Summary of Mathematics doctoral thesis: Iterative method for solving two point boundary value problems for fourth order differential equations and systems

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The objective of the thesis is to develop the iterative method and combining it with other methods to study qualitative and especially the method of solving some two-point boundary problems for the fourth-order differential equations and systems, arising in beam bending theory without using condition of growth rate at infinity, Nagumo condition, etc. of the right-hand side function.

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Nội dung Text: Summary of Mathematics doctoral thesis: Iterative method for solving two point boundary value problems for fourth order differential equations and systems

MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND ECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ……..….***………… NGÔ THỊ KIM QUY ITERATIVE METHOD FOR SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS AND SYSTEMS Major : Applied Mathematics Code: 62 46 01 12 SUMMARY OF MATHEMATICS DOCTORAL THESIS Hanoi – 2017
This thesis was completed at: Graduate University of Science and Technology Vietnam Academy of Science and Technology Supervisor 1: Prof. Dr. Dang Quang A Supervisor 2: Assoc. Prof. Dr. Ha Tien Ngoan Reviewer 1: … Reviewer 2: … Reviewer 3: …. The Dissertation will be officially presented in front of the Doctoral Dissertation Grading Committee, meating at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At ................Date..................Month...................Year 201… The Dissertation is avaiable at: 1. Library of Graduate University of Science and Technology 2. National Library of Vietnam
INTRODUCTION 1. Motivation of the thesis Many problems in physics, mechanics and some other fields are described by differential equations or systems of differential equations with different bound- ary conditions. It is possible to classify the fourth-order differential equations into two forms: fully fourth-order differential equations and non-fully fourth- order ones. A fourth-order differential equation whose right-hand side function contains an unknown function and its derivatives of all order (from first to third order) is called a fully fourth-order differential equation. Otherwise, the equation is called a non-fully fourth-order differential equation. The boundary value problems for differential equations have attracted the attention of scientists such as Alve, Amster, Bai, Li, Ma, Feng, Minh´os, etc. Some Vietnamese mathematicians and mechanics, namely, Dang Quang A, Pham Ky Anh, Nguyen Van Dao, Nguyen Dong Anh, Le Xuan Can, Nguyen Huu Cong, Le Luong Tai, etc. also studied methods for solving the boundary value problems for differential equations. Among the differential equations, the nonlinear fourth-order differential equation has been of great interest recently as it is the mathematical model of many problems in mechanics. Here we take a look at some of the boundary value problems for the nonlinear fourth-order differential equations. Firstly, consider the problem of elastic beams as described by the nonlinear fourth-order differential equation u(4) (x) = f (x, u(x), u00 (x)) (0.0.2) or u(4) (x) = f (x, u(x), u0 (x)) (0.0.3) where u is the deflection of the beam, 0 ≤ x ≤ L. The conditions at two ends of beams are given in dependence of the constraints of the problems. There have been many research results on the qualitative aspects of the problems such as existence, uniqueness and positivity of solutions. Noteworthy is the works of Alves et al. (2009), Amster et al. (2008), Bai (2004), Li (2010), Ma et al. (1997), ..., where the upper and lower solution method, the variational method, the methods of fixed point theorems are used. In these works the conditions of the boundedness of the right-hand function or of its growth rate at infinity is indispensable. In the articles mentioned above, the fourth-order differential equation does not contain third-order derivative. For the last ten years, the fully fourth-order 1
