Skip to main content
SpringerLink
Log in

On a Non–Local Problem for a Multi–Term Fractional Diffusion-Wave Equation

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions.

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

Similar content being viewed by others

References

  1. L.D. Abreu, P. Balazs, M. de Gosson, Z. Mouayn, Discrete coherent states for higher Landau levels. Ann. Physics 363 (2015), 337–353.

    MathSciNet  MATH  Google Scholar 

  2. P. Agarwal, E. Karimov, M. Mamchuev, M. Ruzhansky, On boundary-value problems for a partial differential equation with Caputo and Bessel operators. In: Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science (2017), 707–718, Appl. Numer. Harmon. Anal., Birkhauser/Springer.

    MATH  Google Scholar 

  3. O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynam. 29 (2002), 145–155.

    MathSciNet  MATH  Google Scholar 

  4. O.P. Agrawal, Response of a diffusion-wave system subjected to deterministic and stochastic fields. Z. Angew. Math. Mech. 83 (2003), 265–274.

    MathSciNet  MATH  Google Scholar 

  5. N. Al-Salti, M. Kirane, B.T. Torebek, On a class of inverse problems for a heat equation with involution perturbation. Hacettepe J. of Math. and Stat. 48, No 3 (2019), 669–681.

    MathSciNet  MATH  Google Scholar 

  6. N.K. Bari, Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Uchenye Zapiski Matematika 148, No 4 (1951), 69–107.

    MathSciNet  Google Scholar 

  7. L. Byszewski, Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt = F(xtuux). J. Appl. Math. Stoch. Anal. 3 (1990), 163–168.

    MathSciNet  MATH  Google Scholar 

  8. L. Bysezewski, Theorem about the existence and uniqueness of solution of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), 494–505.

    MathSciNet  Google Scholar 

  9. L. Bysezewski, Uniqueness of solutions of parabolic semilinear nonlocal boundary problems. J. Math. Anal. Appl. 165 (1992), 472–478.

    MathSciNet  Google Scholar 

  10. J. Chabrowski, On non-local problems for parabolic equations. Nagoya Math. J. 93 (1984), 109–131.

    MathSciNet  MATH  Google Scholar 

  11. J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338 (2008), 1364–1377.

    MathSciNet  MATH  Google Scholar 

  12. L.J. Corwin, F.P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications. In: Cambridge Studies in Advanced Math.. Cambridge Univ. Press, Cambridge, 18. Basic Theory and Examples (1990).

    Google Scholar 

  13. M. Dehghan, Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition. Appl. Math. Comput. 147 (2004), 321–331.

    MathSciNet  MATH  Google Scholar 

  14. M. Dehghan, Implicit collocation technique for heat equationwith non-classic initial condition. Int. J. Nonlin. Sci. Numer. Simul. 7 (2006), 447–450.

    Google Scholar 

  15. J. Delgado, M. Ruzhansky, N. Tokmagambetov, Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary. J. Math. Pures Appl. 107, No 6 (2017), 758–783.

    MathSciNet  MATH  Google Scholar 

  16. I.H. Dimovski, Convolutional Calculus. Bulgarian Academy of Sciences, Sofia (1982); Kluwer Acad. Publ. Dordrecht etc. (1990).

    MATH  Google Scholar 

  17. V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie Groups. Ser. Progress in Mathematics 314, Birkhäuser/Springer [Open Access book] (2016).

    MATH  Google Scholar 

  18. V. Fischer and M. Ruzhansky, Sobolev spaces on graded groups. Ann. Inst. Fourier 67, No 4 (2017), 1671–1723.

    MathSciNet  MATH  Google Scholar 

  19. V. Fock, Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Z. Phys. A 47, No 5–6 (1928), 446–448.

    MATH  Google Scholar 

  20. G.B. Folland and E.M. Stein, Hardy Spaces on Homogeneous Groups, Ser. Mathematical Notes 28, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo (1982).

    MATH  Google Scholar 

  21. I.M. Gelfand, Some questions of analysis and differential equations. Amer. Math. Soc. Transl. 26 (1963), 201–219.

    MathSciNet  Google Scholar 

  22. R. Gorenflo, F. Mainardi, Signalling problem and Dirichlet-Neumann map for time-fractional diffusion-wave equation. Matimyas Mat. 21 (1998), 109–118.

