Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation

The main goal in this paper is to study asymptotic behavior in Lp(RN ) for the solutions of the fractional version of the discrete in time N-dimensional diffusion equation, which involves the Caputo fractional h-difference operator. The techniques to prove the results are based in new subordination...

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Autores:: Abadias, Luciano
Alvarez, Edgardo
Díaz , Stiven

Tipo de recurso:: Article of journal

Fecha de publicación:: 2021

Institución:: Corporación Universidad de la Costa

Repositorio:: REDICUC - Repositorio CUC

Idioma:: eng

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network_name_str	REDICUC - Repositorio CUC
repository_id_str
dc.title.eng.fl_str_mv	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
title	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
spellingShingle	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation Subordination formula Scaled Wright function Fractional difference equations Large-time behavior Decay of solutions Discrete fundamental solution
title_short	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
title_full	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
title_fullStr	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
title_full_unstemmed	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
title_sort	Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation
dc.creator.fl_str_mv	Abadias, Luciano Alvarez, Edgardo Díaz , Stiven
dc.contributor.author.spa.fl_str_mv	Abadias, Luciano Alvarez, Edgardo Díaz , Stiven
dc.contributor.corporatename.spa.fl_str_mv	Corporación Universidad de la Costa
dc.subject.proposal.eng.fl_str_mv	Subordination formula Scaled Wright function Fractional difference equations Large-time behavior Decay of solutions Discrete fundamental solution
topic	Subordination formula Scaled Wright function Fractional difference equations Large-time behavior Decay of solutions Discrete fundamental solution
description	The main goal in this paper is to study asymptotic behavior in Lp(RN ) for the solutions of the fractional version of the discrete in time N-dimensional diffusion equation, which involves the Caputo fractional h-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.
publishDate	2021
dc.date.issued.none.fl_str_mv	2021-10-14
dc.date.accessioned.none.fl_str_mv	2022-06-07T17:41:39Z
dc.date.available.none.fl_str_mv	2022-10-14 2022-06-07T17:41:39Z
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dc.identifier.url.spa.fl_str_mv	https://doi.org/10.1016/j.jmaa.2021.125741
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identifier_str_mv	0022-247X 10.1016/j.jmaa.2021.125741 1096-0813 Corporación Universidad de la Costa REDICUC - Repositorio CUC
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dc.language.iso.none.fl_str_mv	eng
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dc.relation.ispartofjournal.spa.fl_str_mv	Journal of Mathematical Analysis and Applications
dc.relation.references.spa.fl_str_mv	[1] L. Abadias, E. Alvarez, Uniform stability for fractional Cauchy problems and applications, Topol. Methods Nonlinear Anal. 52 (2) (2018) 707–728. [2] L. Abadias, E. Alvarez, Asymptotic behaviour for the discrete in time heat equation, Manuscript available at https:// arxiv.org/pdf/2102.11109.pdf. [3] L. Abadias, M. De León, J.L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl. 449 (1) (2017) 734–755. [4] L. Abadias, C. Lizama, Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal. 95 (6) (2016) 1347–1369. [5] L. Abadias, P. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces 2015 (2015) 158145, https://doi.org/10.1155/2015/158145. [6] M. Aigner, Diskrete Mathematik, 6th ed., Friedr. Vieweg & Sohn, 2006. [7] E. Alvarez, S. Diaz, C. Lizama, C-semigroups, subordination principle and the Lévy α-stable distribution on discrete time, Commun. Contemp. Math. (2020). [8] E.G. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, University Press Facilities, Eindhoven University of Technology, 2001. [9] O. Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017) 109–131. [10] O. Ciaurri, L. Roncal, P.R. Stinga, J.L. Torrea, J.L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math. 330 (2018) 688–738. [11] E.B. Davies, Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal. 80 (1) (1988) 16–32. [12] E.B. Davies, Lp spectral theory of higher-order elliptic differential operators, Bull. Lond. Math. Soc. 29 (5) (1997) 513–546. [13] M. Del Pino, J. Dolbeault, Asymptotic behavior of nonlinear diffusions, Math. Res. Lett. 10 (4) (2003) 551–557. [14] J. Duoandikoetxea, J. