Advances in Operator Theory

The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications

Allaberen Ashyralyev and Abdulgafur Taskin

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Abstract

In this study, the structure of fractional spaces generated by the two-dimensional neutron transport operator $A$ defined by formula $Au=\omega_{1}\frac{\partial u}{\partial x}+\omega _{2}\frac{\partial u}{\partial y}$ is investigated. The positivity of $A$ in $C\left( \mathbb{R}^{2}\right)$ and $L_{p}\left( \mathbb{R}^{2}\right)$, $1\leq p \lt \infty$, is established. It is established that, for any $0 \lt \alpha \lt 1$ and $1\leq p \lt \infty$, the norms of spaces $E_{\alpha ,p}\left( L_{p}\left( \mathbb{R}^{2}\right), A\right)$ and $E_{\alpha }\left( C\left( \mathbb{R}^{2}\right), A\right) , W_{p}^{\alpha } \left( \mathbb{R}^{2}\right)$ and $C^{\alpha }\left( \mathbb{R}^{2}\right)$ are equivalent, respectively. The positivity of the neutron transport operator in Hölder space $C^{\alpha }\left( \mathbb{R}^{2}\right)$ and Slobodeckij space $W_{p}^{\alpha }\left( \mathbb{R}^{2}\right)$ is proved. In applications, theorems on the stability of Cauchy problem for the neutron transport equation in Hölder and Slobodeckij spaces are provided.

Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 140-155.

Dates
Received: 12 November 2017
Accepted: 18 April 2018
First available in Project Euclid: 10 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1525917619

Digital Object Identifier
doi:10.15352/aot.1711-1261

Mathematical Reviews number (MathSciNet)
MR3867338

Zentralblatt MATH identifier
06946447

Subjects
Primary: 47B65: Positive operators and order-bounded operators
Secondary: 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx} 35K30: Initial value problems for higher-order parabolic equations 34B27: Green functions

Keywords
Neutron transport operator fractional space Slobodeckij space positive operator

Citation

Ashyralyev, Allaberen; Taskin, Abdulgafur. The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications. Adv. Oper. Theory 4 (2019), no. 1, 140--155. doi:10.15352/aot.1711-1261. https://projecteuclid.org/euclid.aot/1525917619


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References

  • S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math. 16 (1963), 121–239.
  • A. Ashyralyev, A survey of results in the theory of fractional spaces generated by positive operators, TWMS J. Pure Appl. Math. 6 (2015), no. 2, 129–157.
  • A. Ashyralyev and S. Akturk, Positivity of a one-dimensional difference operator in the half-line and its applications, Appl. Comput. Math. 14 (2015), no. 2, 204–220.
  • A. Ashyralyev and A. Taskin, Structure of fractional spaces generated by the two dimensional neutron transport operator, AIP Conf. Proc. 1759 (2016), 661–665.
  • A. Ashyralyev and F. S. Tetikoglu, A note on fractional spaces generated by the positive operator with periodic conditions and applications, Bound. Value Probl. 2015, 2015:31, 17 pp.
  • A. Ashyralyev and P. E. Sobolevskii, Well-posedness of parabolic difference equations, Operator Theory: Advances and Applications vol. 69, Birkhäuser, Verlag, Basel, Boston, Berlin, 1994.
  • A. Ashyralyev, N. Nalbant, and Y. Sozen, Structure of fractional spaces generated by second order difference operators, J. Franklin Inst. 351 (2014), no. 2, 713–731.
  • H. O. Fattorini, Second order linear differential equations in Banach spaces, Elsevier Science Publishing Company, North-Holland, Amsterdam, 1985.
  • S. G. Krein, Linear differential equations in Banach space, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 29. American Mathematical Society, Providence, R.I., 1971.
  • V. I. Lebedova and P. E. Sobolevskii, Spectral properties of the transfer operator with constant coefficients in $L_{p}\left( R^{n}\right) (~1\leq p<\infty) $ spaces, Voronezh. Gosud. Univ. 1983, 54 pages. Deposited VINITI, 02.06.1983, no. 2958-83, (Russian) 1983.
  • V. I. Lebedova, Spectral properties of the transfer operator of neutron in $C(\Omega, C(R^{n}))$ spaces, Qualitative and Approximate Methods for Solving Operator Equations, Yaroslavil, (Russian) 9 (1984), 44–51.
  • E. Lewis and W. Miller Computational methods of neutron transport, American Nuclear Society, USA, 1993.
  • G. I. Marchuk and V. I. Lebedev, Numerical methods in the theory of neutron transport, Taylor and Francis, USA, 1986.
  • M. Mokhtar-Kharroubi, Mathematical topics in neutron transport theory, New aspects, World Scientific, Singapore and River Edge, N.J., 1997.
  • V. Shakhmurov and H. Musaev, Maximal regular convolution-differential equations in weighted Besov spaces, Appl. Comput. Math. 16 (2017), no. 2, 190–200.
  • P. E. Sobolevskii, Some properties of the solutions of differential equations in fractional spaces, Trudy Nauchn.-Issled. Inst. Mat. Voronezh. Gos. Univ., (Russian) 74 (1975), 68–76.
  • H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.
  • V. I. Zhukova, Spectral properties of the transfer operator, Trudy Vsyesoyuznoy Nauchno-Prakticheskoy Konferensii, Chita 5 (2000), no. 1, 170–174.
  • V. I. Zhukova and L. N. Gamolya, Investigation of spectral properties of the transfer operator, Dalnovostochniy Matematicheskiy Zhurnal, (Russian) 5 (2004), no. 1, 158–164.