    MathSciNet  MATH  Google Scholar 

  23. R. Gorenflo, F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 299 (2009), 400–415.

    MathSciNet  MATH  Google Scholar 

  24. A. Haimi and H. Hedenmalm, The polyanalytic Ginibre ensembles. J. Stat. Phys. 153, No 1 (2013), 10–47.

    MathSciNet  MATH  Google Scholar 

  25. B. Helffer and J. Nourrigat, Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué. Comm. Partial Diff. Equations 4, No 8 (1979), 899–958.

    MATH  Google Scholar 

  26. B. Helffer and D. Robert, Asymptotique des niveaux ďénergie pour des hamiltoniens à un degré de liberté. Duke Math. J. 49, No 4 (1982), 853–868.

    MathSciNet  MATH  Google Scholar 

  27. A. Hulanicki, J. W. Jenkins, J. Ludwig, Minimum eigenvalues for positive, Rockland operators. Proc. Amer. Math. Soc. 94 (1985), 718–720.

    MathSciNet  MATH  Google Scholar 

  28. H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Computers and Math. with Appl. 64 (2012), 3377–3388.

    MathSciNet  MATH  Google Scholar 

  29. E. Karimov, M. Mamchuev, M. Ruzhansky, Non-local initial problem for second order time-fractional and space-singular equation. Hokkaido Math. J.. To appear (2018).

    Google Scholar 

  30. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations., Elsevier, North-Holland (2006).

    MATH  Google Scholar 

  31. M. Kirane, B. Samet, B.T. Torebek, Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data. Electr. J. Diff. Equations 2017 (2017), 1–13.

    MathSciNet  MATH  Google Scholar 

  32. M. Kirane, B.T. Torebek, Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations. Fract. Calc. Appl. Anal. 22, No 2 (2019), 358–378; DOI: 10.1515/fca-2019-0022; https://www.degruyter.com/view/journals/fca/22/2/fca.22.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  33. A. Kubica, M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fract. Calc. Appl. Anal. 21, No 2 (2018), 276–311; DOI: 10.1515/fca-2018-0018; https://www.degruyter.com/view/journals/fca/21/2/fca.21.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  34. L. Landau, Diamagnetismus der Metalle. Z. Phys. A 64, No 9–10 (1930), 629–637.

    MATH  Google Scholar 

  35. Z. Li, Y. Liu, M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl. Math. Comput. 257 (2015), 381–397.

    MathSciNet  MATH  Google Scholar 

  36. Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Computers and Math. with Appl. 73 (2017), 96–108.

    MathSciNet  MATH  Google Scholar 

  37. Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24 (1999), 207–233.

    MathSciNet  MATH  Google Scholar 

  38. Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223.

    MathSciNet  MATH  Google Scholar 

  39. Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538–548.

    MathSciNet  MATH  Google Scholar 

  40. F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. A. Carpinteri, F. Mainardi (Eds.) Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York (1997), 291–348.

    MATH  Google Scholar 

  41. A.M. Nakhushev, Fractional Calculus and its Applications., Fizmatlit, Moscow (2003).

    MATH  Google Scholar 

  42. M.A. Naimark, Linear Differential Operators. Ungar, N. York (1968).

    MATH  Google Scholar 

  43. F. Nicola, L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces. Ser. Pseudo-Differential Operators. Theory and Applications 4, Birkhäuser Verlag, Basel (2010).

    MATH  Google Scholar 

  44. R.R. Nigmatullin, The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133 (1986), 425–430.

    Google Scholar 

  45. K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York (1974).

    MATH  Google Scholar 

  46. M. Reed, B. Simon, Methods of Modern Mathematical Physics. Ser. Functional Analysis 1 (Revised and Enlarged Ed.), Academic Press (1980).

    MATH  Google Scholar 

  47. C. Rockland, Hypoellipticity on the Heisenberg group-representation-theoretic criteria. Trans. Amer. Math. Soc. 240 (1978), 1–52.

