Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernels

Abstract

We generalize the Lomov's regularization method for partial differential equations with integral operators, whose kernel contains a rapidly varying exponential factor. We study the case when the upper limit of the integral operator coincides with the differentiation variable. For such problems we develop an algorithm for constructing regularized asymptotics. In contrast to the work by Imanaliev M.I., where for analogous problems with slowly varying kernel only the passage to the limit studied as the small parameter tended to zero, here we construct an asymptotic solution of any order (with respect to the parameter). We note that the Lomov's regularization method was used mainly for ordinary singularly perturbed integro-differential equations (see detailed bibliography at the end of the article). In one of the authors' papers the case of a partial differential equation with slowly varying kernel was considered. The development of this method for partial differential equations with rapidly changing kernel was not made before. The type of the upper limit of an integral operator in such equations generates two fundamentally different situations. The most difficult situation is when the upper limit of the integration operator does not coincide with the differentiation variable. As studies have shown, in this case, the integral operator can have characteristic values, and for the construction of the asymptotics, more strict conditions on the initial data of the problem are required. It is clear that these difficulties also arise in the study of an integro-differential system with a rapidly changing kernels, therefore in this paper the case of the dependence of the upper limit of an integral operator on the variable $ x$ is deliberately avoided. In addition, it is assumed that the same regularity is observed in a rapidly decreasing kernel exponent integral operator. Any deviations from these (seemingly insignificant) limitations greatly complicate the problem from the point of view of constructing its asymptotic solution. We expect that in our further works in this direction we will succeed to weak these restrictions.