Summary of Phd thesis: Solving some nonlinear boundary value problems for fourth order differential equations

The thesis proposes a simple but very effective method to study the unique solvability and an iterative method for solving five boundary value problems for nonlinear fourth order ordinary differential equations with different types of boundary conditions and two boundary value problems for a biharmonic equation and a biharmonic equation of Kirchhoff type by using the reduction of these problems to the operator equations for the function to be sought or an intermediate function.
Nội dung trích xuất từ tài liệu:
Summary of Phd thesis: Solving some nonlinear boundary value problems for fourth order differential equationsMINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGYGRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ----------------------------------- NGUYEN THANH HUONG SOLVING SOME NONLINEAR BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS Major: Applied Mathematics Code: 9 46 01 12 SUMMARY OF PHD THESIS Hanoi – 2019This thesis has been completed:Graduate University of Science and Technology – Vietnam Academy of Science and TechnologySupervisor 1: Prof. Dr. Dang Quang ASupervisor 2: Dr. Vu Vinh QuangReviewer 1:Reviewer 2:Reviewer 3:The thesis will be defended at the Board of Examiners of GraduateUniversity of Science and Technology – Vietnam Academy ofScience and Technology at ............................ on..............................The thesis can be explored at:- Library of Graduate University of Science and Technology- National Library of Vietnam INTRODUCTION1. Motivation of the thesis Many phenomena in physics, mechanics and other fields are modeled byboundary value problems for ordinary differential equations or partial differentialequations with different boundary conditions. The qualitative research as well asthe method of solving these problems are always the topics attracting the attentionof domestic and foreign scientists such as R.P. Agawarl, E. Alves, P. Amster, Z.Bai, Y. Li, T.F. Ma, H. Feng, F. Minh´os, Y.M. Wang, Dang Quang A, PhamKy Anh, Nguyen Dong Anh, Nguyen Huu Cong, Nguyen Van Dao, Le Luong Tai.The existence, the uniqueness, the positivity of solutions and the iterative methodfor solving some boundary value problems for fourth order ordinary differentialequations or partial differential equations have been considered in the works ofDang Quang A et al. (2006, 2010, 2016-2018). Pham Ky Anh (1982, 1986)has also some research works on the solvability, the structure of solution sets,the approximate method of nonlinear periodic boundary value problems. Theexistence of solutions, positive solutions of the beam problems are considered inthe works of T.F. Ma (2000, 2003, 2004, 2007, 2010). Theory and numericalsolution of general boundary problems have been mentioned in R.P. Agarwal(1986), Uri M. Ascher (1995), Herbert B. Keller (1987), M. Ronto (2000). Among boundary problems, the boundary problem for fourth order ordinarydifferential equations and partial differential equations are received great interestby researchers because they are mathematical models of many problems in me-chanics such as the bending of beams and plates. It is possible to classify thefourth order differential equations into two forms: local fourth order differentialequations and nonlocal ones. A fourth order differential equation containing inte-gral terms is called a nonlocal equation or a Kirchhoff type equation. Otherwise, itis called a local equation. Below, we will review some typical methods for studyingboundary value problems for fourth order nonlinear differential equations. The first method is the variational method, a common method of studying theexistence of solutions of nonlinear boundary value problems. With the idea ofreducing the original problem to finding critical points of a suitable functional,the critical point theorems are used in the study of the existence of these criticalpoints. There are many works using the variational method (see T.F. Ma (2000, 12003, 2004), R. Pei (2010), F. Wang and Y. An (2012), S. Heidarkhani (2016),John R. Graef (2016), S. Dhar and L. Kong (2018)). However, it must be notedthat, using the variational method, most of authors consider the existence ofsolutions, the existence of multiple solutions of the problem (it is possible toconsider the uniqueness of the solution in the case of convex functionals) but thereare no examples of existing solutions, and the method for solving the problem hasnot been considered. The next method is the upper and lower solutions method. The main results ofthis method when applying to nonlinear boundary value problems are as follows:If the problem has upper and the lower solutions, the problem has at least onesolution and this solution is in the range of the upper and the lower solutionunder some additional assumptions. In addition, we can construct two monotonesequences with the first approximation being the upper and the lower solutionconverge to the maximal and minimal solutions of the problem. In the case ofmaximal and minimal solutions coincide, the problem has a unique solution. We can mention some typical works using the upper a ...