A Generalization of the Regularization Method to the Singularly Perturbed Integro-Differential Equations With Partial Derivatives

Abstract

We generalize the Lomov’s regularization method to partial integro-differential equations. It turns out that the procedure for regularization and the construction of a regularized asymptotic solution essentially depend 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 the vector case. In both cases, we develop an algorithm for constructing a regularized asymptotic solution 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 initialization problem about allocation of a class of input data, in which the passage to the limit takes place on the whole considered period of time, including the area of boundary layer.