Opuscula Math. 43, no. 4 (2023), 475-491
https://doi.org/10.7494/OpMath.2023.43.4.475
Opuscula Mathematica
The heat equation on time scales
Abstract. 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.
Keywords: heat equation, time scales, Fourier transform.
Mathematics Subject Classification: 39A14, 34N05, 42A38.
- G.E. Andrews, R. Askey, R. Ranjan, Special Functions, Cambridge University Press, Cambridge, 2000.
- 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.
- M. Bohner, G.Sh. Guseinov, The convolution on time scales, Abstr. Appl. Anal. Art. 2007 Art. ID 58373, 24 pp. https://doi.org/10.1155/2007/58373
- 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. http://doi.org/10.1080/10652469.2010.548335
- M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2001.
- S. Cheng, Partial Difference Equations, Taylor & Francis, London, 2003.
- T. Cuchta, Discrete Analogues of Some Classical Special Functions, Missouri University of Science, 2015.
- 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. https://doi.org/10.1504/IJDSDE.2021.117356
- T. Cuchta, D. Grow, N. Wintz, A dynamic matrix exponential via a matrix cylinder transformation, J. Math. Anal. Appl. 478 (2019), 733-751. https://doi.org/10.1016/j.jmaa.2019.06.048
- 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. https://doi.org/10.1007/s00034-010-9196-2
- J.M. Davis, I.A. Gravagne, R.J. Marks, Time scale discrete Fourier transforms, 2010 42nd Southeastern Symposium On System Theory (SSST), 102-110. https://doi.org/10.1109/SSST.2010.5442859
- L.C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010.
- R. Floreanini, L. Vinet, Symmetries of the \(q\)-difference heat equation, Lett. Math. Phys. 32 (1994), no. 1, 37-44. https://doi.org/10.1007/BF00761122
- 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. https://doi.org/10.1016/j.aml.2014.06.002
- S. Georgiev, V. Darvish, The generalized Fourier convolution on time scales, Integral Transforms Spec. Funct. 34 (2022), no. 3, 1-17. https://doi.org/10.1080/10652469.2022.2105323
- S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18-56. https://doi.org/10.1007/BF03323153
- S. Hilger, Special functions, Laplace and Fourier transform on measure chains, Dynam. Systems Appl. 8 (1999), 471-488.
- 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. https://doi.org/10.1080/1023619021000000960
- B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2006), no. 2, 391-415. https://doi.org/10.1016/j.cam.2005.02.011
- B. Jackson, J. Davis, An ergodic approach to Laplace transforms on time scales, J. Math. Anal. Appl. 502 (2021), 125231. https://doi.org/10.1016/j.jmaa.2021.125231
- B. Karpuz, Analyticity of the complex time scale exponential, Complex Anal. Oper. Theory 11 (2017), no. 1, 21-34. https://doi.org/10.1007/s11785-016-0562-3
- 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.
- 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. https://doi.org/10.1016/j.mcm.2005.08.007
- A. Slavík, Asymptotic behavior of solutions to the semidiscrete diffusion equation, Appl. Math. Lett. 106 (2020), 106392, 7 pp. https://doi.org/10.1016/j.aml.2020.106392
- A. Slavík, P. Stehlík, Explicit solutions to dynamic diffusion-type equations and their time integrals, Appl. Math. Comput. 234 (2014), 486-505. https://doi.org/10.1016/j.amc.2014.01.176
- A. Slavík, P. Stehlík, Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl. 427 (2015), 525-545. https://doi.org/10.1016/j.jmaa.2015.02.056
- P. Williams, Fractional calculus on time scales with Taylor’s theorem, Fract. Calc. Appl. Anal. 15 (2012), no. 4, 616-638. https://doi.org/10.2478/s13540-012-0043-y
- Tom Cuchta
- Department of Mathematics, Marshall University, 1 John Marshall Drive, Huntington, WV 25755 USA
- Rui A.C. Ferreira (corresponding author)
- Grupo Física-Matemática, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
- Communicated by Petr Stehlík.
- Received: 2022-11-02.
- Revised: 2023-03-24.
- Accepted: 2023-04-13.
- Published online: 2023-06-13.
