American Institute of Mathematical Sciences

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

Hahn's symmetric quantum variational calculus

Artur M. C. Brito da Cruz 1, , Natália Martins 2, and  Delfim F. M. Torres 3,

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.
Keywords: Euler--Lagrange difference equations, quantum calculus., calculus of variations, Leitmann's principle, Hahn's symmetric calculus.
Mathematics Subject Classification: 39A13, 49K0.
Citation: Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & 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, (2009).   Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[5]

G. Boole, "Calculus of Finite Differences,", Edited by J. F. Moulton 4th ed, (1957).   Google Scholar

[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.  doi: 10.1016/j.na.2011.01.015.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1007/s10898-007-9171-z.  Google Scholar

[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.   Google Scholar

[11]

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

[12]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals,", Emended edition, (2010).   Google Scholar

[13]

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

[14]

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

[15]

V. Kac and P. Cheung, "Quantum Calculus,", Universitext, (2002).  doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.  doi: 10.1016/j.amc.2010.01.015.  Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

show all references

References:
[1]

K. A. Aldwoah, "Generalized Time Scales and Associated Difference Equations,", Ph.D. thesis, (2009).   Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[5]

G. Boole, "Calculus of Finite Differences,", Edited by J. F. Moulton 4th ed, (1957).   Google Scholar

[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.  doi: 10.1016/j.na.2011.01.015.  Google Scholar

[7]

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

[8]

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

[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.  doi: 10.1007/s10898-007-9171-z.  Google Scholar

[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.   Google Scholar

[11]

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

[12]

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals,", Emended edition, (2010).   Google Scholar

[13]

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

[14]

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

[15]

V. Kac and P. Cheung, "Quantum Calculus,", Universitext, (2002).  doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.  doi: 10.1016/j.amc.2010.01.015.  Google Scholar

[24]

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

[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.   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

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