Sự tồn tại, duy nhất nghiệm và phương pháp lặp giải một số bài toán biên cho phương trình vi phân phi tuyến cấp ba - Đặng Quang Long

BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ ----------------------------- ĐẶNG QUANG LONG SỰ TỒN TẠI, DUY NHẤT NGHIỆM VÀ PHƯƠNG PHÁP LẶP GIẢI MỘT SỐ BÀI TOÁN BIÊN CHO PHƯƠNG TRÌNH VI PHÂN PHI TUYẾN CẤP BA LUẬN ÁN TIẾN SĨ NGÀNH TOÁN HỌC HÀ NỘI – 2023 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ ----------------------------- Đặng Quang Long SỰ TỒN TẠI, DUY NHẤT NGHIỆM VÀ PHƯƠNG PHÁP LẶP GIẢI MỘT SỐ BÀI TOÁN BIÊN CHO PHƯƠNG TRÌNH VI PHÂN PHI TUYẾN CẤP BA Chuyên ngành: Toán ứng dụng Mã số: 9 46 01 12 LUẬN ÁN TIẾN SĨ NGÀNH TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: GS. Nguyễn Đông Anh Hà Nội – Năm 2023 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY THE EXISTENCE, UNIQUENESS AND ITERATIVE METHODS FOR SOME NONLINEAR BOUNDARY VALUE PROBLEMS OF THIRD ORDER DIFFERENTIAL EQUATIONS by DANG QUANG LONG Supervisor: Prof. NGUYEN DONG ANH Presented to the Graduate University of Sciences and Technology in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY HANOI - 2023 DECLARATION OF AUTHORSHIP I hereby declare that this thesis was carried out by myself under the guidance and supervision of Prof. Nguyen Dong Anh.

The results in it are original, genuine and have not been published by any other author. The numerical experiments performed in MATLAB are honest and precise. The joint-authored publications have been granted permission to be used in this thesis by the co-authors. The author Dang Quang Long i ACKNOWLEDGMENTS I would like to express my deepest gratitude to my supervisor Prof.

Nguyen Dong Anh. His immense knowledge and kind guidance have helped me tremendously in the completion of this thesis. I would like to show my appreciation to the Graduate University of Sciences and Technology and Institute of Technology, Vietnam Academy of Science and Technology for their generous support during the years of my PhD program. Last but not least, this thesis would not have been possible without the support and encouragement from my family, friends and colleagues.

I would like to give a special thanks to my dear father for his invaluable professional advices. The author ii List of Figures 2.1 The graph of the approximate solution in Example 2.2 The graph of the approximate solution in Example 2.3 The graph of the approximate solution in Example 2.4 The graph of the approximate solution in Example 2.5 The graph of the approximate solution in Example 2.6 The graph of the approximate solution in Example 2.7 The graph of the approximate solution in Example 2.8 The graph of the approximate solution in Example 2.1 The graph of the approximate solution in Example 3.2 The graph of the approximate solution in Example 3.3 The graph of the approximate solution in Example 3.4 The graph of the approximate solution in Example 3.5 The graph of the approximate solution in Example 3.1 The graph of the approximate solution in Example 4.2 The graph of the approximate solution in Example 4. 91 iii List of Tables 2.1 The convergence in Example 2.2 The convergence in Example 2.3 The convergence in Example 2.4 The results in [35] for the problem in Example 2.5 The convergence in Example 2.6 The convergence in Example 2.7 The convergence in Example 2.8 The results in [36] for the problem in Example 2.9 The convergence in Example 2.10 The convergence in Example 2.11 The convergence in Example 2.1 The convergence in Example 3.2 The convergence in Example 3.3 The convergence in Example 3.4 The convergence in Example 3.5 The convergence in Example 3.6 The convergence in Example 3.1 The convergence in Example 4.1 for stopping criterion kUm − uk ≤ h2 .2 The convergence in Example 4.1 for stopping criterion kΦm −Φm−1 k ≤ 10−10 79 4.3 The convergence in Example 4.4 The convergence in Example 4. 91 iv Contents Introduction.

Some fixed point theorems. Schauder Fixed-Point Theorem. Krasnoselskii Fixed-Point Theorem. Banach Fixed-Point Theorem.

Some quadrature formulas. Existence results and iterative method for two-point third order nonlinear BVPs. Existence results and continuous iterative method for third order nonlinear BVPs 17 2. Some particular cases and examples.

Numerical methods for third order nonlinear BVPs. Discrete iterative method 1. Discrete iterative method 2. On some extensions of the problem.

Existence results and iterative method for some nonlinear ODEs with integral boundary conditions. Existence results and iterative method for fully third order nonlinear integral boundary value problems. Existence results and iterative method for fully fourth order nonlinear integral boundary value problems. Iterative method on continuous level.

Discrete iterative method. Existence results and iterative method for integro-differential and functional differential equations. Existence results and iterative method for integro-differential equation. Existence results and iterative method for functional differential equation.

Existence and uniqueness of solution. Solution method and its convergence. 92 List of works of the author related to the thesis. 94 vi Introduction Overview of research situation and the necessity of the re- search Numerous problems in the fields of mechanics, physics, biology, environment, etc.

are reduced to boundary value problems for high order nonlinear ordinary differential equations (ODE), integro-differential equations (IDE) and functional differential equa- tions (FDE). The study of qualitative aspects of these problems such as the existence, uniqueness and properties of solutions, and the methods for finding the solutions al- ways are of interests of mathematicians and engineers. One can find exact solutions of the problems in a very small number of special cases. In general, one needs to seek their approximations by approximate methods, mainly numerical methods.

