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Operational methods and solutions of boundary-value problems with periodic data

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Abstract

We introduce suitable exponential operators and use the multi-dimensional polynomials considered by Hermite and subsequently studied by P. Appell and J. Kampé de Fériet, H. W. Gould and A. T. Hopper, and G. Dattoli et al. in order to obtain explicit solutions of classical boundary-value problems with periodic data.

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References

  1. P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynômes d'Hermite, Gauthier-Villars, Paris (1926).

    MATH  Google Scholar 

  2. Y. Ben Cheikh, “Decomposition of some complex functions with respect to the cyclic group of order n,” Appl. Math. Inform., 4, 30–53 (1999).

    MATH  MathSciNet  Google Scholar 

  3. G. Bretti and P. E. Ricci, “Multidimensional extensions of the Bernoulli and Appell polynomials,” Taiwanese J. Math., 8, 415–428 (2004).

    MATH  MathSciNet  Google Scholar 

  4. C. Cassisa, P. E. Ricci, and I. Tavkhelidze, “Operational identities for circular and hyperbolic functions and their generalizations,” Georgian Math. J., 10, 45–56 (2003).

    MATH  MathSciNet  Google Scholar 

  5. C. Cassisa, P. E. Ricci, and I. Tavkhelidze, “An operatorial approach to solutions of boundary-value problems in the half-plane,” J. Concr. Appl. Math., 1, 37–62 (2003).

    MATH  MathSciNet  Google Scholar 

  6. C. Cassisa, P. E. Ricci, and I. Tavkhelidze, “Exponential operators for solving evolution problems with degeneration,” J. Appl. Funct. Anal., 1, 33–50 (2006).

    MATH  MathSciNet  Google Scholar 

  7. C. Cassisa, P. E. Ricci, and I. Tavkhelidze, “Exponential operators and solution of pseudo-classical evolution problems,” J. Concr. Appl. Math., 4, 33–45 (2006).

    MATH  MathSciNet  Google Scholar 

  8. G. Dattoli, A. Torre, P. L. Ottaviani, and L. Vázquez, “Evolution operator equations: integration with algebraic and finite difference methods. Application to physical problems in classical and quantum mechanics,” Riv. Nuovo Cim., 2 (1997).

  9. G. Dattoli and A. Torre, “Operational methods and two variable Laguerre polynomials,” Atti Acc. Sc. Torino, 132, 1–7 (1998).

    MathSciNet  Google Scholar 

  10. G. Dattoli, A. Torre, and C. Carpanese, “Operational rules and arbitrary order Hermite generating functions,” J. Math. Anal. Appl., 227, 98–111 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Dattoli, H. M. Srivastava, and P. E. Ricci, “Two-index multidimensional Gegenbauer polynomials and integral representations,” Math. Comp. Modelling, 37, 283–291 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Erdély, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York (1953).

    Google Scholar 

  13. H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Math. J., 29, 51–62 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  14. D. T. Haimo and C. Markett, “A representation theory for solutions of a higher order heat equation, I,” J. Math. Anal. Appl., 168, 89–107 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  15. D. T. Haimo and C. Markett, “A representation theory for solutions of a higher order heat equation, II,” J. Math. Anal. Appl., 168, 289–305 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Maroscia and P. E. Ricci, “Hermite-Kampé de Fériet polynomials and solutions of boundary-value problems in the half-space,” J. Concr. Appl. Math., 3, 9–29 (2005).

    MATH  MathSciNet  Google Scholar 

  17. G. Maroscia and P. E. Ricci, “Explicit solutions of multidimensional pseudo-classical boundary-value problems in the half-space,” Math. Comp. Modelling, 40, 667–698 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  18. P. E. Ricci, “Le funzioni pseudo-iperboliche e pseudo-trigonometriche,” Pubbl. Ist. Mat. Appl., 12, 37–49 (1978).

    MathSciNet  Google Scholar 

  19. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Wiley, New York (1984).

    MATH  Google Scholar 

  20. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Moscow (1999).

  21. D. V. Widder, An Introduction to Transform Theory, Academic Press, New York (1971).

    MATH  Google Scholar 

  22. D. V. Widder, The Heat Equation, Academic Press, New York (1975).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma La Sapienza, Roma, Australia

    C. Cassisa & P. E. Ricci

  2. I. Vekua Institute of Applied Mathematics, Tbilisi, Georgia

    I. Tavkhelidze

Authors
  1. C. Cassisa
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  2. P. E. Ricci
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  3. I. Tavkhelidze
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Corresponding author

Correspondence to C. Cassisa.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.

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Cassisa, C., Ricci, P.E. & Tavkhelidze, I. Operational methods and solutions of boundary-value problems with periodic data. J Math Sci 157, 85–97 (2009). https://doi.org/10.1007/s10958-009-9314-5

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  • DOI: https://doi.org/10.1007/s10958-009-9314-5

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