differential equations, namely the equation u(4) (x) = f (x, u(x), u0 (x), u00 (x), u000 (x)) (0.0.6) has attracted the interest of many authors (Ehme et al. (2002), Feng et al. (2009), Li et al. (2013), Li (2016), Minh et al. (2009), Pei et al. (2011), ...). The main results in the these papers are the study of the existence, uniqueness and positivity of the solution. The tools used are Leray-Schauder’s degree theory (see Pei et al. (2011)), the Schauder fixed point theorem based on the monotone method in the present of lower and the upper solutions (see Bai (2007) ), Ehme et al. (2002), Feng et al. (2009), Minh´os et al. (2009)) or Fourier analysis (see Li et al. (2013)). However, in all of the articles mentioned above, the authors need a very important assumption that the function f : [0, 1]× R4 → R satisfies the Nagumo condition and some other conditions of monotonicity and growth at infinity. It should be emphasized that in the monotone method the assumption of the presence of lower and upper solutions is always needed and the finding of them is not easy. The system of fourth-order differential equations have not been studied much, such as Kang et al. (2012), L¨ u et al. (2005), Zhu et al. (2010), in which the authors considered the equations containing only even-order deriva- tives associated with the simply supported boundary conditions. Under very complicated conditions, by using a fixed point index theorem in cones, the au- thors obtained the existence of positive solutions. But it should be emphasized that the obtained results are of pure theoretical character because no examples of existing solutions are shown. Minh´os and Coxe (2017, 2018) for the first time considered the system of coupled fully fourth-order of differential equations. The authors have provided sufficient conditions for solving the system by using the lower and upper solu- tions method and the Schauder fixed point theorem. Demonstrating this result is very cumbersome and complicated and requires Nagumo conditions for the functions f and h. Although significant achievements have been made in investigating the solv- ability of nonlinear boundary value problems, the development of applied fields such as mechanics, physics, biology, etc. always yields complex new problems for the equations as well as boundary conditions. These problems are impor- tant in science and practice. In addition, in the articles mentioned above, the conditions given are complex and difficult to verify. A very important assump- tion is that the right-hand side function satisfies the Nagumo condition and some other conditions of monotonous and growth properties at infinity. For the monotone method, the assumption of the presence of lower and upper solutions is always needed and the finding of them is not easy. Moreover, some articles do not have illustrative examples for theoretical results. Thus, continuing the qualitative and quantitative study of new problems for the fourth-order differ- ential equations and systems with different boundary conditions is crucial in research and practice. For these reasons, we decide a subject for this dissertation with the title ”Iterative method for solving two-point boundary value problems for fourth-order differential equations and systems”. 2
2. Objectives and scope of the thesis The objective of the thesis is to develop the iterative method and combining it with other methods to study qualitative and especially the method of solving some two-point boundary problems for the fourth-order differential equations and systems, arising in beam bending theory without using condition of growth rate at infinity, Nagumo condition, etc. of the right-hand side function. 3. Research methodology and content of the thesis Using the approach of reducing the original nonlinear boundary value prob- lems to operator equations for right-hand side functions, along with the tools of analytical mathematics, functional analysis, differential equation theory, we study the existence, uniqueness and some properties for the solutions of some problems for fully or non-fully nonlinear fourth-order differential equations and systems. Also on the basis of the operator equation, we construct an iterative method for finding the solutions of the problems of problems and prove the convergence of the method. Some examples are given, where exact solutions are known or unknown, to demonstrate the applicability of the obtained theoretical results and the efficiency of the iterative method. 4. The major contribution of the thesis The thesis proposes a method for researching qualitative aspects and an iterative method for solving boundary value problems for fully or non-fully nonlinear fourth-order differential equations and systems by using the reduc- tion of them to the operator equations for the right-hand side functions. The results are: • Establish the existence, uniqueness and some properties for the solutions of problems under some easily verified conditions. • Propose an iterative methods for solving these problems and prove the con- vergence of the iterative process. • Give some examples illustrating the applicability of the obtained theoretical results including examples where the existence or uniqueness is not guaranteed by other authors because these examples do not satisfy the conditions in their theorems. • Computational experiments illustrate the effectiveness of iterative methods. The thesis is written on the basis of articles [1]-[6] in the list of works of the author related to the thesis. 5. The structure of the thesis Besides the introduction, conclusion and references, the contents of the thesis are presented in three chapters. 3
