2013, 3(1): 77-94. doi: 10.3934/naco.2013.3.77

Hahn's symmetric quantum variational calculus

1. 

Escola Superior de Tecnologia de Setúbal, Estefanilha, 2910-761 Setúbal, Portugal

2. 

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

3. 

CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  December 2011 Revised  November 2012 Published  January 2013

We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler--Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.
Citation: Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77
References:
[1]

K. A. Aldwoah, "Generalized Time Scales and Associated Difference Equations," Ph.D. thesis, Cairo University, 2009.

[2]

K. A. Aldwoah, A. B. Malinowska and D. F. M. Torres, The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 93-116.

[3]

R. Almeida and D. F. M. Torres, Leitmann's direct method for fractional optimization problems, Appl. Math. Comput., 217 (2010), 956-962. doi: 10.1016/j.amc.2010.03.085.

[4]

R. Almeida and D. F. M. Torres, Nondifferentiable variational principles in terms of a quantum operator, Math. Methods Appl. Sci., 34 (2011), 2231-2241. doi: 10.1002/mma.1523.

[5]

G. Boole, "Calculus of Finite Differences," Edited by J. F. Moulton 4th ed, Chelsea Publishing Co. New York, 1957.

[6]

A. M. C. Brito da Cruz, N. Martins and D. F. M. Torres, Higher-order Hahn's quantum variational calculus, Nonlinear Anal., 75 (2012), 1147-1157. doi: 10.1016/j.na.2011.01.015.

[7]

A. M. C. Brito da Cruz and N. Martins, The q-symmetric variational calculus, Comput. Math. Appl., 64 (2012), 2241-2250. doi: 10.1016/j.camwa.2012.01.076.

[8]

D. A. Carlson and G. Leitmann, Coordinate transformation method for the extremization of multiple integrals, J. Optim. Theory Appl., 127 (2005), 523-533. doi: 10.1007/s10957-005-7500-2.

[9]

D. A. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations, J. Global Optim., 40 (2008), 41-50. doi: 10.1007/s10898-007-9171-z.

[10]

J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems, Topol. Methods Nonlinear Anal., 33 (2009), 217-231.

[11]

T. Ernst, The different tongues of q-calculus, Proc. Est. Acad. Sci., 57 (2008), 81-99. doi: 10.3176/proc.2008.2.03.

[12]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals," Emended edition, Dover, Mineola, NY, 2010.

[13]

W. Hahn, Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachr., 2 (1949), 4-34. doi: 10.1002/mana.19490020103.

[14]

F. H. Jackson, q-Difference equations, Amer. J. Math., 32 (1910), 305-314. doi: 10.2307/2370183.

[15]

V. Kac and P. Cheung, "Quantum Calculus," Universitext, Springer, New York, 2002. doi: 10.1007/978-1-4613-0071-7.

[16]

R. Koekoek, P. A. Lesky and R. F. Swarttouw, "Hypergeometric Orthogonal Polynomials and Their Q-Analogues," Springer Monographs in Mathematics, Springer, Berlin, 2010. doi: 10.1007/978-3-642-05014-5.

[17]

A. Lavagno and G. Gervino, Quantum mechanics in q-deformed calculus, J. Phys.: Conf. Ser., 174 (2009), 012071, 8 pp. doi: 10.1088/1742-6596/174/1/012071.

[18]

A. Lavagno and P. Narayana Swamy, q-deformed structures and nonextensive statistics: a comparative study, Phys. A, 305 (2002), 310-315. doi: 10.1016/S0378-4371(01)00680-X.

[19]

G. Leitmann, A note on absolute extrema of certain integrals, Internat. J. Non-Linear Mech., 2 (1967), 55-59. doi: 10.1016/0020-7462(67)90018-2.

[20]

G. Leitmann, On a class of direct optimization problems, J. Optim. Theory Appl., 108 (2001), 467-481. doi: 10.1023/A:1017507006157.

[21]

G. Leitmann, Some extensions to a direct optimization method, J. Optim. Theory Appl., 111 (2001), 1-6. doi: 10.1023/A:1017560112706.

[22]

G. Leitmann, On a method of direct optimization, Vychisl. Tekhnol., 7 (2002), 63-67.

[23]

A. B. Malinowska and 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., 217 (2010), 1158-1162. doi: 10.1016/j.amc.2010.01.015.

[24]

A. B. Malinowska and D. F. M. Torres, The Hahn quantum variational calculus, J. Optim. Theory Appl., 147 (2010), 419-442. doi: 10.1007/s10957-010-9730-1.

[25]

N. Martins and D. F. M. Torres, L'Hôpital-type rules for monotonicity with application to quantum calculus, Int. J. Math. Comput., 10 (2011), 99-106.

[26]

N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus, Comput. Math. Appl., 64 (2012), 2166-2175. doi: 10.1016/j.camwa.2011.12.006.

[27]

D. N. Page, Information in black hole radiation, Phys. Rev. Lett., 71 (1993), 3743-3746. doi: 10.1103/PhysRevLett.71.3743.

[28]

D. F. M. Torres and G. Leitmann, Contrasting two transformation-based methods for obtaining absolute extrema, J. Optim. Theory Appl., 137 (2008), 53-59. doi: 10.1007/s10957-007-9292-z.

