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MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY - NGUYEN THANH HUONG SOLVING SOME NONLINEAR BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS Major: Applied Mathematics Code: 46 01 12 SUMMARY OF PHD THESIS Hanoi – 2019 This thesis has been completed: Graduate University of Science and Technology – Vietnam Academy of Science and Technology Supervisor 1: Prof Dr Dang Quang A Supervisor 2: Dr Vu Vinh Quang Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended at the Board of Examiners of Graduate University of Science and Technology – Vietnam Academy of Science and Technology at on The thesis can be explored at: - Library of Graduate University of Science and Technology - National Library of Vietnam INTRODUCTION Motivation of the thesis Many phenomena in physics, mechanics and other fields are modeled by boundary value problems for ordinary differential equations or partial differential equations with different boundary conditions The qualitative research as well as the method of solving these problems are always the topics attracting the attention of 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, Pham Ky Anh, Nguyen Dong Anh, Nguyen Huu Cong, Nguyen Van Dao, Le Luong Tai The existence, the uniqueness, the positivity of solutions and the iterative method for solving some boundary value problems for fourth order ordinary differential equations or partial differential equations have been considered in the works of Dang 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 The existence of solutions, positive solutions of the beam problems are considered in the works of T.F Ma (2000, 2003, 2004, 2007, 2010) Theory and numerical solution 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 ordinary differential equations and partial differential equations are received great interest by researchers because they are mathematical models of many problems in mechanics such as the bending of beams and plates It is possible to classify the fourth order differential equations into two forms: local fourth order differential equations and nonlocal ones A fourth order differential equation containing integral terms is called a nonlocal equation or a Kirchhoff type equation Otherwise, it is called a local equation Below, we will review some typical methods for studying boundary value problems for fourth order nonlinear differential equations The first method is the variational method, a common method of studying the existence of solutions of nonlinear boundary value problems With the idea of reducing 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 critical points There are many works using the variational method (see T.F Ma (2000, 2003, 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 noted that, using the variational method, most of authors consider the existence of solutions, the existence of multiple solutions of the problem (it is possible to consider the uniqueness of the solution in the case of convex functionals) but there are no examples of existing solutions, and the method for solving the problem has not been considered The next method is the upper and lower solutions method The main results of this 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 one solution and this solution is in the range of the upper and the lower solution under some additional assumptions In addition, we can construct two monotone sequences with the first approximation being the upper and the lower solution converge to the maximal and minimal solutions of the problem In the case of maximal and minimal solutions coincide, the problem has a unique solution We can mention some typical works using the upper and lower solutions method when studying boundary value problems for nonlinear fourth order differential equations as follows: J Ehme (2002), Z Bai (2004, 2007), Y.M Wang (2006, 2007), H Feng (2009), F Minh´os (2009) From the above works, we find that the upper and lower solutions method can establish the existence, the uniqueness of solution, and construct the iterative sequences converging to the solution with the very important assumption that these solutions exist but the finding of them is not easy In addition, they need other assumptions about the right-hand side function such as the growth at infinity or the Nagumo condition Except for the mentioned methods, scientists also use the fixed point methods in studying