Chapter 1 presents some preparatory knowledge including some fixed point theorems; the monotone method for solving boundary value problem of differ- ential equations; Green function for some problems and numerical methods for solving differential equations. The basic knowledge presented in Chapter 1 will play a very important role, as the basis for the results which will be presented in Chapter 2 and Chapter 3. In Chapter 2, by using the reduction of the nonlinear boundary value prob- lems to the operator equation of the right-hand side function rather than of the unknown function, we have established the existence, uniqueness and properties of solutions for fully or non-fully fourth-order nonlinear differential equations. Also on the basis of the operator equation, we construct the iteration meth- ods for solving the problems and prove the convergence of the methods. Some examples are given, where exact solutions are known or are not known, demon- strating the applicability of the obtained theoretical results and the efficiency of the iterative method. Continuing the developent of the techniques in Chapter 2, in Chapter 3, for the system of coupled fully or non-fully nonlinear fourth-order differential equations, we also obtain the results of existence, uniqueness and convergence of the iterative method. These results further enrich and confirm the effective- ness of the approach of reducing nonlinear boundary problems to the operator equations for right-hand side functions. In the thesis, the theoretical results are verified by numerical experiments which are programmed in MATLAB 7.0 on a computer with Intel Core i3 pro- cessor and 4GB RAM. 4
Chapter 1 Preliminary knowledge This chapter presents some preparation knowledge needed for subsequent chapters referenced from the literatures Ladde (1985), Melnikov et al. (2012), Samarskii et al. (1989), Zeidler (1986). 1.1 Some fixed point theorems In this section, we present three fixed point theorems applied in the study of the existence, the unique solution of differential equations: Banach fixed point theorem, Brouwer fixed point theorem and Schauder fixed point theorem. 1.2 Monotone method for solving boundary value problem for differential equations One of the more common methods of qualitative research (existence, unique- ness) of the solution and the approximate solution of the differential equation is the monotone method. The method has attracted the attention of researchers in recent years. This method is popular because it not only provides a way to prove the theorems that exist, but also leads to different results, which is an effective technique for studying the qualitative properties of the solution. Suppose there exists an ordered pair of lower and upper solution α and β, that is, α and β are smooth functions with α ≤ β. Based on the property of lower and upper solution, one establishes that the sequence αk is monotone non- decreasing and the sequence βk is monotone nonincresing, and both sequences converge to a solution (say u and u) of the problem. The monotone property of these sequences leads to the relation α ≤ α1 ≤ α2 ≤ ... ≤ αk ≤ ... ≤ u ≤ u ≤ ... ≤ βk ≤ ... ≤ β2 ≤ β1 ≤ β. When u = u, there is a unique solution in the sector hα, βi, otherwise the problem has lower extreme and upper extreme solutions. 1.3 Green function for some problems Green function has broad application in the study of boundary value prob- lems. In particular, the Green function is an important tool for indicating the 5
existence and uniqueness of solutions to problems. Consider the problem of linear boundary value dn y dn−1 y L[y(x)] ≡ p0 (x) n + p1 (x) n−1 + ... + pn (x)y = 0, (1.3.1) dx dx n−1 k k i d y(a) i d y(b) X Mi (y(a), y(b)) ≡ αk k + βk k = 0, i = 1, ...n, (1.3.2) k=0 dx dx where pi (x), i = 0, ...n are continuous functions on (a, b), the leading coefficient p0 (x) must be non-zero in all points in (a, b). Definition 1.4. (Melnikov et al. (2012)) The function G(x, t) is said to be the Greens function for the boundaryvalue problem (1.3.1)-(1.3.2), if, as a function of its first variable x, it meets the following defining criteria, for any t ∈ (a, b) : (i) On both intervals [a, t) and (t, b], G(x, t) is a continuous function having continuous derivatives