Solution With Inner Transition Layer Of Singular Perturbed Problem For Second Order Differential-Difference Equation

Thesis devoted to the investigation of one problem in the class of the problem for differential-difference equations (differential equations with divergent argument), the highest derivative of the equation in problem contains the small parameter.The problems with small parameters combine into the class of problems-singular perturbed problems. Such problems are arising in modeling and investigation of physical, chemical and biological processes and effects. A lot of outstanding scientists in theory of singular perturbations, such as Tikhonov A.N., Vasyl’eva A.B., Butuzov V.F. and others, are investigating these problems successfully for a long time.Particular feature of singular perturbed problem is the fact that singular solution (solution of the equation where small parameter equals to zero) not satisfies to the all initial conditions; consequently in some areas near boundary points there’s occurs the rapid changing of solution, i.e. arise boundary layers.Solution of considering in thesis boundary-value singular perturbed problem for differential-difference equations belongs to the so-called contrast structures, i.e. the solution besides the boundary layers which localized in neighborhood of boundary also has the inner layers which localized in neighborhood of inner points of area.The construction of solution’s asymptotic expansion of investigation problem is based on the method of boundary functions and on the method of smooth connection. In theses there are given a description of the algorithm of asymptotic construction. As a result there’re getting not only continuous but also smooth solution of investigating problem.