Abstract:
In the rectangle $D=\big\{(x,y): a<x<a_{1}, b<y<b_{1}\big\}$ with boundary lines $\Gamma_{1}=\{y=b, a<x<a_{1}\}$, $\Gamma_{2}=\{x=a, b<y<b_{1}\}$ we study a overdetermined system of the Volterra-type integral equations with special lines, which consists of a two-dimensional integral equation and two one-dimensional integral equations. The solution of the overdetermined system of the Volterra-type integral equations with special lines is sought in the class of continuous functions in the rectangle $D$ and vanishing on the lines $\Gamma_{1}$, $\Gamma_{2}$. In the case when the main equation of the studied overdetermined system of integral equations is the first equation of the system and the coefficients of the two-dimensional integral equation are related and not related to each other in a special way, we obtain the conditions of jointness of the equations of the system. In the case when the coefficients of the first equation of the overdetermined system are not related, the solution of the overdetermined system of integral equations is sought in the form of generalized power series. We obtain explicit solutions of the overdetermined system of equations, which, depending on the sign of the coefficients, may contain arbitrary constants, determine the conditions of coexistence of the equations of the system, study the properties of the solutions, set and solve Cauchy-type problems, where the conditions are set on singular manifolds.
Key words:overdetermined system of equations, Volterra type integral equation, special lines, arbitrary constants.
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