Investigation Of Some Singularly Perturbed Problems With Discontinuous Right-side

Singularly perturbed problems are faced in many different fields of science such as physics,chemistry,biology,engineering,etc.Boundary function method is a very powerful tool for solution singularly perturbed problems.In this dissertation we applied theoretical approach that has been proposed by Vasil'eva A.B.She was a former student of famous Russian mathematician Tikhonov A.N.He considered as the father of singular perturbation theory in USSR(Russia).Later,this method has been improved and applied by many other scientists from all over the world for solving different classes of singularly perturbed problems with small parameter.In recent years,many other approaches were developed,nevertheless boundary functions method is still applied for analysis and implementation new algorithms for solving new classes of singularly perturbed problems with high level of accuracy.The main goal of this research was application of boundary functions method for implementation algorithms that will allow construct asymptotic solutions of some certain types of second-order differential equations with small parameters.Thus,demonstrate that boundary functions method generalize the idea of construction asymptotic solution,rather than a unique algorithm for every class of singularly perturbed problems.Based on it in combination with other methods including numerical methods it became possible to implement new techniques for solving already well-known problems and extend that are already exist by solving new classes of SPP.There are four chapters in the dissertation.In 1 Chapter,given theoretical background,explanations and motivation of scientific research.Given review of singular perturbation theory development in Russia along with the basic terms and foundations,concepts,definitions that are relevant to the conducted research.Given an explanation of boundary functions method idea and theorem formulated by Vasil'eva A.B.In 2 Chapter,boundary function method applied to second-order semi-linear singularly perturbed problem with discontinuity and Neumann boundary conditions.Two different cases are analyzed and for given examples provided step-by-step algorithm for construction asymptotic solution in case of contrast structures.Two different types of contrast structures a spike-like and a step-like are demonstrated in this dissertation.Given algorithm for defining internal transition point,since it is unknown at initial statement of a problem.At the end of the chapter provided a proof of asymptotic solution existence and the uniform validity of the asymptotic expansion.In Chapter 3,considered second-order Neumann boundary value problems.In the first case,we have considered problem with small parameter only in front of second-order derivative,in the second case with small parameter in front of both first and second-order derivatives with Neumann boundary conditions.In both cases,formulated conditions that required to be satisfied for successful construction of solution's asymptotic expansion,expressed algorithm of asymptotic solution construction,proved existence of solution and it's asymptotic expansion.Along with the formal asymptotic solution provided refinement that allow to increase accuracy and precision of calculations to the required level of accuracy.As the result formulated theorem that summarized results.Chapter 4,devoted to investigation and application of boundary functions method for construction asymptotic solution of second order quasi-linear Neumann boundary value problem.Here we demonstrate that solution has internal transition layer,step-by-step calculated every part of asymptotic expansion and estimated residual members.In conclusion,we generalize results of research described applications of boundary function method for solving varieties of different classes singularly perturbed problems.