; Momani, S.: Solving systems of fractional differential equations by homotopy–perturbation method, Physics letters A 372, 451-459 (2008) · Zbl 1217.81080 · doi:10.1016/j.physleta.2007.07.059
[19]J. Cang, Y. Tan, H. Xu, S–J. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos, Solitons and Fractals 40 (2009) 1–9. · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[20]Abdulaziz, O.; Hashim, I.; Momani, S.: Application of homotopy–perturbation method to fractional ivps, Journal of computational and applied mathematics 216, 574-584 (2008) · Zbl 1142.65104 · doi:10.1016/j.cam.2007.06.010
[21]Sweilam, N. H.; Khader, M. M.; Al–Bar, R. F.: Numerical studies for a multi-order fractional differential equation, Physics letters A 371, 26-33 (2007) · Zbl 1209.65116 · doi:10.1016/j.physleta.2007.06.016
[22]Arıko&gbreve, A.; Lu; Özkol, I.: Solution of fractional differential equations by using differential transform method, Chaos, solitons and fractals 34, 1473-1481 (2007)
[23]Oturanç, G.; Kurnaz, A.; Keskin, Y.: A new analytical approximate method for the solution of fractional differential equations, International journal of computer mathematics 85, 131-142 (2008) · Zbl 1131.65114 · doi:10.1080/00207160701405477
[24]Ertürk, V. S.; Momani, S.: Solving systems of fractional differential equations using differential transform method, Journal of computational and applied mathematics 215, 142-151 (2008) · Zbl 1141.65088 · doi:10.1016/j.cam.2007.03.029
[25]Karamete, A.; Sezer, M.: A Taylor collocation method for the solution of linear integro-differential equations, International journal of computer mathematics 79, 987-1000 (2002) · Zbl 1006.65144 · doi:10.1080/00207160212116
[26]Gülsu, M.; Sezer, M.; Güney, Z.: Approximate solution of general high-order linear nonhomogeneous difference equations by means of Taylor collocation method, Applied mathematics and computation 173, 683-693 (2006) · Zbl 1088.65113 · doi:10.1016/j.amc.2005.04.048
[27]Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, 223-276 (1997)
[28]Dubois, F.; Mengue, S.: Mixed collocation for fractional differential equations, Numerical algorithms 34, 303-311 (2003) · Zbl 1038.65059 · doi:10.1023/B:NUMA.0000005367.21295.05
[29]Kumar, P.; Agrawal, O. P.: An approximate method for numerical solution of fractional differential equations, Signal processing 86, 2602-2610 (2006) · Zbl 1172.94436 · doi:10.1016/j.sigpro.2006.02.007
[30]Odibat, Z. M.; Shawagfeh, N. T.: Generalized Taylor’s formula, Applied mathematics and computation 186, 286-293 (2007) · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102
[31]Muslim, M.: Existence and approximation of solutions to fractional differential equations, Mathematical and computer modelling 49, 1164-1172 (2009) · Zbl 1165.34304 · doi:10.1016/j.mcm.2008.07.013
[32]Sezer, M.; Yalçinbaş, A.; Gülsu, M.: A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term, International journal of computer mathematics 85, 1055-1063 (2008) · Zbl 1145.65048 · doi:10.1080/00207160701466784