A Regularization Method for Nonlinear Integro-differential Equations with a Zero Operator of the Differential Part and with Several Rapidly Varying Kernels

Abstract

A nonlinear integro-differential equation with a zero operator in the differential part and with several rapidly varying kernels is considered. The article is a continuation of the research carried out previously for one rapidly varying kernel. The main ideas of this generalization and subtleties encountered in elaborating an appropriate regularization method algorithm are fully seen in the case of two rapidly varyingkernels; therefore, to reduce the amount of calculations, exactly this case is considered. A similar problem with one spectral value of the integral operator kernel was considered in one of our papers. In this case, the singularities in the solution of the problem are only described by the kernel spectral value. However, the effect of a zero differential operator manifests itself in the fact that in a first approximation, the asymptotic behavior of the solution of the problem will not include the boundary layer functions, and the limit operator itself will become degenerated (but not zero). The solvability conditions of the corresponding iterative problems, as in the linear case, will not be in the form of differential (as was in the problems with a nonzero operator of the differential part), but in the form of integro-differential equations, with the nonlinearity playing an essential role in the derivation of these equations. It should be noted that in contrast to the linear case, the right side of the problem under study does not contain heterogeneity of the corresponding linear problem. As was shown earlier, its presence in the problem would lead to the occurrence of terms with negative powers of a small parameter in the asymptotic solution, and in the nonlinear case there would be an infinite number of such powers, and the corresponding formal asymptotic solution would have the form of the Laurent series. This would make the development of an algorithm for asymptotic solutions quite problematic; therefore, wishing to remain within the framework of asymptotic solutions such as Taylor series, we excluded inhomogeneity in this study. In addition, in the nonlinear case, so-called resonances may arise, which significantly complicate the development of the corresponding algorithm of the regularization method. In this article, the non-resonance case is considered. Supposedly, an alternative option (a more complex resonance problem) will be the subject of the future studies.