Asymptotic Solutions to a Singularly Perturbed Partial Integro-Differential Equation with a Rapidly Oscillating Right-Hand Side (European Journal of Pure and Applied Mathematics, 2025)
The paper considers the Cauchy problem for a singularly perturbed integro-differential partial differential equation with a rapidly oscillating right-hand side. When considering such problems, it turned out that the existing technique for regularizing singularly perturbed equations is not effective and requires significant rethinking. The development of a new technique for constructing regularized asymptotic solutions for integro-differential equations with partial derivatives and exponential inhomogeneity constitutes the main content of this work. The problem was regularized and the normal and unique solvability of general iterative problems was proved. The asymptotic convergence of formal solutions is proved and a solution to the first iterative problem is constructed.
Keywords: singular perturbation, partial integro differential equation, rapidly oscillating right-hand side, solvability of iterative problems, regularization of an integral.