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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, Issue 3, Pages 3–12
(Mi vuu331)
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This article is cited in 3 scientific papers (total in 3 papers)
MATHEMATICS
On some boundary value problems for a third order loaded integro-differential equation with real parameters
U. I. Baltaeva Urgench State University, Urgench, Uzbekistan
Abstract:
We consider a linear loaded integro-differential equation with hyperbolic operator
$$
\frac\partial{\partial x}\left(u_{xx}-u_{yy}-\lambda u\right)=\mu\sum_{i=1}^na_i(x)D_{0x}^{\alpha _i}u_y(x,0),
$$
and loaded integro-differential equation with mixed operator
$$
\frac\partial{\partial x}\left(u_{xx}-\frac{1-\operatorname{sgn}y}2u_{yy}-\frac{1+\operatorname{sgn}y}2u_y-\lambda u\right)=\mu\sum_{i=1}^na_i(x)D_{0x}^{\alpha_i}u_y(x,0),
$$
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.
Received: 07.04.2012
Citation:
U. I. Baltaeva, “On some boundary value problems for a third order loaded integro-differential equation with real parameters”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 3, 3–12
Linking options:
https://www.mathnet.ru/eng/vuu331 https://www.mathnet.ru/eng/vuu/y2012/i3/p3
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