zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
A collocation-shooting method for solving fractional boundary value problems. (English) Zbl 1222.65078
Summary: We discuss the numerical solution of special class of fractional boundary value problems of order 2. The method of solution is based on a conjugating collocation and spline analysis combined with shooting method. A theoretical analysis about the existence and uniqueness of exact solution for the present class is proved. Two examples involving Bagley-Torvik equation subject to boundary conditions are also presented; numerical results illustrate the accuracy of the present scheme.
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
45J05Integro-ordinary differential equations
[1]Ahmed, E.; Elgazzar, A. S.: On fractional order differential equations model for nonlocal epidemics, Phys A 379, No. 2, 607-614 (2007)
[2]Ahmed, E.; El-Sayed, A. M. A.; Elsaka, H. A. A.: Equilibrium points, stability, and numerical solutions of fractional-order predator-prey and rabies models, J math anal appl 325, No. 1, 542-553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087
[3]Al-Mdallal, Q. M.: An efficient method for solving fractional Sturm-Liouville problems, Chaos, solitons fractals 40, No. 1, 183-189 (2009) · Zbl 1197.65097 · doi:10.1016/j.chaos.2007.07.041
[4]Al-Mdallal, Q. M.: On the numerical solution of fractional Sturm-Liouville problems, Int J comput math 99999, 1 (2009)
[5]Attili, B.; Syam, M.: Efficient shooting method for solving two point boundary value problems, Chaos, solitons fractals 35, No. 5, 895-903 (2008) · Zbl 1132.65067 · doi:10.1016/j.chaos.2006.05.094
[6]Babenko Y. Non integer differential equation in engineering, Conference Bordeaux 3 – 8 July 1994.
[7]Bagley, R. L.; Torvik, P. J.: On the appearance of the fractional derivative in the behavior of real materials, ASME J appl mech 51, No. 2, 294-298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615
[8]Beyer, H.; Kempfle, S.: Definition of physically consistent damping laws with fractional derivatives, Zeitschrift fr angewandte Mathematik und mechanik 75, No. 8, 623-635 (1995) · Zbl 0865.70014 · doi:10.1002/zamm.19950750820
[9]Blank L. Numerical treatment of differential equations of fractional order. Numerical Analysis Report 287, Manchester Center for Numerical Computational Mathematics; 1996.
[10]Daftardar-Geiji, V.; Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations, J math anal appl 301, No. 2, 508-518 (2005) · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039
[11]Daftardar-Geiji, V.; Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving capato derivatives, J math anal appl 328, 1026-1033 (2007) · Zbl 1115.34006 · doi:10.1016/j.jmaa.2006.06.007
[12]Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order, Elect translat numer anal 5, 1-6 (1997) · Zbl 0890.65071 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[13]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[14]Ghorbani, A.; Alavi, A.: Applications of he’s variational iteration method to solve semidifferential equations of nth order, Math prob eng 2008, 1-9 (2008) · Zbl 1155.65380 · doi:10.1155/2008/627983
[15]Ha, S. N.: A nonlinear shooting method for two point boundary value problems, Comput math appl 42, 1411-1420 (2001) · Zbl 0999.65077 · doi:10.1016/S0898-1221(01)00250-4
[16]Keller, H. B.: Numerical methods for two point boundary value problems, (1992)
[17]Lia M, Jimenezc S, Niea N, Tanga Y,Vazqueze L. Solving Two-point boundary value problems of fractional differential equations by spline collocation methods. Available from lt;http://www.cc.ac.cn/2009researchreport/0903.pdf; 2009. p. 1 – 10.
[18]Liw, F.; Zhuang, P.; Anh, V.; Turner, J.: A fractional-order implicit difference approximation for the space-time fractional diffusion equation, Anzjamj 47, C48-C68 (2006)
[19]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal: theory, meth appl 69, No. 10, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[20]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Letters 21, No. 8, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[21]Lubich, C.: Discretized fractional calculus, SIAM J math anal 17, No. 3, 704-719 (1986) · Zbl 0624.65015 · doi:10.1137/0517050
[22]Lubich, C.: Convoluton quadrature and discretized operational calculus (part II), Numerisch Mathematik 52, No. 4, 413-425 (1988) · Zbl 0643.65094 · doi:10.1007/BF01462237
[23]Mainardi F. Fractional relaxiation and fractional diffusion equations: mathematical aspects. In: Proceedings of the 14th IMACS world congress, vol. 1; 1994. p. 329 – 32.
[24]Mainardi, F.: The fundamental solution for the fractional diffusion-wave equation, Appl math lett 9, No. 6, 23-28 (1996) · Zbl 0879.35036 · doi:10.1016/0893-9659(96)00089-4
[25]Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, solutions fractals 7, 1461-1477 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[26]Michalski, M. W.: Derivatives of noninteger order and their applications, Dissertationes mathematicae 328, 1-47 (1993) · Zbl 0880.26007
[27]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[28]Obaidat, Z.; Momani, S.: A generalized differential transform for linear partial differential equations of fractional order, Appl math lett 21, 194-199 (2008) · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[29]Ochmann, M.; Makarov, S.: Representation of the absorption of nonlinear waves by fractional derivatives, J acoust soc am 94, No. 6, 3392-3399 (1993)
[30]Podlubny, I.: Fractional differential equations, (1999)
[31]Ray, S. Saha; Bera, R. K.: Analytical solution of the bagley torvik equation by Adomian decomposition method, Appl math comput 168, 398-410 (2005) · Zbl 1109.65072 · doi:10.1016/j.amc.2004.09.006
[32]Shuqin, Z.: Existence of solution for a boundary value problem of fractional order, Acta Mathematica scientia 26B, No. 2, 220-228 (2006) · Zbl 1106.34010 · doi:10.1016/S0252-9602(06)60044-1
[33]Su, X.: Boundary value problems for a coupled system of nonlinear fractional differential equations, Appl math lett 22, 64-69 (2009) · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001
[34]Rawashdeh, E. A.: Numerical solution of semidifferential equation by collocation method, Appl math comput 174, 869-876 (2006) · Zbl 1090.65097 · doi:10.1016/j.amc.2005.05.029
[35]El-Wakil, S.; Elhambaly, A.; Abdou, M. A.: Adomian decomposition method for solving fractional nonlinear differential equations, Appl math comput 182, 313-324 (2006) · Zbl 1106.65115 · doi:10.1016/j.amc.2006.02.055