Explicit solutions of a two-dimensional integral equation of Volterr type with boundary singularity and strongly singular line when the roots of the characteristic equations are real, different and equal

Abstract

The paper studies a two-dimensional integral equation of Volterra type with singular and strongly singular
boundary lines. The solution of a two-dimensional integral equation of the Volterra type with singular kernels is sought in
the class of continuous functions that vanish on the boundary lines.
In the case when the roots of the characteristic equations are real, different and equal, the parameters of the equations
are related to each other in a certain way, depending on the roots of the characteristic equations and the sign of the
parameters of the integral equation, explicit solutions of the integral equation are found.
It is proved that the solutions of a two-dimensional integral equation, depending on the sign of the parameters, can
contain from one to four arbitrary continuous functions. The cases are determined when the solution to the integral
equation is unique.