A. N. Tynda, D. R. Vechkasov

Penza State University (Penza, Russian Federation)

Abstract. The paper is devoted to the study of Volterra equations with an integral operator containing the trace of the unknown solution in the form of its values at certain points of the set. A number of classical boundary value problems of elliptic, hyperbolic, parabolic, and mixed types are reduced to equations with loads. In particular, the Goursat boundary value problem for a hyperbolic type equation is equivalent to a loaded Volterra integral equation of the second kind. The issues of numerical solution of loaded functional equations in integral form have not been sufficiently studied in the literature. Within the framework of this work, the conditions for the existence and uniqueness of solutions to integral equations with local loads are established. A collocation-type numerical method is constructed based on the approximation of the solution by polynomial splines of variable order. The orders of the polynomials that make up the spline are determined adaptively and are consistent with the maximum step in each section of the node grid, which is constructed taking into account the distribution of unknown loads. In the process of discretization, in order to determine the coefficients of the system of equations, the integrals are approximated by Gauss quadrature sums and a general system of linear algebraic equations is formed with respect to all unknown parameters of the spline. The convergence of such an approximation is proved, a number of numerical results are given, confirming the effectiveness of the proposed approach.

Key Words: Volterra integral equations, loaded operator, existence of a solution, spline approximation, convergence, error estimate

For citation: A. N. Tynda, D. R. Vechkasov. Numerical and theoretical study of Volterra integral equations with a locally loaded operator. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:1(2026), 67–78. DOI: https://doi.org/10.15507/2079-6900.28.202601.67-78

Information about the authors:

Aleksandr N. Tynda, Ph. D. (Phys. and Math.), Head of the Department of Higher and Applied Mathematics, Penza State University (40 Krasnaya street, Penza, 440026, Russia), ORCID: http://orcid.org/0000-0001-6023-9847, tyndaan@mail.ru

Danila R. Vechkasov, Master’s student of the Department of Higher and Applied Mathematics, Penza State University (40 Krasnaya street, Penza, 440026, Russia), ORCID: https://orcid.org/0009-0004-8349-9173, vechkasov.danila@yandex.r