    MathSciNet  MATH  Google Scholar 

  48. L.P. Rothschild, E.M. Stein, Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976), 247–320.

    MathSciNet  MATH  Google Scholar 

  49. M. Ruzhansky, D. Suragan, N. Yessirkegenov, Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces. Fract. Calc. Appl. Anal. 21, No 3 (2018), 577–612; DOI: 10.1515/fca-2018-0032; https://www.degruyter.com/view/journals/fca/21/2/fca.21.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  50. M. Ruzhansky, N. Tokmagambetov, Nonharmonic analysis of boundary value problems. Int. Math. Res. Not. IMRN 12 (2016), 3548–3615.

    MathSciNet  MATH  Google Scholar 

  51. M. Ruzhansky, N. Tokmagambetov, Nonharmonic analysis of boundary value problems without WZ condition. Math. Model. Nat. Phenom. 12 (2017), 115–140.

    MathSciNet  MATH  Google Scholar 

  52. M. Ruzhansky, N. Tokmagambetov, Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. Lett. Math. Phys. 107 (2017), 591–618.

    MathSciNet  MATH  Google Scholar 

  53. M. Ruzhansky, N. Tokmagambetov, On a very weak solution of the wave equation for a Hamiltonian in a singular electromagnetic field. Math. Notes 103 (2018), 856–858.

    MathSciNet  MATH  Google Scholar 

  54. M. Ruzhansky, N. Tokmagambetov, Wave equation for operators with discrete spectrum and irregular propagation speed. Arch. Ration. Mech. Anal. 226, No 3 (2017), 1161–1207.

    MathSciNet  MATH  Google Scholar 

  55. M. Ruzhansky, N. Tokmagambetov, B.T. Torebek, Bitsadze-Samarskii type problem for the integro-differential diffusion-wave equation on the Heisenberg group. Integr. Transf. Spec. Funct. 31 (2020), 1–9.

    MathSciNet  MATH  Google Scholar 

  56. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382 (2011), 426–447.

    MathSciNet  MATH  Google Scholar 

  57. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam (1993).

    MATH  Google Scholar 

  58. W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144.

    MathSciNet  MATH  Google Scholar 

  59. N. Tokmagambetov, T.B. Torebek, Fractional Analogue of Sturm–Liouville Operator. Documenta Math. 21 (2016), 1503–1514.

    MathSciNet  MATH  Google Scholar 

  60. N. Tokmagambetov, B.T. Torebek, Green’s formula for integro–differential operators. J. Math. Anal. Appl. 468, No 1 (2018), 473–479.

    MathSciNet  MATH  Google Scholar 

  61. N. Tokmagambetov, B.T. Torebek, Fractional Sturm–Liouville equations: Self–adjoint extensions. Complex Anal. and Operator Theory 13, No 5 (2019), 2259–2267.

    MathSciNet  MATH  Google Scholar 

  62. N. Tokmagambetov, B.T. Torebek, Anomalous diffusion phenomena with conservation law for the fractional kinetic process. Math. Methods in Appl. Sci. 41, No 17 (2018), 8161–8170.

    MathSciNet  MATH  Google Scholar 

  63. B.T. Torebek, R. Tapdigoglu, Some inverse problems for the nonlocal heat equation with Caputo fractional derivative. Math. Methods in Appl. Sci. 40, No 18 (2017), 6468–6479.

    MathSciNet  MATH  Google Scholar 

  64. W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1986), 2782–2785.

    MathSciNet  MATH  Google Scholar 

  65. R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj 52 (2009), 1–18.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281 Building S8 B, 9000, Ghent, Belgium

    Michael Ruzhansky & Berikbol T. Torebek

  2. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky & Niyaz Tokmagambetov

  3. Al–Farabi Kazakh National University, 71 Al–Farabi ave., Almaty, 050040, Kazakhstan

    Niyaz Tokmagambetov & Berikbol T. Torebek

  4. Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty, 050010, Kazakhstan

    Niyaz Tokmagambetov & Berikbol T. Torebek

Authors
  1. Michael Ruzhansky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Niyaz Tokmagambetov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Berikbol T. Torebek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Ruzhansky.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ruzhansky, M., Tokmagambetov, N. & Torebek, B.T. On a Non–Local Problem for a Multi–Term Fractional Diffusion-Wave Equation. Fract Calc Appl Anal 23, 324–355 (2020). https://doi.org/10.1515/fca-2020-0016

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2020-0016

MSC 2010

Key Words and Phrases

Search

Navigation