up to nth order, and satisfies the governing equation in (1.3.1) on (a, t) and (t, b), i.e.: L[G(x, t)] = 0, x ∈ (a, t); L[G(x, t)] = 0, x ∈ (t, b). (ii) G(x, t) satisfies the boundary conditions in (1.3.2), i.e.: Mi (G(a, t), G(b, t)) = 0, i = 1, ..., n. (iii) For x = t, G(x, t) and all its derivatives up to (n − 2) are continuous x ∂ k G(x, t) ∂ k G(x, t) lim − lim− = 0, k = 0, ..., n − 2. x→t+ ∂xk x→t ∂xk (iv) The (n − 1)th derivative of G(x, t) is discontinuous when x = t, providing ∂ n−1 G(x, t) ∂ n−1 G(x, t) 1 lim+ − lim = − . x→t ∂xn−1 x→t− ∂xn−1 p0 (t) The following theorem specifies the conditions for existence and uniqueness of the Greens function. Theorem 1.6. (Melnikov et al. (2012)) (Existence and uniqueness). If the ho- mogeneous boundary-value problem in (1.3.1)-(1.3.2) has only a trivial solution, then there exists an unique Greens function associated with the problem. Consider the linear inhomogeneous equation dn y dn−1 y L[y(x)] ≡ p0 (x) + p 1 (x) + ... + pn (x)y = −f (x), (1.3.3) dxn dxn−1 subject to the homogeneous boundary conditions n−1 dk y(a) k i d y(b) X Mi (y(a), y(b)) ≡ αki k + βk k = 0, i = 1, ...n. (1.3.4) k=0 dx dx 6
where the coefficients pj (x) and the right-hand side term f (x) in the governing equation are continuous functions, with p0 (x) 6= 0 trn (a, b), and Mi represent linearly independent forms with constant coefficients. The following theorem establishes the relation between the uniqueness of solutions of (1.3.3)-(1.3.4) in terms of the Greens function, constructed for the corresponding homogeneous boundary value problem. Theorem 1.7. (Melnikov et al. (2012)) If the homogeneous boundary-value problem corresponding to (1.3.3)-(1.3.4)has only the trivial solution, thennthe unique solution for (1.3.3)-(1.3.4) has a unique solution can be expressed by the integral Z b y(x) = G(x, t)f (t)dt, a whose kernel G(x, t) is the Greens function of the corresponding homogeneous problem. 1.4 Numerical method for solving differential equations To solve the boundary value problems for differntial equations, one can find their exact solutions in a very small number of special cases. In general, one needs to seek their approximations by approximation methods. For nonlinear equations, the use of approximation methods is almost inevitable. In solving differential equations for differential equations, one can find their exact solutions in a very small number of special cases. In general, one needs to seek their approximations by approximation methods. For nonlinear equations, the use of approximation methods is almost inevitable. Difference method is one of the numerical methods for approximating differential equations. The general idea of difference method is to reduce a differential problem to a discrete problem on a grid of points leading to solving a linear algebraic system of equations. The boundary value problem for the second-order differential equations, by the three-point difference method, leads soving of the system of equations with tridiagonal matrix. One of the effective direct methods of solving this problem is the progonka method (a special type of elemination method). In Section 1.4 we present in detail the method to solve tridiagonal systems(see Samarskii et al. (1989)). 7
Chapter 2 Iterative method for solving boundary value problems for the nonlinear fourth-order differential equations Boundary value problems for the nonlinear fourth order equations with dif- ferent boundary conditions have been studied in a number of articles in recent years. Existence of solutions these problems is established using the Leray- Schauder theory (Pei et al. (2011)), Schauder fixed point based on the mono- tone method in the present of lower and upper solutions, for example, Bai (2007), Ehme et al. (2002), Feng et al. (2009), Minh´os et al. (2009) or Fourier analysis (Li et al. (2013)). In these works the conditions of the boundedness of the right-hand side function or of its growth rate at infinity is indispensable. In the articles above, the authors give the original problem of the operator equation for the unknown function u(x). Differently from that approach, in the articles [1]-[4], we reduce the initial problem of the operator equation for the right-hand side function ϕ(x) = f (x, u(x), v(x), ...). This idea originates from an earlier paper by Dang Quang A (2006) when studying the Neumann problem for harmonic equations. The result is that we have established the existence and uniqueness of solution and the convergence of an iterative method for solving the original problem without the above assumptions. Instead of the condition for the right-hand side function in the whole space of variables