Abstract

We consider a linear loaded integrodifferential 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 integrodifferential 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 integrodifferential operator (in the sense of RiemannLiouville), $a_{i} (x)$ 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 integrodifferential equation with hyperbolic and parabolichyperbolic 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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