Below we review some important topics in the above field of nonlinear boundary value problems and justify why we select problems for studying in this thesis. a) Existence of solutions and numerical methods for two-point third order nonlinear boundary value problems High order differential equations, especially third order and fourth order differen- tial equations describe many problems of mechanics, physics and engineering such as bending of beams, heat conduction, underground water flow, thermoelasticity, plasma physics and so on [1, 2, 3, 4]. The study of qualitative aspects and solution methods for linear problems, when the equations and boundary conditions are linear, is basically re- solved. In recent years, ones draw a great attention to nonlinear differential equations.

There are numerous researches on the existence and solution methods for fourth order nonlinear boundary value problems. It is worthy to mention some typical works con- cerning the existence of solutions and positive solutions, the multiplicity of solutions, and analytical and numerical methods for finding solutions [5, 6, 7, 8, 9, 10]. Among the contributions to the study of fourth order nonlinear boundary value problems, there are some results of Vietnamese authors (see, e. Concerning the not fully or fully third order differential equations u000 (t) = f (t, u(t), u0 (t), u00 (t)), 0<t<1 (0.1) there are also many researches.

A number of works are devoted to the existence, unique- ness and positivity of solutions of the problems with different boundary conditions. The methods for investigating qualitative aspects of the problems are diverse, including the method of lower and upper solutions and monotone technique [7, 15, 16, 17, 18, 19], Leray-Schauder continuation principle [20], fixed point theory on cones [21], etc. It should be emphasized that in the above works there is an essential assumption that the function f (t, x, y, z) : [0, 1] × R3 → R satisfies a Nagumo-type condition on the last two variables [22], or linear growth in x, y, z at infinity [20], or some complicated conditions including monotone increase in each of x and y [23], or one-sided Lipschitz 1 condition in x for f = f (t, x) [19] and in x, y for f = f (t, x, y) [17]. Sun et al.

in [24] studied the existence of monotone positive solution of the BVP for the case f = f (u(t)) under conditions which are difficult to be verified. Differently from the above approaches to the third order boundary value problems, very recently Kelevedjiev and Todorov [25] using barrier strips type conditions gave suf- ficient conditions guaranteeing positive or non-negative, monotone, convex or concave solutions. It should be said that in the mentioned works, no examples of solutions are shown although the sufficient conditions are satisfied and the verification of them is difficult. Therefore, it is desired to overcome the above shortcoming, namely, to construct easily verified sufficient conditions and show examples when these conditions are satisfied and solutions in these examples.

For solving third order linear and nonlinear boundary value problems for the equa- tion (0.1) having in mind that the problems under consideration have solutions, there is a great number of methods including analytical and numerical methods. Below we briefly review these methods via some typical works. First we mention some works where analytical methods are used. Specifically, in [26] the authors proposed an it- erative method based on embedding Green’s functions into well-known fixed point iterations, including Picard’s and Krasnoselskii–Mann’s schemes.

The uniform con- vergence is proved but the method is very difficult to realize because it requires to calculate integrals of the product the Green function of the problem with the func- tion f (t, un (t), u0n (t), u00n (t)) at each iteration. In [27, 28] the Adomian decomposition method and its modification are applied. Recently, in 2020, He [29] suggests a simple but effective way to the third-order ordinary differential equations by the Taylor series technique. In general, for solving the BVPs for nonlinear third order equations numer- ical methods are widely used.

Namely, Al Said et al. [30] have solved a third order two point BVP using cubic splines. Noor et al. [31] generated second order method based on quartic splines.

Other authors [32, 33] generated finite difference schemes using fourth degree B-spline and quintic polynomial spline for this problem subject to other boundary conditions. El-Danaf [34] constructed a new spline method based on quartic nonpolynomial spline functions that has a polynomial part and a trigonomet- ric part to develop numerical methods for a linear differential equation. Recently, in 2016 Pandey [35] solved the problem for the case f = f (t, u) by the use of quartic polynomial splines. The convergence of the method of at least O(h2 ) for the linear case f = f (t) was proved.

In the following year, this author in [36] proposed two difference schemes for the general case f = f (t, u(t), u0 (t), u00 (t)) and also established the second order accuracy for the linear case. In 2019, Chaurasia et al. [37] used ex- ponential amalgamation of cubic spline functions to form a novel numerical method of second-order accuracy. It should be emphasized that all of above mentioned authors only drew attention to the construction of the discrete analogue of the equation (0.1) associated with some boundary conditions and estimated the error of the obtained solu- tion assuming that the nonlinear system of algebraic equations can be solved by known iterative methods.

Thus, they did not take into account the errors arising in the last iterative methods. Motivated by the above facts we wish to construct iterative numerical methods of competitive accuracy or more accurate compared with some existing methods, and importantly, to obtain the total error combining the error of iterative process and the error of discretization of continuous problems at each iteration. 2 b) Boundary value problems with integral boundary conditions Recently, boundary value problems for nonlinear differential equations with integral boundary conditions have attracted attention from many researchers. They consti- tute a very interesting and important class of problems because they arise in many applied fields such as heat conduction, chemical engineering, underground water flow, thermoelasticity and plasma physics.