[29]

D. Youm, q-deformed conformal quantum mechanics, Phys. Rev. D 62 (2000), 095009, 5 pages. doi: 10.1103/PhysRevD.62.095009.

show all references

References:
[1]

K. A. Aldwoah, "Generalized Time Scales and Associated Difference Equations," Ph.D. thesis, Cairo University, 2009.

[2]

K. A. Aldwoah, A. B. Malinowska and D. F. M. Torres, The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 93-116.

[3]

R. Almeida and D. F. M. Torres, Leitmann's direct method for fractional optimization problems, Appl. Math. Comput., 217 (2010), 956-962. doi: 10.1016/j.amc.2010.03.085.

[4]

R. Almeida and D. F. M. Torres, Nondifferentiable variational principles in terms of a quantum operator, Math. Methods Appl. Sci., 34 (2011), 2231-2241. doi: 10.1002/mma.1523.

[5]

G. Boole, "Calculus of Finite Differences," Edited by J. F. Moulton 4th ed, Chelsea Publishing Co. New York, 1957.

[6]

A. M. C. Brito da Cruz, N. Martins and D. F. M. Torres, Higher-order Hahn's quantum variational calculus, Nonlinear Anal., 75 (2012), 1147-1157. doi: 10.1016/j.na.2011.01.015.

[7]

A. M. C. Brito da Cruz and N. Martins, The q-symmetric variational calculus, Comput. Math. Appl., 64 (2012), 2241-2250. doi: 10.1016/j.camwa.2012.01.076.

[8]

D. A. Carlson and G. Leitmann, Coordinate transformation method for the extremization of multiple integrals, J. Optim. Theory Appl., 127 (2005), 523-533. doi: 10.1007/s10957-005-7500-2.

[9]

D. A. Carlson and G. Leitmann, Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations, J. Global Optim., 40 (2008), 41-50. doi: 10.1007/s10898-007-9171-z.

[10]

J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems, Topol. Methods Nonlinear Anal., 33 (2009), 217-231.

[11]

T. Ernst, The different tongues of q-calculus, Proc. Est. Acad. Sci., 57 (2008), 81-99. doi: 10.3176/proc.2008.2.03.

[12]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals," Emended edition, Dover, Mineola, NY, 2010.

[13]

W. Hahn, Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachr., 2 (1949), 4-34. doi: 10.1002/mana.19490020103.

[14]

F. H. Jackson, q-Difference equations, Amer. J. Math., 32 (1910), 305-314. doi: 10.2307/2370183.

[15]

V. Kac and P. Cheung, "Quantum Calculus," Universitext, Springer, New York, 2002. doi: 10.1007/978-1-4613-0071-7.

[16]

R. Koekoek, P. A. Lesky and R. F. Swarttouw, "Hypergeometric Orthogonal Polynomials and Their Q-Analogues," Springer Monographs in Mathematics, Springer, Berlin, 2010. doi: 10.1007/978-3-642-05014-5.

[17]

A. Lavagno and G. Gervino, Quantum mechanics in q-deformed calculus, J. Phys.: Conf. Ser., 174 (2009), 012071, 8 pp. doi: 10.1088/1742-6596/174/1/012071.

[18]

A. Lavagno and P. Narayana Swamy, q-deformed structures and nonextensive statistics: a comparative study, Phys. A, 305 (2002), 310-315. doi: 10.1016/S0378-4371(01)00680-X.

[19]

G. Leitmann, A note on absolute extrema of certain integrals, Internat. J. Non-Linear Mech., 2 (1967), 55-59. doi: 10.1016/0020-7462(67)90018-2.

[20]

G. Leitmann, On a class of direct optimization problems, J. Optim. Theory Appl., 108 (2001), 467-481. doi: 10.1023/A:1017507006157.

[21]

G. Leitmann, Some extensions to a direct optimization method, J. Optim. Theory Appl., 111 (2001), 1-6. doi: 10.1023/A:1017560112706.

[22]

G. Leitmann, On a method of direct optimization, Vychisl. Tekhnol., 7 (2002), 63-67.

[23]

A. B. Malinowska and 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., 217 (2010), 1158-1162. doi: 10.1016/j.amc.2010.01.015.

[24]

A. B. Malinowska and D. F. M. Torres, The Hahn quantum variational calculus, J. Optim. Theory Appl., 147 (2010), 419-442. doi: 10.1007/s10957-010-9730-1.

[25]

N. Martins and D. F. M. Torres, L'Hôpital-type rules for monotonicity with application to quantum calculus, Int. J. Math. Comput., 10 (2011), 99-106.

[26]

N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus, Comput. Math. Appl., 64 (2012), 2166-2175. doi: 10.1016/j.camwa.2011.12.006.

[27]

D. N. Page, Information in black hole radiation, Phys. Rev. Lett., 71 (1993), 3743-3746. doi: 10.1103/PhysRevLett.71.3743.

[28]

D. F. M. Torres and G. Leitmann, Contrasting two transformation-based methods for obtaining absolute extrema, J. Optim. Theory Appl., 137 (2008), 53-59. doi: 10.1007/s10957-007-9292-z.

[29]

D. Youm, q-deformed conformal quantum mechanics, Phys. Rev. D 62 (2000), 095009, 5 pages. doi: 10.1103/PhysRevD.62.095009.

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