Weekly Ill-posed integral geometry problems of Volterra type in three-dimensional space

Yıl 2024, Cilt: 26 Sayı: 2, 472 – 478, 15.07.2024

Akram Begmatov Alisher Ismoilov

https://doi.org/10.25092/baunfbed.1462616

Öz

In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space.

Anahtar Kelimeler

Integral geometry problem, Fourier transform, uniqueness theorem, Volterra integral equation.

Kaynakça

• Lavrentyev M.M. and Savelyev L.Y., Operator Theory and Ill-Posed Problems. Moscow: Publ House of the Inst Math (2010).
• Romanov V. G. “Reconstructing a function by means of integrals along a family of curves”, Soviet Math. Dokl., 8:5, 923-925 (1967).
• Romanov V.G. Some inverse problems for hyperbolic equations. — Novosibirsk: Nauka, 164 p. (1972). (in Russian).
• Buchheim A.L. On Some Problems of Integral Geometry. Siberian Math J, 13 (1),34 (1972).
• Yon F. Plane waves and spherical means as applied to partial differential equations. – M.: Izd-vo inostr. lit., (1958), 158 p.
• Lavrentiev M.M. Inverse problems and special operator equations of the first kind // Mezhdunar. mat. kongress v v Nitstse, 1970. – M.: Nauka, S. 130-136 (1972). (in Russian).
• Begmatov Akram H. “Two classes of weakly ill-posed problems of integral geometry on the plane”, Siberian Math. J., 36:2, 213–218 (1995).
• Begmatov Akram H. “The integral geometry problem for a family of cones in the n-dimensional space”, Siberian Math. J., 37:3, 430–435 (1996).
• Begmatov Akram. H. “Volterra problems of integral geometry in the plane for curves with singularities”, Siberian Math. J., 38:4, 723-737 (1997).
• Begmatov Akram Kh., Ismoilov A.S. Restoring the function set by integrals for the family of parabolas on the plane // Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, Vol. 3, issue 2., pp. 246-254 2020.
• Begmatov A.Kh., Ismoilov A.S. Оn a problem of integral geometry over a family of parabolas with perturbation. Journal of the Balkan Tribological Association 27 (4), 497-509 (2021).
• Begmatov A.Kh., Ismoilov A.S., Khudayberdiev D.G. Weakly ill-posed problems of integral geometry on the plane with perturbation. Journal of the Balkan Tribological Association, Vol. 29, No 3, 273–289 (2023).
• Tricomi F. Integral equations / F. Trikomi. – M.: Izdatel'stvo inostrannoy literatury, (1960) 301 p.

Üç boyutlu uzayda Volterra tipindeki, zayıf nokorrekt integral geometri problemi

Yıl 2024, Cilt: 26 Sayı: 2, 472 – 478, 15.07.2024

Akram Begmatov Alisher Ismoilov

https://doi.org/10.25092/baunfbed.1462616

Öz

Bu makalede küre ailesinden uzaydaki bır fonksiyonnu kurtarılma problemi ele alınmaktadır. Volterranin önce birinci sonra ikinci tür integral denklemine getırmek yoluyla kanıtlanır. Furier değiştirme yöntemleri de kullanılmaktadır. Üç boyutlu uzayda Volterra tipindeki operatör denklemlerinin bazı yeni sınıfları için teklik teoremleri kanıtlanmıştır.

Anahtar Kelimeler

İntegral geometri problemi, Fourier değiştirilmesi, teklik teoremi, Volterra integral denklemi

Kaynakça

• Lavrentyev M.M. and Savelyev L.Y., Operator Theory and Ill-Posed Problems. Moscow: Publ House of the Inst Math (2010).
• Romanov V. G. “Reconstructing a function by means of integrals along a family of curves”, Soviet Math. Dokl., 8:5, 923-925 (1967).
• Romanov V.G. Some inverse problems for hyperbolic equations. — Novosibirsk: Nauka, 164 p. (1972). (in Russian).
• Buchheim A.L. On Some Problems of Integral Geometry. Siberian Math J, 13 (1),34 (1972).
• Yon F. Plane waves and spherical means as applied to partial differential equations. – M.: Izd-vo inostr. lit., (1958), 158 p.
• Lavrentiev M.M. Inverse problems and special operator equations of the first kind // Mezhdunar. mat. kongress v v Nitstse, 1970. – M.: Nauka, S. 130-136 (1972). (in Russian).
• Begmatov Akram H. “Two classes of weakly ill-posed problems of integral geometry on the plane”, Siberian Math. J., 36:2, 213–218 (1995).
• Begmatov Akram H. “The integral geometry problem for a family of cones in the n-dimensional space”, Siberian Math. J., 37:3, 430–435 (1996).
• Begmatov Akram. H. “Volterra problems of integral geometry in the plane for curves with singularities”, Siberian Math. J., 38:4, 723-737 (1997).
• Begmatov Akram Kh., Ismoilov A.S. Restoring the function set by integrals for the family of parabolas on the plane // Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, Vol. 3, issue 2., pp. 246-254 2020.
• Begmatov A.Kh., Ismoilov A.S. Оn a problem of integral geometry over a family of parabolas with perturbation. Journal of the Balkan Tribological Association 27 (4), 497-509 (2021).
• Begmatov A.Kh., Ismoilov A.S., Khudayberdiev D.G. Weakly ill-posed problems of integral geometry on the plane with perturbation. Journal of the Balkan Tribological Association, Vol. 29, No 3, 273–289 (2023).
• Tricomi F. Integral equations / F. Trikomi. – M.: Izdatel'stvo inostrannoy literatury, (1960) 301 p.

Toplam 13 adet kaynakça vardır.

Ayrıntılar

Kaynak Göster

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