Model three-dimensional Volterra type integral equations with boundary singular, weak singular and strong singular kernels

L. N. Rajabova   ; M. B. Khushvakhtzoda

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 2, pp. 103-112

Résumé

In this paper, we study a three-dimensional model Volterra type integral equation with boundary weakly special, special and strongly special kernels in the domain $\Omega=\{(x,y,z):\ 0\leq a<x<\infty,\ 0\leq b<y<b_{0},\ 0\leq c<z<c_{0}\}$, which we will call a rectangular pipe. In the case when the coefficients of the equation are interconnected, the solution of the equation is sought in the class of continuous functions in $\Omega$ vanishing with a certain asymptotic behavior on special domains. It is proved that, under certain conditions, the problem of finding a solution to a three-dimensional integral equation of the Volterra type with boundary weakly special, special and strongly special kernels is reduced to solving one-dimensional integral equations of the Volterra type with special boundary kernels. Note that when solving this integral equation, connections of these equations with first-order differential equations with weakly singular, singular and strongly singular coefficients are used. Then it is established that there is no need to require differentiability from the obtained solution and the right-hand side, it is sufficient that the right-hand side of the three-dimensional integral equation with boundary special, weakly special, and strongly special kernels is continuous and vanishes with certain asymptotics on special domains. It is proved that, depending on the sign of the coefficients of the equation, the explicit solution of a three-dimensional Volterra-type model integral equation with special kernels can contain from one to three arbitrary functions of two variables, and the case is also determined when the solution of the integral equation is unique.

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