A generalization of the regularization method to the singularly perturbed integro-differential equations with partial derivatives

We generalize Lomov's regularization method to partial integrodifferential equations. It turns out that the procedure for regularization and the construction of a regularized asymptotic solutions essentially depends on the type of the integral operator. The most difficult is the case, when the upper limit of the integral is not a variable of differentiation. In this paper, we consider its scalar option. For the integral operator with the upper limit coinciding with the variable of differentiation, we investigate vector case. In both cases, we develop an algorithm for constructing a regularized asymptotic solutions and carry out its full substantiation. Based on the analysis of the principal term of the asymptotic solution, we study the limit in solution of the original problem (with the small parameter tending to zero) and solve the so-called task initialization allocation class of input data, in which the passage to the limit takes place on all the considered period of time, including the area boundary layer.