Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 1(17). C. 22-32. ISSN 2079-6641

DOI: 10.18454/2079-6641-2017-17-1-22-32

MSC 76W05, 86A25

ABOUT A PROBLEM FOR THE DEGENERATING MIXED TYPE EQUATION FRACTIONAL DERIVATIVE

B. I. Islomov¹, N. K. Ochilova²

¹National University of Uzbekistan, 100125, Tashkent, Vuzgorodok, Universitetskaya str.4, Uzbekistan
²Tashkent financial institute,100000, Tashkent, Amir Temur-57.Uzbekistan
E-mail: nargiz.ochilova@gmail.com

The existence and the uniqueness of solution of local problem for degenerating mixed type equation is investigated. Considering parabolic-hyperbolic equation involve the Caputo fractional derivative. The uniqueness of solution is proved using the method of the extremume principle and integral energy, the existence is proved by the method of integral equations.

Keywords: boundary value problem, degenerating equation, parabolic-hyperbolic type, Gauss hypergeometric function, Cauchy problem, existence and uniqueness of solution, a principle an extremum, method of integral equations, Caputo fractional derivative.

© Islomov B. I., Ochilova N. K., 2017

УДК 517.956

О ЗАДАЧЕ ДЛЯ ВЫРОЖДАЮЩЕГОСЯ УРАВНЕНИЯ СМЕШАННОГО ТИПА С ДРОБНОЙ ПРОИЗВОДНОЙ

Б. И. Исломов¹, Н. К. Очилова²

¹Национальный университетет Узбекистана, 100125, Ташкент, Вуз-городок, ул. Университетская 4, Республика Узбекистан
²Ташкенский финансовый институт, 100000, Ташкент, ул. Амира Тимура, 57, Республика Узбекистан
E-mail: nargiz.ochilova@gmail.com

Исследуется существование и единственность решения локальной задачи для вырождающегося уравнения смешанного типа. Рассматривается параболическо-гиперболическое уравнение с дробной производной Капуто. Единственность решения доказана с использованием экстремального принципа и интеграла энергии, существование доказано методом интегральных уравнений.

Ключевые слова: краевая задача, вырождающееся уравнение, параболо-гиперболический тип, гипергеометрическая функция Гаусса, задача Коши, существование и единственность решения, принцип экстремума, метод интегральных уравнений, дробная производная Капуто.

© Исломов Б. И., Очилова Н. К., 2017

References

  1. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations. V. 204, North-Holland Mathematics Studies, Amsterdam, 2006.
  2. Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  3. Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999.
  4. Samko S. G., Kilbas A. A., Marichev O. I., Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, 1993.
  5. Marichev O. I., Kilbas A. A., Repin A. A., Boundary value problems for partial differential equations with discounting coefficients. (In Russian), Izdat. Samar. Gos.Ekonom. Univ., Samara, 2008.
  6. Repin O. A., Boundary value problems with shift for equations of hyperbolic and mixed type. (In Russian), Saratov Univ., Saratov, 1992.
  7. Abdullaev O. Kh., “About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives”, International journal of differential equations, 2016, 9815796.
  8. Kilbas A. A. and Repin O. A., “An analog of the Bitsadze-Samarskii problem for a mixed type equation with a fractional derivative”, Differential equations, 39:5 (2003), 674–680.
  9. Kilbas A. A., Repin O. A., “An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative”, Fractional Calculus & Applied Analysis, 13:1 (2010), 69–84.
  10. Pskhu A.V., Uravneniye v chasnykh proizvodnykh drobnogo poryadka. (Russian) [Partial differential equation of fractional order], Nauka, Moscow, 2005, 200 pp.
  11. Ochilova N. K., “Study the unique solvability of boundary value problem of Frankl for mixed-type equation degenerate on the boundary and within the region”, Vestnik KRAUNC. Fiz.-Mat. Nauki — Bulletin KRASEC. Phys. & Math. Sci., 2014, №1(8), 20-32.
  12. Smirnov M. M., Mixed type equations, Nauka, Moscow, 2000.
  13. Pskhu A.V., “Solution of boundary value problems fractional diffusion equation by the Green function method”, Differential equation, 39 (2003), 1509-1513.

 

References (GOST)

  1. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam. (2006).
  2. Miller K.S., Ross B. An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, (1993).
  3. Podlubny I. Fractional Differential Equations, Academic Press, New York, (1999).
  4. Samko S. G., Kilbas A. A., Marichev O. I. Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, PA, (1993).
  5. Marichev O. I., Kilbas A. A., Repin A. A. Boundary value problems for partial differential equations with discounting coefficients. (In Russian). Izdat.Samar.Gos.Ekonom. Univ., Samara (2008)
  6. Repin O. A. Boundary value problems with shift for equations of hyperbolic and mixed type. (In Russian). Saratov Univ., Saratov (1992)
  7. Abdullaev O.Kh. About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives”, International journal of differential equations., vol. 2016, Article ID 9815796, 12 pages.
  8. Kilbas A. A. and Repin O. A. An analog of the Bitsadze-Samarskii problem for a mixed type equation with a fractional derivative,” Differential equations. (2003). vol. 39, no. 5, pp. 674–680.
  9. Kilbas A. A., Repin O. A. “An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative,” Fractional Calculus & Applied Analysis. (2010) vol. 13, no. 1, pp. 69–84.
  10. Pskhu A.V. Uravneniye v chasnykh proizvodnykh drobnogo poryadka. (Russian) [Partial differential equation of fractional order]. Nauka. Moscow. 2005. 200 p.
  11. Ochilova N. K. Study the unique solvability of boundary value problem of Frankl for mixedtype equation degenerate on the boundary and within the region. Vestnik KRAUNC. Fiz.- Mat. Nauki — Bulletin KRASEC. Phys. & Math. Sci. 2014. №1(8). pp. 20-32.
  12. Smirnov M. M. Mixed type equations. Moscow. Nauka. (2000).
  13. Pskhu A.V. Solution of boundary value problems fractional diffusion equation by the Green function method. Differential equation, 39. (2003), pp. 1509-1513.

Для цитирования: Islomov B. I., Ochilova N. K. About a problem for the degenerating mixed type equation fractional derivative // Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 1(17). C. 22-32. DOI: 10.18454/2079-6641-2017-17-1-22-32

For citation: Islomov B. I., Ochilova N. K. About a problem for the degenerating mixed type equation fractional derivative, Vestnik KRAUNC. Fiz.-mat. nauki. 2017, 17: 1, 22-32. DOI: 10.18454/2079-6641-2017-17-1-22-32

Поступила в редакцию / Original article submitted: 25.12.2016


islomov

     Islamov  Bozor  – Dr. Si. (Phys & Math), Professor, Department of Differential equations and mathematical physics, National University of Uzbekistan named by Mirzo Ulugbek, Tashkent, Rep  ublic of Uzbekistan.

   Исломов Бозор – доктор физико-математических наук, профессор кафедры дифференциальных уравнений и математической физики, Национальный Университет Узбекистана имени Мирзо Улугбека, г. Ташкент, республика Узбекистан.

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Ochilova

  Ochilova Nargiza Komilovna — assistant professor of mathematical analysis of physical and mathematical fakultetaTashkenskogo Pedagogical University. Nizami, Tashkent, Republic of Uzbekistan.

 Очилова Наргиза Комиловна — доцент кафедры математического анализа физико-математического факультетаТашкенского педагогического университета им. Низами, г. Ташкент, Република Узбекистана.

     

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