EXPLICIT SOLUTION OF ONE-TYPE INTEGRAL VOLTERRA EQUATION ON THE SYMMETRIC INTERVAL WITH A SUM-DIFFERENCE KERNEL

Abstract

Many problems in the theory and practice are reduced to solving the first-kind integral equations with a “weak” kernel, i. e. the kernel goes to the infinity of integrable order when arguments are matching. The success of investigation of such problems often depends on the solution of the explicit equation corresponding to the problem. In some cases, it is possible to get such a solution. In our case, we consider the first-kind equation with a kernel, which represents the square root of a fractional-linear function, on a symmetric interval. Given the equation symmetry, this equation can be reduced to an equivalent two-equation system, each of which is reduced to the Abel equation solution and its generalizations. The solution is written in explicit form. The examples are presented.