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Section
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Mathematics
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Title
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On some boundary value problems for a third order loaded integro-differential equation with real parameters
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Author(-s)
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Baltaeva U.I.a
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Affiliations
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Urgench State Universitya
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Abstract
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We consider a linear loaded integro-differential equation with hyperbolic operator $${\frac{{\partial}} {{\partial x}}}\left( {u_{xx} - u_{yy} - \lambda u} \right) = \mu {\sum\limits_{i = 1}^{n} {a_{i} (x)}}D_{0x}^{\alpha _{i}} u_{y} (x,0),$$ and loaded integro-differential equation with mixed operator $${\frac{{\partial}} {{\partial x}}}\left( {u_{xx} - {\frac{{1 -{\rm sgn}\, y}}{{2}}}u_{yy} - {\frac{{1 + {\rm sgn}\, y}}{{2}}}u_{y} - \lambda u}\right) = \mu {\sum\limits_{i = 1}^{n} {a_{i} (x)}} D_{0x}^{\alpha_{i}} u_{y} (x,0),$$ where $D_{0x}^{\alpha _{i} }$ is integro-differential operator (in the sense of Riemann-Liouville), $a_{i} (x)$ are coefficients, $\lambda ,\mu$ are given real parameters, and $\lambda > 0.$
In this paper, the unique solvability of the boundary value problems (of a type similar to the Darboux problem and the Tricomi problem) of a loaded third order integro-differential equation with hyperbolic and parabolic-hyperbolic operators is proved by method of integral equations. The problem is similarly reduced to a Volterra integral equation with a shift. Under sufficient conditions for given functions and coefficients the unique solvability is proved for the solution of obtained integral equations.
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Keywords
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loaded equation, equations of mixed type, integro-differential equation, integral equation with a shift, Bessel functions
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UDC
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517.956
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MSC
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35M10, 35L35
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DOI
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10.20537/vm120301
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Received
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7 April 2012
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Language
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Russian
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Citation
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Baltaeva U.I. On some boundary value problems for a third order loaded integro-differential equation with real parameters, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 3, pp. 3-12.
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References
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- Nakhushev A.M. Darboux problem for a one degenerating loaded integro-differential equation of the second order, Differ. Uravn., 1976, vol. 12, no. 1, pp. 103-108.
- Nakhushev A.M. Uravneniya matematicheskoi biologii (The equations of mathematical biology), Moscow: Vysshaya shkola, 1995, 301 p.
- Kaziev V.M. About Darboux problem for a one loaded integro-differential equation of the second order, Differ. Uravn., 1978, vol. 14, no. 1, pp. 181-184.
- Dzhuraev T.D. Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipa (Boundary value problems for the equation of mixed and mixed-composite type), Tashkent: Fan, 1971, 240 p.
- Dzhurayev T.D., Sopuev A., Mamazhonov M. Kraevye zadachi dlya uravnenii parabolo-giperbolicheskogo tipa (Boundary value problems for the parabolic-hyperbolic type equations), Tashkent: Fan, 1986, 576 p.
- Salakhitdinov M.S. Uravnenie smeshanno-sostavnogo tipa (Equation of mixed-composite type), Tashkent: Fan, 1974, 156 p.
- Mikhlin S.G. Lektsii po lineinym integral`nym uravneniyam (Lecture on linear integral equations), Мoscow: Fizmatgiz, 1959, 224 p.
- Sabitov K.B. Construction of solution explicit form of Darboux problem for telegraph equation and application for inversion of integral equations, Differ. Uravn., 1990, vol. 26, no. 6, pp. 1023-1032.
- Baltaeva U.I. On some boundary value problems for a third order loaded equation with a parabolic-hyperbolic operator, Uzbek. Mat. Journal, Tashkent, 2007, no. 3, pp. 26-37.
- Baltaeva U.I. Boundary value problems for the loaded third order equations of the mixed type, Cand. Sci. (Phys.–Math.) Dissertation, Tashkent, 2008, 111 p.
- Baltaeva U.I., Islomov B. Boundary value problems for the loaded third order differential equations of the hyperbolic and mixed types, Ufim. Mat. J., 2011, vol. 3, no. 3, pp. 15-25.
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