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The heat equation on time scales

Treść / Zawartość
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.
Słowa kluczowe
Rocznik
Strony
475--491
Opis fizyczny
Bibliogr. 27 poz., wykr.
Twórcy
autor
  • Department of Mathematics, Marshall University, 1 John Marshall Drive, Huntington, WV 25755, USA
  • Grupo Física-Matemática, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Bibliografia
  • [1] G.E. Andrews, R. Askey, R. Ranjan, Special Functions, Cambridge University Press, Cambridge, 2000.
  • [2] N.R.O. Bastos, D. Mozyrska, D.F.M. Torres, Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform, Int. J. Math. Comput. 11 (2011), J11, 1–9.
  • [3] M. Bohner, G.Sh. Guseinov, The convolution on time scales, Abstr. Appl. Anal. Art. 2007, Art. ID 58373, 24 pp.
  • [4] M. Bohner, G. Guseinov, B. Karpuz, Properties of the Laplace transform on time scales with arbitrary graininess, Integral Transforms Spec. Funct. 22 (2011), no. 11, 785–800.
  • [5] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2001.
  • [6] S. Cheng, Partial Difference Equations, Taylor & Francis, London, 2003.
  • [7] T. Cuchta, Discrete Analogues of Some Classical Special Functions, Missouri University of Science, 2015.
  • [8] T. Cuchta, S. Georgiev, Analysis of the bilateral Laplace transform on time scales with applications, Int. J. Dyn. Syst. Differ. Equ. 11 (2021), no. 3–4, 255–274.
  • [9] T. Cuchta, D. Grow, N. Wintz, A dynamic matrix exponential via a matrix cylinder transformation, J. Math. Anal. Appl. 479 (2019), 733–751.
  • [10] J. Davis, I. Gravagne, R. Marks, Bilateral Laplace transforms on time scales: convergence, convolution, and the characterization of stationary stochastic time series, Circuits, Systems And Signal Processing 29 (2010), no. 6, 1141–1165.
  • [11] J.M. Davis, I.A. Gravagne, R.J. Marks, Time scale discrete Fourier transforms, 2010 42nd Southeastern Symposium On System Theory (SSST), 102–110.
  • [12] L.C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010.
  • [13] R. Floreanini, L. Vinet, Symmetries of the q-difference heat equation, Lett. Math. Phys. 32 (1994), no. 1, 37–44.
  • [14] M. Friesl, A. Slavík, P. Stehlík, Discrete-space partial dynamic equations on time scales and applications to stochastic processes, Appl. Math. Lett. 37 (2014), 86–90.
  • [15] S. Georgiev, V. Darvish, The generalized Fourier convolution on time scales, Integral Transforms Spec. Funct. 34 (2022), no. 3, 1–17.
  • [16] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1–2, 18–56.
  • [17] S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Dynam. Systems Appl. 8 (1999), 471–488.
  • [18] S. Hilger, An application of calculus on measure chains to Fourier theory and Heisenberg’s uncertainty principle, J. Difference Equ. Appl. 8 (2002), 897–936.
  • [19] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2006), no. 2, 391–415.
  • [20] B. Jackson, J. Davis, An ergodic approach to Laplace transforms on time scales, J. Math. Anal. Appl. 502 (2021), 125231.
  • [21] B. Karpuz, Analyticity of the complex time scale exponential, Complex Anal. Oper. Theory 11 (2017), no. 1, 21–34.
  • [22] J. Mallet-Paret, Spatial patterns, spatial chaos, and traveling waves in lattice differential equations, Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), 105–129, [in:] Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45, North-Holland, Amsterdam, 1996.
  • [23] R. Marks II, I.A. Gravagne, J.M. Davis, J.J. DaCunha, Nonregressivity in switched linear circuits and mechanical systems, Math. Comput. Modelling 43 (2006), no. 11–12, 1383–1392.
  • [24] A. Slavík, Asymptotic behavior of solutions to the semidiscrete diffusion equation, Appl. Math. Lett. 106 (2020), 106392, 7 pp.
  • [25] A. Slavík, P. Stehlík, Explicit solutions to dynamic diffusion-type equations and their time integrals, Appl. Math. Comput. 234 (2014), 486–505.
  • [26] A. Slavík, P. Stehlík, Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl. 427 (2015), 525–545.
  • [27] P. Williams, Fractional calculus on time scales with Taylor’s theorem, Fract. Calc. Appl. Anal. 15 (2012), no. 4, 616–638.
Uwagi
Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-4a6b1d7c-8b61-488a-9e0b-d378f568d33d
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