Articles in "Mathematics"

Summary: The paper examines different types of hypersingular integrals with the Cauchy kernel on a segment and a unit circle and defines them using specific methods. It presents more general definitions for one-dimensional hypersingular integrals with the Cauchy kernel based on Hadamard's integral in the sense of a finite part. The paper also establishes the existence theorems of these hypersingular integrals and formulas, which demonstrates the accuracy of the resulting integrals that are applied in various applications and engineering problem-solving. The proposed formulas are straightforward to calculate, making the new approximate method reliable and easy to apply and the obtained numerical results demonstrate the stability and efficiency of the approach.

Summary: The study investigates the inverse scattering problem for the Schrodinger operator with complex potentials, considering indefinite discontinuous coefficients on the axis. Using the integral representation of the Jost solutions on the real and imaginary axes, solved the direct scattering problem. An additional study of the operator's spectrum was conducted, scattering data was introduced, and the eigenfunction expansion was obtained. Integral equations derived play a crucial role in solving the inverse problem and finally prove the uniqueness theorem for the solution. © 2024

Summary: Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem.