Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

Online
ISSN
1569-3945
See all formats and pricing
More options …
Volume 27, Issue 6

Issues

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Michael Ruzhansky
  • Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium; and School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Niyaz TokmagambetovORCID iD: https://orcid.org/0000-0001-5725-9740
  • Corresponding author
  • Al-Farabi Kazakh National University, 71 Al-Farabi ave., Almaty 050040; and Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty 050010, Kazakhstan; and Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
  • orcid.org/0000-0001-5725-9740
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Berikbol T. Torebek
  • Al-Farabi Kazakh National University, 71 Al-Farabi ave., Almaty 050040; and Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty 050010, Kazakhstan; and Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-10-15 | DOI: https://doi.org/10.1515/jiip-2019-0031

Abstract

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm–Liouville problems, differential models with involution, fractional Sturm–Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a “cooling function”, and how the involution normally slows down the cooling speed of the rod.

Keywords: Heat equation; time-fractional diffusion equation; inverse problem; self-adjoint operator; Rockland operator

MSC 2010: 35K90; 42A85; 44A35

References

  • [1]

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

  • [2]

    N. K. Bari, Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4) (1951), 69–107. Google Scholar

  • [3]

    K. Beauchard and P. Cannarsa, Heat equation on the Heisenberg group: Observability and applications, J. Differential Equations 262 (2017), no. 8, 4475–4521. CrossrefGoogle Scholar

  • [4]

    L. A. Caffarelli and Y. Sire, On some pointwise inequalities involving nonlocal operators, Harmonic Analysis, Partial Differential Equations and Applications, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham (2017), 1–18. Google Scholar

  • [5]

    J. R. Cannon and P. DuChateau, Structural identification of an unknown source term in a heat equation, Inverse Problems 14 (1998), no. 3, 535–551. CrossrefGoogle Scholar

  • [6]

    M. Chatzakou, J. Delgado and M. Ruzhansky, On a class of anharmonic oscillators, preprint (2018), https://arxiv.org/abs/1811.12566.

  • [7]

    J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems 25 (2009), no. 11, Article ID 115002. Google Scholar

  • [8]

    L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications. Part I: Basic Theory and Examples, Cambridge Stud. Adv. Math. 18, Cambridge University Press, Cambridge, 1990. Google Scholar

  • [9]

    P. M. de Carvalho-Neto and R. Fehlberg, Júnior, Conditions for the absence of blowing up solutions to fractional differential equations, Acta Appl. Math. 154 (2018), 15–29. CrossrefGoogle Scholar

  • [10]

    J. Delgado, M. Ruzhansky and N. Tokmagambetov, Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary, J. Math. Pures Appl. (9) 107 (2017), no. 6, 758–783. CrossrefGoogle Scholar

  • [11]

    B. K. Driver and T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2005), no. 2, 340–365. CrossrefGoogle Scholar

  • [12]

    V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie groups, Progr. Math. 314, Birkhäuser/Springer, Cham, 2016. Google Scholar

  • [13]

    V. Fischer and M. Ruzhansky, Sobolev spaces on graded Lie groups, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 4, 1671–1723. CrossrefGoogle Scholar

  • [14]

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

  • [15]

    G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, 1982. Google Scholar

  • [16]

    K. M. Furati, O. S. Iyiola and M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. Comput. 249 (2014), 24–31. Google Scholar

  • [17]

    I. M. Gel’fand, Some questions of analysis and differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 201–219. Google Scholar

  • [18]

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

  • [19]

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

  • [20]

    B. Helffer and D. Robert, Asymptotique des niveaux d’énergie pour des hamiltoniens à un degré de liberté, Duke Math. J. 49 (1982), no. 4, 853–868. Google Scholar

  • [21]

    A. Hulanicki, J. W. Jenkins and J. Ludwig, Minimum eigenvalues for positive, Rockland operators, Proc. Amer. Math. Soc. 94 (1985), no. 4, 718–720. CrossrefGoogle Scholar

  • [22]

    M. I. Ismailov and M. Çiçek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Appl. Math. Model. 40 (2016), no. 7–8, 4891–4899. CrossrefGoogle Scholar

  • [23]

    B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems 31 (2015), no. 3, Article ID 035003. Google Scholar

  • [24]

    N. Juillet, Diffusion by optimal transport in Heisenberg groups, Calc. Var. Partial Differential Equations 50 (2014), no. 3–4, 693–721. CrossrefGoogle Scholar

  • [25]

    I. A. Kaliev and M. M. Sabitova, Problems of determining the temperature and density of heat sources from the initial and final temperatures, J. Appl. Indust. Math. 4 (2010), no. 3, 332–339. CrossrefGoogle Scholar

  • [26]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science B.V., Amsterdam, 2006. Google Scholar

  • [27]

    M. Kirane and S. A. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. Comput. 218 (2011), no. 1, 163–170. Google Scholar

  • [28]

    M. Kirane, B. Samet and B. T. Torebek, Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data, Electron. J. Differential Equations 2017 (2017), Paper No. 257. Google Scholar

  • [29]

