A semidiscrete scheme for evolution equations with memory

Filippo Dell'Oro; Olivier Goubet; Youcef Mammeri; Vittorino Pata

doi:10.3934/dcds.2019247

Article Contents

2019, Volume 39, Issue 10: 5637-5658. Doi: 10.3934/dcds.2019247

This issue Previous Article Standing and travelling waves in a parabolic-hyperbolic system Next Article Slices of parameter spaces of generalized Nevanlinna functions

A semidiscrete scheme for evolution equations with memory

1.
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy
2.
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France

^* Corresponding author: Filippo Dell'Oro
^* Corresponding author: Filippo Dell'Oro

Received: May 2018

Revised: January 2019

Published: July 2019

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We introduce a new mathematical framework for the time discretization of evolution equations with memory. As a model, we focus on an abstract version of the equation

$ \partial_t u(t) - \int_0^\infty g(s) \Delta u(t-s)\, {{\rm{d}}} s = 0 $

with Dirichlet boundary conditions, modeling hereditary heat conduction with Gurtin-Pipkin thermal law. Well-posedness and exponential stability of the discrete scheme are shown, as well as the convergence to the solutions of the continuous problem when the time-step parameter vanishes.

Keywords:

Mathematics Subject Classification: Primary: 65J08, 45K05; Secondary: 65M12, 45M10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	V. V. Chepyzhov, E. Mainini and V. Pata, Stability of abstract linear semigroups arising from heat conduction with memory, Asymptot. Anal., 50 (2006), 269–291.
[2]	M. Conti, V. Danese, C. Giorgi and V. Pata, A model of viscoelasticity with time-dependent memory kernels, Amer. J. Math., 140 (2018), 349–389. doi: 10.1353/ajm.2018.0008.
[3]	M. Conti, E. Marchini and V. Pata, Exponential stability for a class of linear hyperbolic equations with hereditary memory, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1555-1565. doi: 10.3934/dcdsb.2013.18.1555.
[4]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.
[5]	R. H. De Staelen and D. Guidetti, On a finite difference scheme for an inverse integro-differential problem using semigroup theory: A functional analytic approach, Numer. Funct. Anal. Optim., 37 (2016), 850-886. doi: 10.1080/01630563.2016.1180630.
[6]	M. Fabrizio, G. Gentili and D. W. Reynolds, On a rigid linear heat conductor with memory, Int. J. Engng. Sci., 36 (1998), 765-782. doi: 10.1016/S0020-7225(97)00123-7.
[7]	G. Gentili and C. Giorgi, Thermodynamic properties and stability for the heat flux equation with linear memory, Quart. Appl. Math., 51 (1993), 342-362. doi: 10.1090/qam/1218373.
[8]	C. Giorgi, M. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133.
[9]	M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (A. Lorenzi and B. Ruf, Eds.), Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 50 (2002), 155–178.
[10]	M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.
[11]	M. Kovács and J. Printems, Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory termm, J. Math. Anal. Appl., 413 (2014), 939-952. doi: 10.1016/j.jmaa.2013.12.034.
[12]	W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B, 35 (1993), 23-70. doi: 10.1017/S0334270000007268.
[13]	W. McLean and V. Thomée, Asymptotic behaviour of numerical solutions of an evolution equation with memory, Asymptot. Anal., 14 (1997), 257-276.
[14]	W. McLean, V. Thomée and L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math., 69 (1996), 49-69. doi: 10.1016/0377-0427(95)00025-9.
[15]	R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 331-332. doi: 10.1016/0022-247X(78)90234-2.
[16]	J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683.
[17]	S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Series in Comp. Meth. in Mech. and Th. Sci. Taylor & Francis, 1980. doi: 10.1201/9781482234213.
[18]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[19]	U. Stefanelli, Well-posedness and time discretization of a nonlinear Volterra integrodifferential equation, J. Integral Equations Appl., 13 (2001), 273-304. doi: 10.1216/jiea/1020254675.
[20]	B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.
[21]	D. Xu, The asymptotic behavior for numerical solution of a Volterra equation, Acta Math. Appl. Sin. Engl. Ser., 19 (2003), 47-58. doi: 10.1007/s10255-003-0080-8.
[22]	D. Xu, Decay properties for the numerical solutions of a partial differential equation with memory, J. Sci. Comput., 62 (2015), 146-178. doi: 10.1007/s10915-014-9850-0.