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## Archive of Issues

Uzbekistan Urgench
Year
2012
Issue
3
Pages
3-12
 Section Mathematics Title On some boundary value problems for a third order loaded integro-differential equation with real parameters Author(-s) Baltaeva U.I.a Affiliations Urgench State Universitya Abstract 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. Keywords loaded equation, equations of mixed type, integro-differential equation, integral equation with a shift, Bessel functions UDC 517.956 MSC 35M10, 35L35 DOI 10.20537/vm120301 Received 7 April 2012 Language Russian Citation 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. References 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. Full text