    L. Landau, Diamagnetismus der Metalle, Z. Phys. A 64 (1930), no. 9–10, 629–637. CrossrefGoogle Scholar

  • [30]

    Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam. 24 (1999), no. 2, 207–233. Google Scholar

  • [31]

    F. Mainardi, Waves and Stability in Continuous Media, World Scientific, Singapore, 2000. Google Scholar

  • [32]

    M. A. Naĭmark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space. With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968. Google Scholar

  • [33]

    H. T. Nguyen, D. L. Le and V. T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model. 40 (2016), no. 19–20, 8244–8264. CrossrefGoogle Scholar

  • [34]

    F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces, Pseudo Diff. Oper. 4, Birkhäuser, Basel, 2010. Google Scholar

  • [35]

    R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phis. Stat. Sol. 133 (1986), 299–318. Google Scholar

  • [36]

    I. Orazov and M. A. Sadybekov, On a class of problems of determining the temperature and density of heat sources given initial and final temperature, Sib. Math. J. 53 (2012), no. 1, 146–151. CrossrefGoogle Scholar

  • [37]

    I. Orazov and M. A. Sadybekov, One nonlocal problem of determination of the temperature and density of heat sources, Russian Math. 56 (2012), no. 2, 60–64. CrossrefGoogle Scholar

  • [38]

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

  • [39]

    L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3–4, 247–320. CrossrefGoogle Scholar

  • [40]

    D. Rottensteiner and M. Ruzhansky, Harmonic and Anharmonic Oscillators on the Heisenberg group, preprint (2018), https://arxiv.org/abs/1812.09620.

  • [41]

    M. Ruzhansky and N. Tokmagambetov, Nonharmonic analysis of boundary value problems, Int. Math. Res. Not. IMRN (2016), no. 12, 3548–3615. Google Scholar

  • [42]

    M. Ruzhansky and N. Tokmagambetov, Nonharmonic analysis of boundary value problems without WZ condition, Math. Model. Nat. Phenom. 12 (2017), no. 1, 115–140. CrossrefGoogle Scholar

  • [43]

    M. Ruzhansky and N. Tokmagambetov, Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field, Lett. Math. Phys. 107 (2017), no. 4, 591–618. CrossrefGoogle Scholar

  • [44]

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

  • [45]

    M. Ruzhansky and N. Tokmagambetov, Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups, J. Differential Equations 265 (2018), no. 10, 5212–5236. CrossrefGoogle Scholar

  • [46]

    M. Ruzhansky and N. Tokmagambetov, On a very weak solution of the wave equation for a Hamiltonian in a singular electromagnetic field, Mat. Zametki 103 (2018), no. 5, 790–793. Google Scholar

  • [47]

    K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields 1 (2011), no. 4, 509–518. CrossrefGoogle Scholar

  • [48]

    T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab. 19 (2014), Paper No. 16. Google Scholar

  • [49]

    N. Tokmagambetov and B. T. Torebek, Fractional analogue of Sturm-Liouville operator, Doc. Math. 21 (2016), 1503–1514. Google Scholar

  • [50]

    N. Tokmagambetov and B. T. Torebek, Green’s formula for integro-differential operators, J. Math. Anal. Appl. 468 (2018), no. 1, 473–479. CrossrefGoogle Scholar

  • [51]

    N. Tokmagambetov and B. T. Torebek, Well-posed problems for the fractional Laplace equation with integral boundary conditions, Electron. J. Differential Equations 2018 (2018), Paper No. 90. Google Scholar

  • [52]

    N. Tokmagambetov and B. T. Torebek, Fractional Sturm-Liouville equations: self-adjoint extensions, Complex Anal. Oper. Theory 13 (2019), no. 5, 2259–2267. CrossrefGoogle Scholar

  • [53]

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

  • [54]

    J. Tyson and J. Wang, Heat content and horizontal mean curvature on the Heisenberg group, Comm. Partial Differential Equations 43 (2018), no. 3, 467–505. CrossrefGoogle Scholar

  • [55]

    V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I: Background and Theory, Nonlinear Physical Science, Springer, Heidelberg, 2013. Google Scholar

  • [56]

    V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume II: Application, Nonlinear Physical Science, Springer, Heidelberg, 2013. Google Scholar

  • [57]

    W. Wang, M. Yamamoto and B. Han, Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation, Inverse Problems 29 (2013), no. 9, Article ID 095009. Google Scholar

  • [58]

    H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen, London, 1931. Google Scholar

  • [59]

    Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems 27 (2011), no. 3, Article ID 035010. Google Scholar

  • [60]

    Z. Q. Zhang and T. Wei, Identifying an unknown source in time-fractional diffusion equation by a truncation method, Appl. Math. Comput. 219 (2013), no. 11, 5972–5983. Google Scholar

About the article

Received: 2019-04-25

Accepted: 2019-09-07

Published Online: 2019-10-15

Published in Print: 2019-12-01


The first author was supported in parts by the FWO Odysseus Project, EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The second author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP05130994. The third author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP05131756. No new data was collected or generated during the course of research.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 6, Pages 891–911, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2019-0031.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in