×

The semidiscrete damped wave equation with a fractional Laplacian. (English) Zbl 1510.35382

Summary: In this paper we completely solve the open problem of finding the fundamental solution of the semidiscrete fractional-spatial damped wave equation. We combine operator theory and Laplace transform methods with properties of Bessel functions to show an explicit representation of the solution when initial conditions are given. Our findings extend known results from the literature and also provide new insights into the qualitative behavior of the solutions for the studied model. As an example, we show the existence of almost periodic solutions as well as their profile in the homogeneous case.

MSC:

35R11 Fractional partial differential equations
39A06 Linear difference equations
26A33 Fractional derivatives and integrals
44A10 Laplace transform
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abadias, Luciano, Large time behaviour for the heat equation on \(\mathbb{Z} \), moments and decay rates, J. Math. Anal. Appl., Paper No. 125137, 25 pp. (2021) · Zbl 1464.39008 · doi:10.1016/j.jmaa.2021.125137
[2] Bateman, H., Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc., 494-512 (1943) · Zbl 0061.20201 · doi:10.1090/S0002-9904-1943-07927-X
[3] Bochner, Salomon, Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino Rend. Sem. Mat., 225-253 (1955/56) · Zbl 0072.17301
[4] Ciaurri, \'{O}scar, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 109-131 (2017) · Zbl 1476.39003 · doi:10.1007/s11854-017-0015-6
[5] Ciaurri, \'{O}scar, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 688-738 (2018) · Zbl 1391.35388 · doi:10.1016/j.aim.2018.03.023
[6] Corduneanu, C., Almost periodic functions, Interscience Tracts in Pure and Applied Mathematics, No. 22, x+237 pp. (1968), Interscience Publishers [John Wiley & Sons], New York-London-Sydney · Zbl 0175.09101
[7] D’Abbicco, M., Diffusion phenomena for the wave equation with structural damping in the \(L^p-L^q\) framework, J. Differential Equations, 2307-2336 (2014) · Zbl 1288.35073 · doi:10.1016/j.jde.2014.01.002
[8] D’Abbicco, M., Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation, J. Pseudo-Differ. Oper. Appl., 261-293 (2016) · Zbl 1350.42045 · doi:10.1007/s11868-015-0141-9
[9] D’Abbicco, Marcello, Asymptotic profile of solutions for strongly damped Klein-Gordon equations, Math. Methods Appl. Sci., 2287-2301 (2019) · Zbl 1416.35151 · doi:10.1002/mma.5508
[10] Erd\'{e}lyi, A., Tables of integral transforms. Vol. I, xx+391 pp. (1954), McGraw-Hill Book Co., Inc., New York-Toronto-London · Zbl 0055.36401
[11] Feintuch, Avraham, Infinite chains of kinematic points, Automatica J. IFAC, 901-908 (2012) · Zbl 1246.93014 · doi:10.1016/j.automatica.2012.02.034
[12] Fitzgibbon, W. E., Limiting behavior of the strongly damped extensible beam equation, Differential Integral Equations, 1067-1076 (1990) · Zbl 0747.35018
[13] Friesl, Michal, Discrete-space partial dynamic equations on time scales and applications to stochastic processes, Appl. Math. Lett., 86-90 (2014) · Zbl 1417.35236 · doi:10.1016/j.aml.2014.06.002
[14] Gonz\'{a}lez-Camus, Jorge, Fundamental solutions for semidiscrete evolution equations via Banach algebras, Adv. Difference Equ., Paper No. 35, 32 pp. (2021) · Zbl 1485.35384 · doi:10.1186/s13662-020-03206-7
[15] Gonz\'{a}lez-Camus, Jorge, Fundamental solutions for discrete dynamical systems involving the fractional Laplacian, Math. Methods Appl. Sci., 4688-4711 (2019) · Zbl 1423.35398 · doi:10.1002/mma.5685
[16] Gradshteyn, I. S., Table of integrals, series, and products, xlviii+1171 pp. (2007), Elsevier/Academic Press, Amsterdam · Zbl 1208.65001
[17] Keyantuo, Valentin, Lattice dynamical systems associated with a fractional Laplacian, Numer. Funct. Anal. Optim., 1315-1343 (2019) · Zbl 1417.49031 · doi:10.1080/01630563.2019.1602542
[18] Lizama, Carlos, H\"{o}lder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 1365-1403 (2018) · Zbl 1397.34034 · doi:10.3934/dcds.2018056
[19] Padgett, J. L., Anomalous diffusion in one-dimensional disordered systems: a discrete fractional Laplacian method, J. Phys. A, 135205, 21 pp. (2020) · Zbl 1514.60113 · doi:10.1088/1751-8121/ab7499
[20] Pearson, Carl E., Asymptotic behavior of solutions to the finite-difference wave equation, Math. Comp., 711-715 (1969) · Zbl 0185.42204 · doi:10.2307/2004957
[21] Ponce, Gustavo, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 399-418 (1985) · Zbl 0576.35023 · doi:10.1016/0362-546X(85)90001-X
[22] Pr\"{u}ss, Jan, Evolutionary integral equations and applications, Monographs in Mathematics, xxvi+366 pp. (1993), Birkh\"{a}user Verlag, Basel · Zbl 0784.45006 · doi:10.1007/978-3-0348-8570-6
[23] N. B. Salem, Space-time fractional diffusion equation associated with Jacobi expansions, Appl. Anal. (2022), To appear.
[24] Slav\'{\i }k, Anton\'{\i }n, Mixing problems with many tanks, Amer. Math. Monthly, 806-821 (2013) · Zbl 1287.34004 · doi:10.4169/amer.math.monthly.120.09.806
[25] Slav\'{\i }k, Anton\'{\i }n, Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl., 525-545 (2015) · Zbl 1338.35449 · doi:10.1016/j.jmaa.2015.02.056
[26] Slav\'{\i }k, Anton\'{\i }n, Asymptotic behavior of solutions to the semidiscrete diffusion equation, Appl. Math. Lett., 106392, 7 pp. (2020) · Zbl 1439.35502 · doi:10.1016/j.aml.2020.106392
[27] Slav\'{\i }k, Anton\'{\i }n, Asymptotic behavior of solutions to the multidimensional semidiscrete diffusion equation, Electron. J. Qual. Theory Differ. Equ., Paper No. 9, 9 pp. (2022) · Zbl 1499.34295
[28] Tarasov, Vasily E., Exact discretization of fractional Laplacian, Comput. Math. Appl., 855-863 (2017) · Zbl 1371.65112 · doi:10.1016/j.camwa.2017.01.012
[29] Triggiani, R., Regularity of some structurally damped problems with point control and with boundary control, J. Math. Anal. Appl., 299-331 (1991) · Zbl 0771.93013 · doi:10.1016/0022-247X(91)90332-T
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.