October  2019, 39(10): 5637-5658. doi: 10.3934/dcds.2019247

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

Received  May 2018 Revised  January 2019 Published  July 2019

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.
Citation: Filippo Dell'Oro, Olivier Goubet, Youcef Mammeri, Vittorino Pata. A semidiscrete scheme for evolution equations with memory. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5637-5658. doi: 10.3934/dcds.2019247
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. Google Scholar

[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. Google Scholar

[3]

M. ContiE. 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. Google Scholar

[4]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar

[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. Google Scholar

[6]

M. FabrizioG. 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. Google Scholar

[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. Google Scholar

[8]

C. GiorgiM. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

W. McLean and V. Thomée, Asymptotic behaviour of numerical solutions of an evolution equation with memory, Asymptot. Anal., 14 (1997), 257-276. Google Scholar

[14]

W. McLeanV. 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. Google Scholar

[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. Google Scholar

[16]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

M. ContiE. 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. Google Scholar

[4]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar

[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. Google Scholar

[6]

M. FabrizioG. 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. Google Scholar

[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. Google Scholar

[8]

C. GiorgiM. G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: A semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

W. McLean and V. Thomée, Asymptotic behaviour of numerical solutions of an evolution equation with memory, Asymptot. Anal., 14 (1997), 257-276. Google Scholar

[14]

W. McLeanV. 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. Google Scholar

[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. Google Scholar

[16]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

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