we only need to consider this function in a bounded domain. This effective approach consists in the reduction of the problem to an operator equation for the right-hand side function instead of the functionu(x) to be sought as the other authors did. The numerical realization of the problems is reduced to the solution of two linear sec- ond order boundary value problems at each iteration. This allows to construct numerical methods of higher order accuracy for the problem. We illustrate the obtained theoretical results on some example, where the exact solution of the problem is known or unknown. The results of this chapter are presented in articles [1]-[4] in the list of works of the author related to the thesis. It should be added that, in the paper by Dang quang A and Truong Ha Hai (2016), the method was developed with the nonlinear fourth-order elliptic equation. 8
2.1 The boundary value problem for the non- fully nonlinear fourth-order differential equa- tion This part focuses on a boundary value problem for the non-fully nonlinear fourth-order differential equation describing the bending equilibrium of a beam on an elastic foundation, whose two ends are simply supported u(4) (x) = f (x, u(x), u00 (x)), 0 < x < 1, (2.1.1) u(0) = u(1) = u00 (0) = u00 (1) = 0. where f : [0, 1] × R2 → R is continuous. It has attracted attention from many authors owing to its importance in mechanics, such as Aftabizadeh (1986), Ma et al. (1997), Bai et al. (2004), Li (2010)... In these articles, the conditions of the boundedness of the right-hand side function or of its growth rate at infinity is indispensable. In the article [2], we also consider the problem (2.1.1). Differently from the approaches of the authors mentioned above, we reduce the initial prob- lem of the operator’s equation to the right-hand side function ϕ = f (x, u, u00 ). we prove that the operator for ϕ under some easily verified conditions on the function f (x, u, v) in a specified bounded domain is contractive. This ensures that the original boundary value problem has a unique solution generated by the fixed point of the operator and the convergence of the iterative method for constructing approximations. The positivity of the solution and the monotony of iterations are also considered. The numerical experiments on these and other examples show the fast convergence of the iterative method. To investigate the problem (2.1.1), for ϕ ∈ C[0, 1], consider the operator equation ϕ = Aϕ, (2.1.2) where A is defined by (Aϕ)(x) = f (x, uϕ (x), vϕ (x)). (2.1.3) Here vϕ (x), uϕ (x) respectively are the solutions of the sequence of problems 00 vϕ = ϕ(x), 0 < x < 1, (2.1.4) vϕ (0) = vϕ (1) = 0, 00 uϕ = vϕ (x), 0 < x < 1, (2.1.5) uϕ (0) = uϕ (1) = 0. Proposition 2.1. (The relation between the solution of the problem (2.1.1) with the solution of the operator equation (2.1.2)). If ϕ(x) is a solution of (2.1.2) where A is determined from (2.1.3)-(2.1.5) then uϕ (x) is a solution of the problem (2.1.1) and vice versa. Lemma 2.1. For the solution of the problems (2.1.4), (2.1.5) there hold the following assertions: 9
(i) 1 1 kvk ≤ kϕk, kuk ≤ kϕk, (2.1.7) 8 64 where k.k is the maximum norm in C[0, 1]. (ii) If ϕ(x) ≥ 0 in [0, 1] then −kϕk/8 ≤ v(x) ≤ 0 end 0 ≤ u(x) ≤ kϕk/64 in [0, 1]. For each number M > 0 denotes M M DM = (x, u, v) | 0 ≤ x ≤ 1, |u| ≤ , |v| ≤ , (2.1.9) 64 8 and by B[O, M ], we denote a closed ball centred at O with the radius M in the space of continuous functions C[0, 1]. Theorem 2.1.(Uniqueness of solution). Suppose that there exist numbers M, L1 , L2 ≥ 0 such that (i) |f (x, u, v)| ≤ M for any (x, u, v) ∈ DM . (2.1.10) (ii) |f (x, u2 , v2 ) − f (x, v1 , u1 )| ≤ L1 |u2 − u1 | + L2 |v2 − v1 | (2.1.11) for any (x, ui , vi ) ∈ DM , i = 1, 2. (iii) 1 q := (L1 + 8L2 ) < 1. (2.1.12) 64 Then the problem (2.1.1) has a unique solution u(x) ∈ C[0, 1], satisfying the estimate kuk ≤ M/64. Consider a particular case of Theorem 2.1. Denote + n M M o DM = (x, u, v) | x ∈ [0, 1], 0 ≤ u ≤ , − ≤ v ≤ 0 . (2.1.16) 64 8 Theorem 2.2.(Positivity of solution). Suppose that there exist numbers M, L1 , L2 ≥ 0 such that (i) + 0 ≤ f (x, u, v) ≤ M for any (x, u, v) ∈ DM . (2.1.17) (ii) |f (x, u2 , v2 ) − f (x, v1 , u1 )| ≤ L1 |u2 − u1 | + L2 |v2 − v1 | (2.1.18) + for any (x, ui , vi ) ∈ DM , i = 1, 2. (iii) 1 q := (L1 + 8L2 ) < 1. (2.1.19) 64 10
