Asymptotic expansion of the solution of a boundary value problem for singularly perturbed integro-differential equations of higher orders

A boundary value problem for singularly perturbed linear integro-differential equations of higher orders with small parameters at several higher derivatives having the phenomena of initial jumps are considered in this work. This means that starting from a certain order number, the derivatives of the solutions at the initial point tend to infinity as the small parameter tends to zero. Previously, such classes of equations have not been studied. A uniform asymptotic expansion of solutions to the problem under consideration is constructed with an arbitrary degree of accuracy with respect to a small parameter. An estimate of the remainder term of the asymptotic expansion of the solution is proved.