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Inverse source problems for positive operators, I : hypoelliptic diffusion and subdiffusion equations

Michael Ruzhansky (UGent) , Niyaz Tokmagambetov (UGent) and Berikbol Torebek (UGent)
(2019) JOURNAL OF INVERSE AND ILL-POSED PROBLEMS. 27(6). p.891-911
Author
Organization
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, UNKNOWN SOURCE-TERM, HEISENBERG-GROUP, TEMPERATURE DISTRIBUTION, WAVE-EQUATION, HEAT-EQUATION, INEQUALITIES, DENSITY

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MLA
Ruzhansky, Michael, et al. “Inverse Source Problems for Positive Operators, I : Hypoelliptic Diffusion and Subdiffusion Equations.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, vol. 27, no. 6, 2019, pp. 891–911, doi:10.1515/jiip-2019-0031.
APA
Ruzhansky, M., Tokmagambetov, N., & Torebek, B. (2019). Inverse source problems for positive operators, I : hypoelliptic diffusion and subdiffusion equations. JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, 27(6), 891–911. https://doi.org/10.1515/jiip-2019-0031
Chicago author-date
Ruzhansky, Michael, Niyaz Tokmagambetov, and Berikbol Torebek. 2019. “Inverse Source Problems for Positive Operators, I : Hypoelliptic Diffusion and Subdiffusion Equations.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS 27 (6): 891–911. https://doi.org/10.1515/jiip-2019-0031.
Chicago author-date (all authors)
Ruzhansky, Michael, Niyaz Tokmagambetov, and Berikbol Torebek. 2019. “Inverse Source Problems for Positive Operators, I : Hypoelliptic Diffusion and Subdiffusion Equations.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS 27 (6): 891–911. doi:10.1515/jiip-2019-0031.
Vancouver
1.
Ruzhansky M, Tokmagambetov N, Torebek B. Inverse source problems for positive operators, I : hypoelliptic diffusion and subdiffusion equations. JOURNAL OF INVERSE AND ILL-POSED PROBLEMS. 2019;27(6):891–911.
IEEE
[1]
M. Ruzhansky, N. Tokmagambetov, and B. Torebek, “Inverse source problems for positive operators, I : hypoelliptic diffusion and subdiffusion equations,” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, vol. 27, no. 6, pp. 891–911, 2019.
@article{8640174,
  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.}},
  author       = {{Ruzhansky, Michael and Tokmagambetov, Niyaz and Torebek, Berikbol}},
  issn         = {{0928-0219}},
  journal      = {{JOURNAL OF INVERSE AND ILL-POSED PROBLEMS}},
  keywords     = {{Heat equation,time-fractional diffusion equation,inverse problem,self-adjoint operator,Rockland operator,UNKNOWN SOURCE-TERM,HEISENBERG-GROUP,TEMPERATURE DISTRIBUTION,WAVE-EQUATION,HEAT-EQUATION,INEQUALITIES,DENSITY}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{891--911}},
  title        = {{Inverse source problems for positive operators, I : hypoelliptic diffusion and subdiffusion equations}},
  url          = {{http://dx.doi.org/10.1515/jiip-2019-0031}},
  volume       = {{27}},
  year         = {{2019}},
}
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