Then the problem (2.1.1) has a unique positive solution u(x) ∈ C[0, 1], satisfying the estimate 0 ≤ u(x) ≤ M/64. Consider the following iterative process: 1. Given ϕ0 (x) ∈ B[O, M ], for example, ϕ0 (x) = f (x, 0, 0). (2.1.20) 2. Knowing ϕk (k = 0, 1, ...) solve consecutively two problems 00 vk = ϕk (x), 0 < x < 1, (2.1.21) vk (0) = vk (1) = 0, 00 uk = vk (x), 0 < x < 1, uk (0) = uk (1) = 0. (2.1.22) 3. Update ϕk+1 = f (x, uk , vk ). (2.1.23) Theorem 2.3. Under the assumptions of Theorem 2.1 (or Theorem 2.2) the above iterative method converges with the rate of geometric progression and there holds the estimate qk ||uk − u|| ≤ ||ϕ1 − ϕ0 ||, (2.1.24) 64(1 − q) where u is the exact solution of the problem (2.1.1) and q is defined by (2.1.12). Lemma 2.2. (Monotony) Assume that all the conditions of Theorem 2.1 are satisfied. In addition, we assume that the function f (x, u, v) is increasing in (1) (2) u and decreasing in v for any (x, u, v) ∈ DM . Then, if ϕ0 , ϕ0 ∈ B[O, M ] (1) (2) are initial approximations and ϕ0 (x) ≤ ϕ0 (x) for any x ∈ [0, 1] then the (1) (2) sequences uk , uk generated by the iterative process satisfy the property (1) (2) uk (x) ≤ uk (x), k = 0, 1, ...; x ∈ [0, 1]. Theorem 2.4. Denote ϕmin = min f (x, u, v), ϕmax = max f (x, u, v). (x,u,v)∈DM (x,u,v)∈DM Under the assumptions of Lemma 2.1, if starting from ϕ0 = ϕmin we obtain the increasing sequence uk , inversely, starting from ϕ0 = ϕmax we obtain the decreasing sequence uk , both of them converge to the exact solution u(x) of the problem. Therefore, if ϕmin ≥ 0 the problem has nonnegative solution, inversely, if ϕmax ≤ 0 the problem has nonpositive solution. We show that the examples in the papers Bai (2004), Li (2010), Ma et al. (1997), Pao (2001) satisfy our conditions, therefore, have a unique solution, while only the existence of a solution is ensured there. In all examples we take 11
the starting approximation ϕ0 = f (x, 0, 0) and use the uniform grid with the number of grid points N = 100. The numerical experiments are performed until ek = kuk − uk−1 k ≤ 10−16 . (2.1.27) For showing the actual rate of convergence of the iterative method we use the ratios r(k) = e(k)/e(k − 1). Example 2.2. (see Bai (2004)). Consider the problem u(4) (x) = −5u00 − (u + 1)2 + sin2 πx + 1, u(0) = u(1) = u00 (0) = u00 (1) = 0. In this example f (x, u, v) = −5v − (u + 1)2 + sin2 πx + 1. We see that the conditions of Theorem 2.1 are satisfied with M = 3.5, L1 = 2.11, L2 = 5 and q ≈ 0.6580. Hence, the problem has a unique solution, and the iterative method converges. The numerical experiment shows that after k = 45 iterations the iterative process stops with e(45) = 5.8981e − 017, and the actual ratio of geometric progression is qact ≈ 0.4858 instead of the theoretically estimated q ≈ 0.6580 as above. The ratios r(k) and some iterations are depicted in Figure 2.1. From the figure we see the most decreasing of rk most as not changed. Figure 2.1: The ratios r(k) (left) and some iterations (right) in Example 2.2 Remark that in Bai (2004), author can only establish the existence but does not guarantee the uniqueness of a solution. As Li (2010), Bai also used the lower solution α = 0 and the upper solution β = sin πx. The sequences of approximations αn and βn in Bai (2004) are generated by solving the equation of the form u(4) + 5u00 + 4u = g(x) at each iteration. This equation is difficult to solve because the differential operator is impossible to decompose into the product of second order differential operators. 12
2.2 The boundary value problems for the fully nonlinear fourth-order differential equations In this section, we focus on the boundary problems for the fully nonlinear fourth-order differential equations with two different types of boundary condi- tions. 2.2.1 The case of boundary conditions of simply supported type Consider the problem u(4) (x) = f (x, u(x), u0 (x), u00 (x), u000 (x)), 0 < x < 1, (2.2.1) u(0) = u(1) = u00 (0) = u00 (1) = 0. where f : [0, 1] × R4 → R is continuous. This problem models the bending equilibrium of a beam on an elastic foundation, whose two ends are simply supported. For the fully fourth order nonlinear boundary value problem (2.2.1), in 2013, Li and Liang established the existence of solution for the problem under the restriction of the linear growth of the function f (x, u, y, v, z) in each variable on the infinity. In the paper [3], we consider the problem (2.2.1), too. Due to the reduction of the problem to an operator equation for the right hand side function, which will be proved to be contractive, we establish the existence and uniqueness of a solution and the convergence of an iterative method for finding the solution. The results are similar to those in the paper [2] with n M DM = (x, u, y, v, z) | 0 ≤ x ≤ 1, |u| ≤ , 64 (2.2.9) M M Mo |y| ≤ , |v| ≤ , |z| ≤ , 16 8 2 + n M DM = (x, u, y, v, z)| 0 ≤ x ≤ 1; 0 ≤ u ≤ ; 64 (2.2.22) M −M Mo |y| ≤ ; ≤ v ≤ 0; |z| ≤ , 16 8 2 We illustrate the obtained theoretical results on some examples, where the right- hand side functions