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)
Discrete-time fractional variational problems. (English) Zbl 1203.94022
Summary: We introduce a discrete-time fractional calculus of variations on the time scale (h) a ,a,h>0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when h tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation.
94A12Signal theory (characterization, reconstruction, filtering, etc.)
26A33Fractional derivatives and integrals (real functions)
62M10Time series, auto-correlation, regression, etc. (statistics)
49J99Existence theory for optimal solutions
[1]Agrawal, O. P.: Formulation of Euler–Lagrange equations for fractional variational problems, J. math. Anal. appl. 272, No. 1, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[2]Agrawal, O. P.: A general finite element formulation for fractional variational problems, J. math. Anal. appl. 337, No. 1, 1-12 (2008) · Zbl 1123.65059 · doi:10.1016/j.jmaa.2007.03.105
[3]Agrawal, O. P.; Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. vib. Control 13, No. 9–10, 1269-1281 (2007) · Zbl 1182.70047 · doi:10.1177/1077546307077467
[4]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, No. 3, 033503 (2010)
[5]Almeida, R.; Torres, D. F. M.: Hölderian variational problems subject to integral constraints, J. math. Anal. appl. 359, No. 2, 674-681 (2009) · Zbl 1169.49016 · doi:10.1016/j.jmaa.2009.06.029
[6]Almeida, R.; Torres, D. F. M.: Isoperimetric problems on time scales with nabla derivatives, J. vib. Control 15, No. 6, 951-958 (2009)
[7]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
[8]Atici, F. M.; Eloe, P. W.: A transform method in discrete fractional calculus, Int. J. Difference equ. 2, No. 2, 165-176 (2007)
[9]Atici, F. M.; Eloe, P. W.: Initial value problems in discrete fractional calculus, Proc. amer. Math. soc. 137, No. 3, 981-989 (2009) · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[10]Baleanu, D.: New applications of fractional variational principles, Rep. math. Phys. 61, No. 2, 199-206 (2008) · Zbl 1166.58304 · doi:10.1016/S0034-4877(08)80007-9
[11]Baleanu, D.; Defterli, O.; Agrawal, O. P.: A central difference numerical scheme for fractional optimal control problems, J. vib. Control 15, No. 4, 583-597 (2009)
[12]Baleanu, D.; Jarad, F.: Discrete variational principles for higher-order Lagrangians, Nuovo cimento soc. Ital. fis. B 120, No. 9, 931-938 (2005)
[13]D. Baleanu, F. Jarad, Difference discrete variational principles, in: Mathematical Analysis and Applications, American Institute of Physics, Melville, NY, 2006 pp. 20–29.
[14]Baleanu, D.; Maaraba, T.; Jarad, F.: Fractional variational principles with delay, J. phys. A 41, No. 31, 315403 (2008) · Zbl 1141.49321 · doi:10.1088/1751-8113/41/31/315403
[15]Baleanu, D.; Muslih, S. I.: Nonconservative systems within fractional generalized derivatives, J. vib. Control 14, No. 9–10, 1301-1311 (2008) · Zbl 1229.70048 · doi:10.1177/1077546307087450
[16]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
[17]Bartosiewicz, Z.; Torres, D. F. M.: Noether’s theorem on time scales, J. math. Anal. appl. 342, No. 2, 1220-1226 (2008)
[18]Bohner, M.: Calculus of variations on time scales, Dynam. syst. Appl. 13, 339-349 (2004) · Zbl 1069.39019
[19]M., Bohner; C., Ferreira R. A.; M., Torres D. F.: Integral inequalities and their applications to the calculus of variations on time scales, Math. inequal. Appl. 13, No. 3, 511-522 (2010) · Zbl 1190.26015 · doi:http://files.ele-math.com/abstracts/mia-13-35-abs.pdf
[20]Bohner, M.; Peterson, A.: Dynamic equations on time scales, (2001)
[21]Cresson, J.; Frederico, G. S. F.; Torres, D. F. M.: Constants of motion for non-differentiable quantum variational problems, Topol. methods nonlinear anal. 33, No. 2, 217-231 (2009) · Zbl 1188.49023
[22]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
[23]El-Nabulsi, R. A.; Torres, D. F. M.: Fractional action like variational problems, J. math. Phys. 49, No. 5, 053521 (2008) · Zbl 1152.81422 · doi:10.1063/1.2929662
[24]R.A.C. Ferreira, D.F.M. Torres, Higher-order calculus of variations on time scales, in: Mathematical Control Theory and Finance, Springer, Berlin, 2008, pp. 149–159. · Zbl 1191.49017 · doi:10.1007/978-3-540-69532-5_9
[25]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
[26]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
[27]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.
