Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator

Abstract

The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.

On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

“…tt         will be exact solution of the problem (2). Defining a solution of the system (34) in the form of series: we obtain the following iteration problems:…”

“…  small parameter. It is required to construct a regularized asymptotic of a solution[1,2] of the problem (1) at for 0.…”

Internal boundary layer for integral-differential equations with zero spectrum of the limit operator and rapidly changing kernel

“…Due to these formulas, result of substitution of the series (10) into the integral operator (8) can be rewritten as follows:…”

“…At the end of the last century articles devoted to the study of singularly perturbed integral-differential systems with rapidly changing kernels began to appear [3,5,6,8,9,10,11,12,13,14,16,17]. Integral operators with rapidly decreasing kernel create in the structure of solutions of the original problem a new type of singularities by a small parameter, which describes the internal boundary layer.…”

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

“…We will establish the formulation in metric spaces ( , ). It might be pointed out that it is usual to state formulations related to differential or dynamic systems and their stability, including those being formulated in a fractional calculus framework, in normed or Banach spaces since their dynamics evolve through time described by their state vectors [14,[29][30][31][32][33][34][35][36][37][38][39]. A possibility to focus on the study of their equilibrium points in a formal and structured fashion as well as their limit solutions, provided that they exist, (for instance, the presence of possible limit cycles) is through fixed point theory since the equilibrium points are fixed points of certain mappings and the limit cycles are repeated portions of limit state space trajectories.…”