do not satisfy the condition of linear growth at infinity, therefore, Li (2013) cannot ensure the existence of a solution of the problems. But as seen above using the theory in [3], we have established the existence and uniqeness of a solution and the convergence of the iterative method. This convergence is also confirmed by numerical experiments. Although having the same boundary conditions, in the boundary problem for non-fully fourth-order nonlinear differential equation (2.1.1), the sequence of solutions is monotonous but in fully fourth-order nonlinear differential equa- tion (2.2.1), the sequence does not have this property because it depends on the properties of the Green function and its derivatives corresponding to the problems. 13
2.2.2 The case of boundary conditions of clamped-free beam type Consider the problem u(4) (x) = f (x, u(x), u0 (x), u00 (x), u000 (x)), 0 < x < 1, (2.2.30) u(0) = u0 (0) = u00 (1) = u000 (1) = 0, which models a cantilever beam in equilibrium state, where f : [0, 1] × R4 → R is continuous. In 2016, under the assumptions that the function f (x, u, y, v, z) is superlin- ear or sublinear growth on u, y, v, z and satisfies a Nagumo-type condition on v and z, he established the existence of positive solutions of the problem (2.2.30). This interesting theoretical result is proved with the use of the theory of the fixed point index in cones in a very compilated way and is illustrated on two examples. In the paper [1], consider the (2.2.30), using the contraction mapping prin- ciple for an operator equation for the right-hand side function, we prove the existence and uniqueness of a solution of the problem. The positivity of solution also is studied. Besides, an iterative method for finding the solution is proposed and investigated. The applicability of our approach and the effectiveness of the iterative method are demonstrated on examples. Examples that do not satisfy the conditions in Li (2016), but the theories we make confirm the uniqueness of the problem. Furthermore, the conditions of our theorem are simpler and easily verified. To investigate the problem (2.2.30), for ϕ ∈ C[0, 1], consider the operator equation ϕ = Aϕ, (2.2.33) where A is defined by (Aϕ)(x) = f (x, uϕ (x), yϕ (x), vϕ (x), zϕ (x)), (2.2.34) where yϕ (x) = u0ϕ (x), zϕ (x) = vϕ0 (x). (2.2.35) Here vϕ (x), uϕ (x) respectively are the solutions of the sequence of problems 00 vϕ (x) = ϕ(x), 0 < x < 1, vϕ (1) = vϕ0 (1) = 0, (2.2.36) 00 uϕ (x) = vϕ (x), 0 < x < 1, uϕ (0) = u0ϕ (0) = 0. (2.2.37) Proposition 2.3. (The relation between the solution of the problem (2.2.30) with the solution of the operator equation (2.2.33)). If ϕ(x) is a solution of (2.2.33) where A is determined from (2.2.34)-(2.2.37) then uϕ (x) is a solution of the problem (2.2.30) and vice versa. The results are similar to those in the paper [2] with M M M DM = {(x, u, y, v, z)| 0 ≤ x ≤ 1, |u| ≤ , |y| ≤ , |v| ≤ , |z| ≤ M }, 8 6 2 (2.2.39) 14
+ n M DM = (x, u, y, v, z)| 0 ≤ x ≤ 1; 0 ≤ u ≤ ; 8 (2.2.57) M M o 0 ≤ y ≤ ; 0 ≤ v ≤ ; −M ≤ z ≤ 0 , 6 2 Consider the following iterative process: 1. Given ϕ0 (x) = f (x, 0, 0, 0, 0). (2.2.59) 2. Knowing ϕk (k = 0, 1, ...) solve consecutively two problems 00 vk = ϕk (x), 0 < x < 1, vk (1) = vk0 (1) = 0, (2.2.60) 00 uk = vk (x), 0 < x < 1, (2.2.61) uk (0) = u0k (0) = 0. 3. Update ϕk+1 = f (x, uk , u0k , vk , vk0 ). (2.2.62) According to the iterative method we have proposed above, numerical solving of the fully fourth order boundary value problem (2.2.30) is reduced to the solution of the sequence of the initial value problem (2.2.61) and the end value problem (2.2.60) for second order ordinary differential equations. Notice that the right- hand side functions of problems (2.2.60) v (2.2.61) depends only the variable x, so the approximate solutions in essence is to approximate the definite integrals. Hence it is possible to construct difference schemes with high order of accuracy although the right-hand side function of the problem is the discrete functions defined at grid points. For numerical realization of the iterative method we use Simpson’s rule of fourth order of accuracy for the problems (2.2.60), (2.2.61) on uniform grids ω h = {xi = ih, i = 0, 1, ..., N ; h = 1/N }. We provide a number of examples illustrating the effectiveness of theoretical results, including examples where Li (2016) does not guarantee the existence of the problem but uses the theory that they I propose to be able to establish the existence and uniqueness of solution and the iterative convergence method. In the context of the thesis, the method we propose applies to the prob- lem of two-point boundary value problem with continuous right-hand function, and for problems with non-continuous right-hand function the problem must be considered in the suitable space. Conclusion of Chapter 2 In this chapter, we study the solvability and iterative solution for fully or non-fully fourth-order nonlinear differential equations using the approach of re- ducing the original nonlinear boundary value problems to operator equations for right-hand side functions. The results are: - Establish the existence, uniqueness and some properties for the solutions of problems under easy to verify conditions. - Propose an iterative methods for solving these problems and prove the con- vergence of the iterative process. 15