[28]Hilscher, R.; Zeidan, V.: Calculus of variations on time scales: weak local piecewise c1rd solutions with variable endpoints, J. math. Anal. appl. 289, No. 1, 143-166 (2004) · Zbl 1043.49004 · doi:10.1016/j.jmaa.2003.09.031
[29]Kelley, W. G.; Peterson, A. C.: Difference equations, (1991) · Zbl 0733.39001
[30]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[31]Jarad, F.; Baleanu, D.: Discrete variational principles for Lagrangians linear in velocities, Rep. math. Phys. 59, No. 1, 33-43 (2007)
[32]Jarad, F.; Baleanu, D.; Maraaba, T.: Hamiltonian formulation of singular Lagrangians on time scales, Chin. phys. Lett. 25, No. 5, 1720-1723 (2008)
[33]Malinowska, A. B.; Torres, D. F. M.: Strong minimizers of the calculus of variations on time scales and the Weierstrass condition, Proc. est. Acad. sci. 58, No. 4, 205-212 (2009) · Zbl 1179.49025 · doi:10.3176/proc.2009.4.02
[34]A.B. Malinowska, D.F.M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci. (2010), in press. DOI: 10.1002/mma.1289.
[35]Malinowska, A. B.; Torres, D. F. M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. math. Appl. 59, No. 9, 3110-3116 (2010) · Zbl 1193.49023 · doi:10.1016/j.camwa.2010.02.032
[36]A.B. Malinowska, D.F.M. Torres, Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales, Appl. Math. Comput. (2010), in press. DOI: 10.1016/j.amc.2010.01.015.
[37]T. Maraaba, F. Jarad, D. Baleanu, Variational optimal-control problems with delayed arguments on time scales, Adv. Difference Equ. (2009), Art. ID 840386, 15pp. · Zbl 1184.49023 · doi:10.1155/2009/840386
[38]Martins, N.; Torres, D. F. M.: Calculus of variations on time scales with nabla derivatives, Nonlinear anal. 71, No. 12, e763-e773 (2009)
[39]K.S. Miller, B. Ross, Fractional difference calculus, in: Univalent Functions, Fractional Calculus, and their Applications (Komacr;riyama, 1988), Horwood, Chichester, 1989, pp. 139–152.
[40]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[41]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
[42]Ortigueira, M. D.: Fractional central differences and derivatives, J. vib. Control 14, No. 9–10, 1255-1266 (2008) · Zbl 1229.26015 · doi:10.1177/1077546307087453
[43]Podlubny, I.: Fractional differential equations, (1999)
[44]Rabei, E. M.; Nawafleh, K. I.; Hijjawi, R. S.; Muslih, S. I.; Baleanu, D.: The Hamilton formalism with fractional derivatives, J. math. Anal. appl. 327, No. 2, 891-897 (2007) · Zbl 1104.70012 · doi:10.1016/j.jmaa.2006.04.076
[45]Rabei, E. M.; Tarawneh, D. M.; Muslih, S. I.; Baleanu, D.: Heisenberg’s equations of motion with fractional derivatives, J. vib. Control 13, No. 9–10, 1239-1247 (2007) · Zbl 1161.81352 · doi:10.1177/1077546307077469
[46]Ross, B.; Samko, S. G.; Love, E. R.: Functions that have no first order derivative might have fractional derivatives of all orders less than one, Real anal. Exchange 20, 140-157 (1994) · Zbl 0820.26002
[47]S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.
[48]Silva, M. F.; Machado, J. A. Tenreiro; Barbosa, R. S.: Using fractional derivatives in joint control of hexapod robots, J. vib. Control 14, No. 9–10, 1473-1485 (2008) · Zbl 1229.70029 · doi:10.1177/1077546307087436
[49]Torres, D. F. M.; Leitmann, G.: Contrasting two transformation-based methods for obtaining absolute extrema, J. optim. Theory appl. 137, No. 1, 53-59 (2008) · Zbl 1147.49002 · doi:10.1007/s10957-007-9292-z
[50]Van Brunt, B.: The calculus of variations, (2004)