Skip to main content
SpringerLink
Log in

Discrete Subdiffusion Equations with Memory

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

In this paper, we study a discrete subdiffusion equation with memory. Based on the backward operator and the theory of fractional resolvent families, we find a discrete fractional resolvent sequence which allows to write the solution to this discrete subdiffusion equation as a variation of constant formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abadías, L., Álvarez, E.: Fractional Cauchy problem with memory effects. Math. Nachr. 293(10), 1846–1872 (2020)

    Article  MathSciNet  Google Scholar 

  2. Abadías, L., Lizama, C.: Almost automorphic mild solutions to fractional partial difference-differential equations. Appl. Anal. 95(6), 1347–1369 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abadías, L., Miana, P.J.: A Subordination Principle on Wright Functions and Regularized Resolvent Families. J. Funct. Spaces. https://doi.org/10.1155/2015/158145

  4. Abadías, L., Lizama, C., Miana, P.J., Velasco, M.: On well-posedness of vector-valued fractional differential-difference equations. Discret. Contin. Dyn. Syst. 39(5), 2679–2708 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62(3), 1602–6111 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atici, F., Eloe, P.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bazhlekova, E.: Fractional evolution equations in Banach spaces. Ph.D. thesis, Eindhoven University of Technology (2001)

  8. Bazhlekova, E.: Subordination principle for a class of fractional order differential equations. Mathematics 3(2), 412–427 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid. Numer. Math. 131, 1–31 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bernardo, F., Cuevas, C., Soto, H.: Qualitative theory for Volterra difference equations. Math. Methods Appl. Sci. 41(14), 5423–5458 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C., Shih, T.: Finite element methods for integrodifferential equations. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  12. Coleman, B., Gurtin, M.: Equipresence and constitutive equation for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967)

    Article  MathSciNet  Google Scholar 

  13. Cuesta, E.: Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., Dynamical Systems and Differential Equations. In: Proceedings of the 6th AIMS International Conference, suppl., pp. 277–285 (2007)

  14. Cuevas, C., Choquehuanca, M., Soto, H.: Asymptotic analysis for Volterra difference equations. Asymptot. Anal. 88(3), 125–164 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Da Prato, G., Iannelli, M.: Linear integrodifferential equations in Banach space. Rend. Sem. Mat. Univ. Padova 62, 207–219 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Diethelm, J., Ford, N., Freed, A., Luchko, Y.: Algorithms for the fractional calculus: A selection of numerical methods. Comput. Methods Appl. Mech. Engrg. 194, 743–773 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. GTM Book Series, vol. 194. Springer, New York (2000)

  18. Goodrich, C.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61(2), 191–202 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goodrich, C., Peterson, A.: Discrete Fractional Calculus. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  20. Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and Applications, vol. 34. Cambridge University Press, Cambridge (1990)

  21. Gurtin, M., Pipkin, A.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  22. Haase, M.: The functional calculus for sectorial operators, Operator Theory: Advances and applications, 169. Birkäuser Verlag, Basel (2006)

    Book  Google Scholar 

  23. Han, Y., Hu, Y., Song, J.: Maximum principle for general controlled systems driven by fractional Brownian motions. Appl. Math. Optim. 67(2), 279–322 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. He, J., Lizama, C., Zhou, Y.: The Cauchy problem for discrete-time fractional evolution equations. J. Comput. Appl. Math. 370, 112683 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Haubold, H., Mathai, A., Saxena, R.: Mittag-Leffler functions and their applications. J. Appl. Math. https://doi.org/10.1155/2011/298628 (2011)

  26. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co. Inc, River Edge, NJ (2000)

    Book  MATH  Google Scholar 

  27. Jin, B., Li, B., Zhou, Z.: Discrete maximal regularity of time-stepping schemes for fractional evolution equations. Numer. Math. 138(1), 101–131 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Keyantuo, V., Lizama, C., Warma, M.: Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal. (2013)

  29. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)

    Book  MATH  Google Scholar 

  30. Krasnoschok, M., Pata, V., Vasylyeva, N.: Semilinear subdiffusion with memory in the one-dimensional case. Nonlinear Anal. TMA 165, 1–17 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Krasnoschok, M., Pata, V., Vasylyeva, N.: Solvability of linear boundary value problems for subdiffusion equations with memory. J. Integral Equ. Appl. 30, 417–445 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krasnoschok, M., Pata, V., Vasylyeva, N.: Semilinear subdiffusion with memory in multidimensional domains. Math. Nachr. 292(7), 1490–1513 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, M., Chen, C., Li, F.: On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, 2702–2726 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lizama, C.: The Poisson distribution, abstract fractional difference equations, and stability. Proc. Am. Math. Soc. 145(9), 3809–3827 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lizama, C., Murillo-Arcila, M.: Maximal regularity in \(l_p\)-spaces for discrete time fractional shifted equations. J. Differ. Equ. 263(6), 3175–3196 (2017)

    Article  MATH  Google Scholar 

  37. Lizama, C., Murillo-Arcila, M.: Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes. Discret. Contin. Dyn. Syst. 40, 509–528 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lizama, C., Ponce, R.: Solutions of abstract integro-differential equations via Poisson transformation. Math. Methods Appl. Sci. 44(3), 2495–2505 (2021)

  39. Lubich, Ch.: Discretize fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lubich, Ch.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52, 129–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction To Mathematical Models. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  42. McLean, W., Thomée, V.: Time discretization of an evolution equation via Laplace transforms. IMA J. Numer. Anal. 24, 439–463 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. McLean, W., Sloan, I., Thomée, V.: Time discretization via Laplace transformation of an integro-differential equation of parabolic type. Numer. Math. 102(3), 497–522 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  46. Ponce, R.: Hölder continuous solutions for fractional differential equations and maximal regularity. J. Differ. Equ. 255, 3284–3304 (2013)

    Article  MATH  Google Scholar 

  47. Ponce, R.: Time discretization of fractional subdiffusion equations via fractional resolvent operators. Comput. Math. Appl. 80, 69–92 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ponce, R.: Subordination Principle for subdiffusion equations with memory. J. Integral Equ. Appl. 32(4), 479–493 (2020)

  49. Prüss, J.: Evolutionary Integral Equations and Applications, Monographs Math., vol. 87. Birkhäuser Verlag, New York (1993)

    Book  MATH  Google Scholar 

  50. Sloan, I., Thomée, V.: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 23(5), 1052–1061 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tamilalagan, P., Balasubramaniam, P.: The solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps via resolvent operators. Appl. Math. Optim. 77(3), 443–462 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. Webb, G.: An abstract semilinear Volterra integrodifferential equation. Proc. Am. Math. Soc. 69(2), 255–260 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  53. Xu, D.: Uniform \(l^1\) convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates. Math. Comput. 83(286), 735–769 (2014)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referees for reading the manuscript and for making suggestions that have improved the previous version of this paper.

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile

    Rodrigo Ponce

Authors
  1. Rodrigo Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rodrigo Ponce.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ponce, R. Discrete Subdiffusion Equations with Memory. Appl Math Optim 84, 3475–3497 (2021). https://doi.org/10.1007/s00245-021-09753-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-021-09753-z

Keywords

Mathematics Subject Classification

Search

Navigation