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)
Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. (English) Zbl 1193.49023
Summary: This paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo.
MSC:
49K21Optimal control problems involving relations other than differential equations
45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
References:
[1]Debnath, L.: Recent applications of fractional calculus to science and engineering, Int. J. Math. math. Sci., No. 54, 3413-3442 (2003) · Zbl 1036.26004 · doi:10.1155/S0161171203301486
[2]Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, Scientific computing in chemical engineering II. Computational fluid dynamics, reaction engineering, and molecular properties, 217-224 (1999)
[3]Ferreira, N. M. Fonseca; Duarte, F. B.; Lima, M. F. M.; Marcos, M. G.; Machado, J. A. Tenreiro: Application of fractional calculus in the dynamical analysis and control of mechanical manipulators, Fract. calc. Appl. anal. 11, No. 1, 91-113 (2008) · Zbl 1159.26303
[4]Hilfer, R.: Fractional diffusion based on Riemann–Liouville fractional derivatives, J. phys. Chem. B 104, No. 16, 3914-3917 (2000)
[5]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[6]Kulish, V. V.; Lage, J. L.: Application of fractional calculus to fluid mechanics, J. fluids eng. 124, No. 3, 803-806 (2002)
[7]Magin, R.: Fractional calculus in bioengineering. Part 1–3, Critical reviews in bioengineering 32 (2004)
[8]Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995)
[9]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[10]Oustaloup, A.; Pommier, V.; Lanusse, P.: Design of a fractional control using performance contours. Application to an electromechanical system, Fract. calc. Appl. anal. 6, No. 1, 1-24 (2003) · Zbl 1036.93023
[11]Machado, J. A. Tenreiro; Barbosa, R. S.: Introduction to the special issue on fractional differentiation and its applications, J. vib. Control 14, No. 9–10, 1253 (2008)
[12]Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics, (2008)
[13]Agrawal, O. P.: Fractional variational calculus and the transversality conditions, J. phys. A 39, No. 33, 10375-10384 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[14]Agrawal, O. P.: Generalized Euler–Lagrange equations and transversality conditions for fvps in terms of the Caputo derivative, J. vib. Control 13, No. 9–10, 1217-1237 (2007) · Zbl 1158.49006 · doi:10.1177/1077546307077472
[15]Agrawal, O. P.: Fractional variational calculus in terms of Riesz fractional derivatives, J. phys. A 40, No. 24, 6287-6303 (2007) · Zbl 1125.26007 · doi:10.1088/1751-8113/40/24/003
[16]Almeida, R.; Malinowska, A. B.; Torres, D. F. M.: A fractional calculus of variations for multiple integrals with application to vibrating string, J. math. Phys. 51 (2010)
[17]Almeida, R.; Torres, D. F. M.: Calculus of variations with fractional derivatives and fractional integrals, Appl. math. Lett. 22, No. 12, 1816-1820 (2009) · Zbl 1183.26005 · doi:10.1016/j.aml.2009.07.002
[18]Atanacković, T. M.; Konjik, S.; Pilipović, S.: Variational problems with fractional derivatives: Euler–Lagrange equations, J. phys. A 41, No. 9, 095201 (2008) · Zbl 1175.49020 · doi:10.1088/1751-8113/41/9/095201
[19]Baleanu, D.; Agrawal, Om.P.: Fractional Hamilton formalism within Caputo’s derivative, Czech J. Phys. 56, No. 10–11, 1087-1092 (2006) · Zbl 1111.37304 · doi:10.1007/s10582-006-0406-x
[20]Baleanu, D.; Muslih, S. I.; Rabei, E. M.: On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear dynam. 53, No. 1–2, 67-74 (2008) · Zbl 1170.70324 · doi:10.1007/s11071-007-9296-0
[21]El-Nabulsi, R. A.; Torres, D. F. M.: Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann–Liouville derivatives of order (α,β), Math. methods appl. Sci. 30, No. 15, 1931-1939 (2007) · Zbl 1177.49036 · doi:10.1002/mma.879
[22]El-Nabulsi, R. A.; Torres, D. F. M.: Fractional actionlike variational problems, J. math. Phys. 49, No. 5, 053521 (2008) · Zbl 1152.81422 · doi:10.1063/1.2929662
[23]Frederico, G. S. F.; Torres, D. F. M.: A formulation of Noether’s theorem for fractional problems of the calculus of variations, J. math. Anal. appl. 334, No. 2, 834-846 (2007) · Zbl 1119.49035 · doi:10.1016/j.jmaa.2007.01.013
[24]Frederico, G. S. F.; Torres, D. F. M.: Fractional conservation laws in optimal control theory, Nonlinear dynam. 53, No. 3, 215-222 (2008) · Zbl 1170.49017 · doi:10.1007/s11071-007-9309-z
[25]G.S.F. Frederico, D.F.M. Torres, Fractional Noether’s theorem in the Riesz–Caputo sense, Appl. Math. Comput. (2010), in press (doi:10.1016/j.amc.2010.01.100).
[26]Klimek, M.: Stationarity-conservation laws for fractional differential equations with variable coefficients, J. phys. A 35, No. 31, 6675-6693 (2002) · Zbl 1039.35005 · doi:10.1088/0305-4470/35/31/311
[27]Muslih, S. I.; Baleanu, D.: Fractional Euler–Lagrange equations of motion in fractional space, J. vib. Control 13, No. 9–10, 1209-1216 (2007) · Zbl 1158.49008 · doi:10.1177/1077546307077473
[28]Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics, Phys. rev. E (3) 53, No. 2, 1890-1899 (1996)
[29]Cruz, P. A. F.; Torres, D. F. M.; Zinober, A. S. I.: A non-classical class of variational problems, Int. J. Math. model. Numer. optim. 1, No. 3, 227-236 (2010) · Zbl 1218.49038 · doi:10.1504/IJMMNO.2010.031750
[30]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[31]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[32]Podlubny, I.: Fractional differential equations, (1999)
[33]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, (1993) · Zbl 0818.26003
[34]Brunetti, R.; Guido, D.; Longo, R.: Modular structure and duality in conformal quantum field theory, Comm. math. Phys. 156, No. 1, 201-219 (1993) · Zbl 0809.46086 · doi:10.1007/BF02096738
[35]A.B. Malinowska, D.F.M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., doi:10.1002/mma.1289.