nonlinear boundary problems By using these methods, the original problem was reduced to the problem of finding fixed points of an operator, then applying the fixed point theorem to this operator (see R.P Agarwal (1984), B Yang (2005), P Amster (2008), T.F Ma (2010), S Yardimci (2014)) It should be emphasized that, in the works that apply the fixed point method to study nonlinear boundary problems, most authors reduce the given problem to the operator equation for the function to be sought Using the fixed point theorems such as ones of Schauder, Leray-Schauder, Krassnosel’skii for this operator, we can only establish the existence of solutions Using the Banach fixed point theorem, we not only establish the existence and uniqueness of solution but also construct an iterative method which converges with the rate of geometric progression However, it must be noted that the selection of the operator and considering this operator on a suitable space so that the assumptions put on the related functions are simple and still ensure the conditions to apply the the fixed point theorem in qualitative research as well as the method of solving nonlinear boundary problems plays very important role One of the popular numerical methods used in the approximation of boundary problems for fourth order ordinary differential equations and partial differential equations is finite difference method (see T.F Ma (2003), R.K Mohanty (2000), J Talwar (2012), Y.M Wang (2007)) By replacing derivatives by difference formulas, the problem is discretized into algebraic systems of equations Solving these systems, we obtain the approximate solution of the problem at grid nodes Note that when using finite difference method to study nonlinear boundary value problems, many works recognize the existence of solutions of the problem and discrete the problem from the beginning This approach has a disadvantage that it is difficult to evaluate the stability, the convergence of the difference scheme and the error between the exact solution and the approximate one When studying nonlinear boundary problems, in addition to the popular methods presented above, we can mention some other methods such as the finite element method, Taylor series method, Fourier series method, Brouwer theory, Leray-Schauder theory We can also combine the above methods to get the full study of both qualitative and quantitative aspects of the problem With the continuous development of science, technology, physics, mechanics, from practical problems in these areas, new boundary problems are posed more and more complex in both equations and boundary conditions Authors will use different methods, approaches and techniques for different problems Each proposed method will have its advantages and disadvantages and it is difficult to confirm that this method is really better than the other method from theory to experiment However, our method will study both quantitative and qualitative aspects of the problems so that the conditions are simple and easy to test We also give some numerical examples which illustrate the effectiveness of proposed method and compare with the results of other authors in some way For these reasons, we decide to choose the title ”Solving some nonlinear boundary value problems for fourth order differential equations” Objectives and scope of the thesis For some nonlinear boundary problems for fourth order ordinary differential equations and partial differential equations which are models of problems in bending theory of beams and plates: - Make qualitative research (the existence, the uniqueness, the positivity of solutions) by using fixed point theorems and maximum principles without infinite growth conditions, the Nagumo condition of the right-hand side function - Construct iterative methods for solving the problems - Give some examples illustrating the applicability of the obtained theoretical results, including examples showing the advantages of the method in the thesis compared with the methods of some other authors Research methodology and content of the thesis - Use the approach of reducing the original nonlinear boundary value problems to operator equations for the function to be sought or an intermediate function with the tools of mathematical analysis, functional analysis, theory of differential equation for studying the existence, the uniqueness and some properties of solutions of some problems for local and nonlocal