- Give some examples illustrating the applicability of the obtained theoretical results including examples where the existence or uniqueness is not guaranteed by other authors because these examples do not satisfy the conditions in their theorems. - Computational experiments illustrate the effectiveness of iterative methods. 16
Chapter 3 Iterative method for solving boundary value problems for the systems of nonlinear fourth-order differential equations In this chapter, we study a method for solving the boundary problems for fully and non-fully nonlinear fourth-order differential equations with two types of boundary conditions. The results of this chapter are presented in articles [5], [6] in the list of works of the author related to the thesis. 3.1 The boundary value problem for a system of non-fully nonlinear fourth order differ- ential equations The problems for the fourth-order differential system have not been studied extensively, such as Kang et al. (2012), Lu et al. (2005), Zhu et al. (2010), in which the authors consider the equation contain only even derivatives. The theoretical pointers of the immobilization of the cone, the authors have obtained the existence of positive solutions. However, the results obtained are purely theoretical because no examples illustrate the existence of solutions. Consider the system of differential equations u (x) = f (x, u(x), v(x), u00 (x), v 00 (x)), 0 < x < 1, (4) (3.1.1) v (4) (x) = h(x, u(x), v(x), u00 (x), v 00 (x)), 0 < x < 1, with boundary conditions u(0) = u(1) = u00 (0) = u00 (1) = 0, v(0) = v(1) = v 00 (0) = v 00 (1) = 0. (3.1.2) where f, h : [0, 1] × R+ × R+ × R− × R− → R+ are continuous functions and u00 , v 00 in f, h are the bending moment terms which represent bending effect. In 2012, Kang et al. has established the existence of the positive solution of (3.1.1)-(3.1.2) with very complex conditions. Differently from the approaches of the other authors, in [5], we continue to develop techniques in articles [1]-[4] for the fourth-order nonlinear differential 17
equation (3.1.1)-(3.1.2). By reducing the problem to an operator equation for the pair of nonlinear terms but not for the pair of the functions to be sought (u, v), we establish the existence, uniqueness of solution under easily verified conditions. We assume that the functions f, h are continuous in a bounded domain of [0, 1] × R8 , which will be specified later. Then without any Nagumo- type conditions. We also investigate the convergence of an iterative method for finding approximate solutions and their monotony. Besides, we also prove the property of sign preserving of the solution and the convergence of an iterative method for finding the solution. Several examples, where exact solutions of the problem are known or not, demonstrate the effectiveness of the obtained theoretical results. ϕ To investigate the problem (3.1.1)-(3.1.2), vi w = ψ , ϕ, ψ ∈ C[0, 1], consider the operator equation w = T w, (3.1.9) where T is defined by (Aw)(x) f (x, uϕ (x), vψ (x), rϕ (x), zψ (x)) T w = (Bw)(x) = h(x, u (x), v (x), r (x), z (x)) . (3.1.10) ϕ ψ ϕ ψ Here rϕ (x), uϕ (x), zψ (x), vψ (x) respectively are the solutions of the sequence of problems 00 rϕ (x) = ϕ(x), 0 < x < 1, (3.1.11) rϕ (0) = rϕ (1) = 0, 00 uϕ (x) = rϕ (x), 0 < x < 1, (3.1.12) uϕ (0) = uϕ (1) = 0. 00 zψ (x) = ψ(x), 0 < x < 1, (3.1.13) zψ (0) = zψ (1) = 0, 00 vψ (x) = zψ (x), 0 < x < 1, (3.1.14) vψ (0) = vψ (1) = 0. Proposition 3.1. (The relation between the solution of the problem (3.1.1)- (3.1.2) with the solution of the operator equation (3.1.9)). If w(x) is a so- lution of (3.1.9), where T is determined from (3.1.10)-(3.1.14), then s(x) = (uϕ (x), vψ (x)) is a solution of the problem (3.1.1)-(3.1.2) and vice versa. For each number M > 0 denote n DM = (x, u, v, r, z)| 0 ≤ x ≤ 1, 5M 5M M Mo (3.1.21) |u| ≤ , |v| ≤ , |r| ≤ , |z| ≤ , 384 384 8 8 and as usual, by B[O, M ] we denote the closed ball centered at O with the radius M in the space F = (C[0, 1])2 , i.e., B[0, M ] = {w ∈ F : kwkF ≤ M } 18