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, funct[r]
(1)MINISTRY OF EDUCATION AND TRAINING
VIETNAM ACADEMY OF 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
(2)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:
(3)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 order differential equations and non-fully fourth-order ones A fourth-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, ≤ 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
(4)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 →
Rsatisfies 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), Lău 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 funcassump-tion satisfies the Nagumo condiassump-tion 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 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
(5)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 equareduc-tions for the right-hand side funcreduc-tions 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 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
(6)Chapter 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 will play a very important role, as the basis for the results which will be presented in Chapter and Chapter
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 methmeth-ods 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
(7)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 sequenceαk 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
(8)existence and uniqueness of solutions to problems Consider the problem of linear boundary value
L[y(x)] ≡ p0(x)
dny
dxn +p1(x)
dn−1y
dxn−1 + +pn(x)y = 0, (1.3.1)
Mi(y(a), y(b)) ≡ n−1
X
k=0
αikd
ky(a)
dxk +β i k
dky(b) dxk
= 0, i = 1, n, (1.3.2) 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 lim
x→t+
∂kG(x, t)
∂xk −xlim→t−
∂kG(x, t)
∂xk = 0, k = 0, , n−2
(iv) The (n−1)th derivative of G(x, t) is discontinuous when x = t, providing
lim
x→t+
∂n−1G(x, t)
∂xn−1 −xlim→t−
∂n−1G(x, t) ∂xn−1 = −
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 L[y(x)] ≡p0(x)
dny
dxn +p1(x)
dn−1y
dxn−1 + + pn(x)y = −f(x), (1.3.3)
subject to the homogeneous boundary conditions
Mi(y(a), y(b)) ≡ n−1
X
k=0
αikd
ky(a)
dxk +β i k
dky(b) dxk
(9)
where the coefficients pj(x) and the right-hand side term f(x) in the governing
equation are continuous functions, with p0(x) 6= 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
y(x) =
Z b
a
G(x, t)f(t)dt,
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))
(10)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)), Leray-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