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (6) (1992) 693–698. [15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, H. Bateman, Higher Transcenden-tal Functions, vol. III, McGraw–Hill, New York, 1953. [16] A. Erdélyi, F.G. Tricomi, The aymptotic expansion of a ratio of Gamma functions, Pac. J. Math. 1 (1951) 133–142. [17] M. Escobedo, E. Zuazua, Large time behavior for convection-diffusion equations in RN , J. Funct. Anal. 100 (1) (1991) 119–161. [18] L.C. Evans, Partial Differential Equations, second ed., Graduate Studies in Mathematics, vol. 19, AMS Publications, Providence, Rhode Island, 2014. [19] J. Fourier, Théorie Analytique de la Chaleur, Reprint of the 1822 original Cambridge Library Collection, Cambridge University Press, Cambridge, 2009. [20] A. Gmira, L. Veron, Asymptotic behaviour of the solution of a semilinear parabolic equation, Monatshefte Math. 94 (1982) 299–311. [21] A. Gmira, L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in RN , J. Funct. Anal. 53 (1984) 258–276. [22] C. Goodrich, C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Isr. J. Math. 236 (2020) 533–589. [23] C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015. [24] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, Academic Press, Inc., San Diego, CA, 2000. [25] A. Grigor’yan, Estimates of heat kernels on Riemannian manifolds, manuscript available at www.ma.ic.ac.uk/~grigor, 1999. [26] J. Kemppainen, J. Siljander, R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differ. Equ. 263 (1) (2017) 149–201. [27] A.A. Kilbas, M. Saigo, H-Transforms, Theory and Applications, Analytical Methods and Special Functions, vol. 9, 2004. [28] S. Kusuoka, D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. Math. 127 (1) (1988) 165–189. [29] P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. Math. 124 (1) (1986) 1–21. [30] C. Lizama, lp-Maximal regularity for fractional difference equations on UMD spaces, Math. Nachr. 288 (17/18) (2015) 2079–2092. [31] C. Lizama, The Poisson distribution, abstract fractional difference equations and stability, Proc. Am. Math. Soc. 145 (9) (2017) 3809–3827. [32] C. Lizama, L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst. 38 (3) (2018) 1365–1403. [33] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. [34] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. [35] D. Mozyrska, M. Wyrwas, The Z-transform method and delta type fractional difference operators, Discrete Dyn. Nat. Soc. 2015 (2015) 852734, https://doi.org/10.1155/2015/852734. [36] S. Mustapha, Gaussian estimates for heat kernels on Lie groups, Math. Proc. Camb. Philos. Soc. 128 (1) (2000) 45–64. [37] J.R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal. 140 (2) (1997) 161–195. [38] R. Ponce, Time discretization of fractional subdiffusion equations via fractional resolvent operators, Comput. Math. Appl. 80 (4) (2020) 69–92. [39] A. Zygmund, Trigonometric Series, Vols. I, II, 2nd ed., Cambridge University Press, New York, 1959.
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dc.rights.spa.fl_str_mv	© 2021 Elsevier Inc. All rights reserved. Atribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)
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spelling	Abadias, LucianoAlvarez, EdgardoDíaz , StivenCorporación Universidad de la Costa2022-06-07T17:41:39Z2022-10-142022-06-07T17:41:39Z2021-10-140022-247Xhttps://hdl.handle.net/11323/9214https://doi.org/10.1016/j.jmaa.2021.12574110.1016/j.jmaa.2021.1257411096-0813Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/The main goal in this paper is to study asymptotic behavior in Lp(RN ) for the solutions of the fractional version of the discrete in time N-dimensional diffusion equation, which involves the Caputo fractional h-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.23 páginasapplication/pdfengAcademic Press Inc.United States© 2021 Elsevier Inc. All rights reserved.Atribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)https://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/embargoedAccesshttp://purl.org/coar/access_right/c_f1cfSubordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equationArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARThttp://purl.org/coar/version/c_970fb48d4fbd8a85https://www-sciencedirect-com.ezproxy.cuc.edu.co/science/article/pii/S0022247X21008209?via%3Dihub#!Journal of Mathematical Analysis and Applications[1] L. Abadias, E. Alvarez, Uniform stability for fractional Cauchy problems and applications, Topol. Methods Nonlinear Anal. 52 (2) (2018) 707–728.[2] L. Abadias, E. Alvarez, Asymptotic behaviour for the discrete in time heat equation, Manuscript available at https:// arxiv.org/pdf/2102.11109.pdf.[3] L. Abadias, M. De León, J.L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl. 449 (1) (2017) 734–755.[4] L. Abadias, C. Lizama, Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal. 95 (6) (2016) 1347–1369.[5] L. Abadias, P. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces 2015 (2015) 158145, https://doi.org/10.1155/2015/158145.[6] M. Aigner, Diskrete Mathematik, 6th ed., Friedr. Vieweg & Sohn, 2006.[7] E. Alvarez, S. Diaz, C. Lizama, C-semigroups, subordination principle and the Lévy α-stable distribution on discrete time, Commun. Contemp. Math. (2020).[8] E.G. