On some boundary value problems for a~third order loaded integro-differential equation with real parameters

U. I. Baltaeva

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2012), pp. 3-12

Résumé

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.

Détail

Keywords: loaded equation, equations of mixed type, integro-differential equation, integral equation with a shift, Bessel functions.