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, University Press Facilities, Eindhoven University of Technology, 2001.[9] O. Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017) 109–131.[10] O. Ciaurri, L. Roncal, P.R. Stinga, J.L. Torrea, J.L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math. 330 (2018) 688–738.[11] E.B. Davies, Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal. 80 (1) (1988) 16–32.[12] E.B. Davies, Lp spectral theory of higher-order elliptic differential operators, Bull. Lond. Math. Soc. 29 (5) (1997) 513–546.[13] M. Del Pino, J. Dolbeault, Asymptotic behavior of nonlinear diffusions, Math. Res. Lett. 10 (4) (2003) 551–557.[14] J. Duoandikoetxea, J. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (6) (1992) 693–698.[15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, H. Bateman, Higher Transcenden-tal Functions, vol. III, McGraw–Hill, New York, 1953.[16] A. Erdélyi, F.G. Tricomi, The aymptotic expansion of a ratio of Gamma functions, Pac. J. Math. 1 (1951) 133–142.[17] M. Escobedo, E. Zuazua, Large time behavior for convection-diffusion equations in RN , J. Funct. Anal. 100 (1) (1991) 119–161.[18] L.C. Evans, Partial Differential Equations, second ed., Graduate Studies in Mathematics, vol. 19, AMS Publications, Providence, Rhode Island, 2014.[19] J. Fourier, Théorie Analytique de la Chaleur, Reprint of the 1822 original Cambridge Library Collection, Cambridge University Press, Cambridge, 2009.[20] A. Gmira, L. Veron, Asymptotic behaviour of the solution of a semilinear parabolic equation, Monatshefte Math. 94 (1982) 299–311.[21] A. Gmira, L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in RN , J. Funct. Anal. 53 (1984) 258–276.[22] C. Goodrich, C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Isr. J. Math. 236 (2020) 533–589.[23] C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer International Publishing, 2015.[24] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, Academic Press, Inc., San Diego, CA, 2000.[25] A. Grigor’yan, Estimates of heat kernels on Riemannian manifolds, manuscript available at www.ma.ic.ac.uk/~grigor, 1999.[26] J. Kemppainen, J. Siljander, R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differ. Equ. 263 (1) (2017) 149–201.[27] A.A. Kilbas, M. Saigo, H-Transforms, Theory and Applications, Analytical Methods and Special Functions, vol. 9, 2004.[28] S. Kusuoka, D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. Math. 127 (1) (1988) 165–189.[29] P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. Math. 124 (1) (1986) 1–21.[30] C. Lizama, lp-Maximal regularity for fractional difference equations on UMD spaces, Math. Nachr. 288 (17/18) (2015) 2079–2092.[31] C. Lizama, The Poisson distribution, abstract fractional difference equations and stability, Proc. Am. Math. Soc. 145 (9) (2017) 3809–3827.[32] C. Lizama, L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst. 38 (3) (2018) 1365–1403.[33] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010.[34] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.[35] D. Mozyrska, M. Wyrwas, The Z-transform method and delta type fractional difference operators, Discrete Dyn. Nat. Soc. 2015 (2015) 852734, https://doi.org/10.1155/2015/852734.[36] S. Mustapha, Gaussian estimates for heat kernels on Lie groups, Math. Proc. Camb. Philos. Soc. 128 (1) (2000) 45–64.[37] J.R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal. 140 (2) (1997) 161–195.[38] R. Ponce, Time discretization of fractional subdiffusion equations via fractional resolvent operators, Comput. Math. Appl. 80 (4) (2020) 69–92.[39] A. Zygmund, Trigonometric Series, Vols. I, II, 2nd ed., Cambridge University Press, New York, 1959.231507Subordination formulaScaled Wright functionFractional difference equationsLarge-time behaviorDecay of solutionsDiscrete fundamental solutionPublicationORIGINAL1-s2.0-S0022247X21008209-main.pdf1-s2.0-S0022247X21008209-main.pdfapplication/pdf448238https://repositorio.cuc.edu.co/bitstreams/b64f8878-8c10-4901-8d23-cea031a5c368/downloadfd5351558757d13ac1e72c3f3145b872MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-83196https://repositorio.cuc.edu.co/bitstreams/1ed23f40-6bd9-4e49-b560-63782f105abe/downloade30e9215131d99561d40d6b0abbe9badMD52TEXT1-s2.0-S0022247X21008209-main.pdf.txt1-s2.0-S0022247X21008209-main.pdf.txttext/plain45701https://repositorio.cuc.edu.co/bitstreams/f6717e00-46aa-4184-a33d-178eacbcc8c7/downloadd5098f6963ee6a3d43dc6254cbf6e136MD53THUMBNAIL1-s2.0-S0022247X21008209-main.pdf.jpg1-s2.0-S0022247X21008209-main.pdf.jpgimage/jpeg13151https://repositorio.cuc.edu.co/bitstreams/799a16c1-5741-44f3-bb1f-c24f210bfa23/downloada4ce4b6961b32761519c43aed4c7e9ceMD5411323/9214oai:repositorio.cuc.edu.co:11323/92142024-09-17 10:49:03.611https://creativecommons.org/licenses/by-nc-nd/4.0/© 2021 Elsevier Inc. 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Subordination principle, Wright functions and large-time behavior for the discrete in time fractional diffusion equation

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