fourth order differential equations - Propose iterative methods for solving these problems and prove the convergence of the iterative processes - Give some examples in both cases of known and unknown solution to illustrate the validity of theoretical results and examine the convergence of iterative methods The major contributions of the thesis 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 Major results: - Establish the existence, the uniqueness of the solutions of problems under some easily verified conditions Consider the positivity of the solution of the boundary problem for fourth order ordinary differential equations with Dirichlet boundary condition, combined boundary conditions and the boundary problem for biharmonic equation - Propose iterative methods for solving these problems and prove the convergence with the rate of geometric progression of the iterative processes - Give some examples for illustrating the applicability of the obtained theoretical results, including examples showing the advantages of the method in the thesis compared with the methods of other authors - Perform experiments for illustrating the effectiveness of iterative methods The thesis is written on the basis of articles [A1]-[A8] in the list of works of the author related to the thesis Besides the introduction, conclusion and references, the contents of the thesis are presented in three chapters The results in the thesis were reported and discussed at: 11th Workshop on Optimization and Scientific Computing, Ba Vi, 24-27/4/2013 4th National Conference on Applied Mathematics, Hanoi, 23-25/12/2015 14th Workshop on Optimization and Scientific Computing, Ba Vi, 21-23/4/2016 Conference of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 12-13/11/2016 10th National Conference on Fundamental and Applied Information Technology Research (FAIR’10), Da Nang, 17-18/8/2017 The second Vietnam International Applied Mathematics Conference (VIAMC 2017), Ho Chi Minh, December 15 to 18, 2017 Scientific Seminar of the Department of Mathematical methods in Information Technology, Institute of Information Technology, Vietnam Academy of Science and Technology Chapter Preliminary knowledge This chapter presents some preparation knowledge needed for subsequent chapters referenced from the literatures of A.N Kolmogorov and S.V Fomin (1957), E Zeidler (1986), A.A Sammarskii (1989, 2001), A Granas and J Dugundji (2003), J Li (2005), Dang Quang A (2009), R.L Burden (2011) • Section 1.1 recalls three fixed point theorems: Brouwer fixed point theorem, Schauder fixed point theorem, Banach fixed point theorem • Section 1.2 presents the definition of the Green function for the boundary value problem for linear differential equations of order n and some specific examples of how to define the Green function of boundary problems for second order and fourth order differential equations with different boundary conditions • Section 1.3 gives some formulas for approximation derivatives and integrals with second order and fourth order accuracy • Section 1.4 presents the formula for approximation of Poisson equation with fourth order accuracy • Section 1.5 mentions the elimination method for three-point equations and the cyclic reduction method for three-point vector equations Chapter The existence and uniqueness of a solution and the iterative method for solving boundary value problems for nonlinear fourth order ordinary equations Chapter investigates 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: simply supported type, Dirichlet boundary condition, combined boundary conditions, nonlinear boundary conditions By using the reduction of these problems to the operator equations for the function to be sought or for an intermediate function, we prove that under some assumptions, which are easy to verify, the operator is contractive Then, the uniqueness of a solution is established, and the iterative method for solving the problem converges This chapter is written on the basis of articles [A2]-[A4], [A6]-[A8] in the list of works of the author related to the thesis 2.1 The boundary value problem for the local nonlinear fourth order differential equation 2.1.1 The case of combined boundary conditions The thesis presents in detail the results of the work [A4] for the problem u(4) (x) = f (x, u(x), u (x), u (x), u (x)), < x < 1, u(0) = 0, u (1) = 0, au (0) − bu (0) = 0, cu (1) + du (1) = 0, (2.1.1) where a, b, c, d ≥ 0, ρ := ad + bc + ac > and f : [0, 1] × R4 → R is a continuous function 2.1.1.1 The existence and uniqueness of a solution For function ϕ(x) ∈ C[0, 1], consider the nonlinear operator A : C[0, 1] → C[0, 1] defined by (Aϕ)(x) = f (x, u(x), u (x), u (x), u (x)), (2.1.2) where u(x) is a solution of the problem u(4) (x) = ϕ(x), < x < 1, u(0) = 0, u (1) = 0, au (0) − bu (0) = 0, cu (1) + du (1) = (2.1.3) Proposition 2.1 A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x) is a solution of the operator equation ϕ = Aϕ if and only if the function u(x) determined from the boundary value problem (2.1.3) satisfiesthe problem (2.1.1) Set v(x) = u (x), the problem (2.1.3) can be decomposed into two second problems v (x) = ϕ(x), < x < 1, av(0) − bv (0) = 0, cv(1) + dv (1) = 0, u (x) = v(x), < x < 1, u(0) = 0, u (1) = Then (Aϕ)(x) = f (x, u(x), y(x), v(x), z(x)), y(x) = u (x), z(x) = v (x) For any number M > 0, we define the set DM = (x, u, y, v, z) | ≤ x ≤ 1, |u| ≤ ρ1 M, |y| ≤ ρ2 M, |v| ≤ ρ3 M, |z| ≤ ρ4 M , where ρ1 = ρ3 = 2ad + bc + 6bd + , 24 12ρ a(d + c/2) ρ + ρ2 = b(d + c/2) , ρ ad + bc + 4bd + , 12 4ρ ρ4 = ac + max(ad, bc) ρ Denote the closed ball in the space C[0, 1] by B[O, M ] Lemma 2.1 Assume that there exist constants M > 0, K1 , K2 , K3 , K4 ≥ such that |f (x, u, y, v, z)| ≤ M for all (x, u, y, v, z) ∈ DM Then, the operator A maps B[O, M ] into itself Furthermore, if |f (x, u2 ,y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )| ≤ K1 |u2 − u1 | + K2 |y2 − y1 | + K3 |v2 − v1 | + K4 |z2 − z1 | (2.1.4) for all (t, ui , yi , vi , zi ) ∈ DM (i = 1, 2) and q = K1 ρ1 + K2 ρ2 + K3 ρ3 + K4 ρ4 < (2.1.5) then A is a contraction operator in B[O, M ] Theorem 2.1 Assume that all the conditions of Lemma 2.1 are satisfied Then the problem (2.1.1) has a unique solution u and u ≤ ρ1 M, u ≤ ρ2 M, u ≤ ρ3 M, u ≤ ρ4 M where u(x) is the solution of the problem u(4) (x) = ϕ(x), u(a) = u(b) = 0, a < x < b, (2.1.8) u (a) = u (b) = Proposition 2.2 If the function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x) is a solution of the operator equation ϕ = Aϕ (2.1.9) then the function u(x) determined from the boundary value problem (2.1.8) solves the problem (2.1.6) Conversely, ifu(x) is a solution of the boundary value problem (2.1.6) then the function ϕ(x) = f (x, u(x), u (x), u (x), u (x)) is a fixed point of the operator A defined above by (2.1.7) and (2.1.8) Thus, the solution of the problem (2.1.6) is reduced to the solution of the operator equation (2.1.9) For any number M > 0, we define the set DM = (x, u, y, v, z) | a ≤ x ≤ b, |u| ≤ C4,0 (b − a)4 M, |y| ≤ C4,1 (b − a)3 M, |v| ≤ C4,2 (b − a)2 M, |z| ≤ C4,3 (b − a)M , √ where C4,0 = 1/384, C4,1 = 1/72 3, C4,2 = 1/12, C4,3 = 1/2 By using Schauder fixed point theorem and Bannach fixed point theorem for the operator A, we establish the existence and uniqueness theorems of the problem (2.1.6) Theorem 2.4 Suppose that the function f is continuous and there exists constant M > such that |f (x, u, y, v, z)| ≤ M for all (x, u, y, v, z) ∈ DM Then, the problem (2.1.6) has at least a solution Theorem 2.5 Suppose that the assumptions of Theorem 2.4 hold Additionally, assume that there exist constants K0 , K1 , K2 , K3 ≥ such that |f (x, u2 , y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )| ≤ K0 |u2 − u1 | + K1 |y2 − y1 | + K2 |v2 − v1 | + K3 |z2 − z1 |, (2.1.10) for all (x, ui , yi , vi , zi ) ∈ DM (i = 1, 2) and Ki C4,k (b − a)4−k < q= k=0 Then the problem (2.1.6) has a unique solution u and u ≤ C4,0 (b − a)4 M, u ≤ C4,1 (b − a)3 M, ≤ C4,2 (b − a)2 M, u ≤ C4,3 (b − a)M u 11 (2.1.11) Denote + DM = (x, u, y, v, z) | a ≤ x ≤ b, ≤ u ≤ C4,0 (b − a)4 M, |y| ≤ C4,1 (b − a)3 M, |v| ≤ C4,2 (b − a)2 M, |z| ≤ C4,3 (b − a)M + the function f is such Theorem 2.6 (Positivity of solution) Suppose that in DM that ≤ f (t, x, y, u, z) ≤ M and the conditions (2.1.10), (2.1.11) of Theorem 2.5 are satisfied The the problem (2.1.6) has a unique nonnegative solution 2.1.2.2 Solution method and numerical examples The iterative method for solving the problem (2.1.6) is proposed as follows: Iterative method 2.1.2 i) Given ϕ0 (x), for example, ϕ0 (x) = f (x, 0, 0, 0, 0) b ii) Knowing ϕk (x), (k = 0, 1, 2, ) calculate uk (x) = a G(x, t)ϕk (t)dt and the (m) derivatives uk (x) of uk (x) (m) uk (x) b = a ∂ m G(x, t) ϕk (t)dt (m = 1, 2, 3) ∂xm iii) Update ϕk+1 (x) = f (x, uk (x), uk (x), uk (x), uk (x)) k q Set pk = ϕ1 − ϕ0 We have the following result: 1−q Theorem 2.7 Under the assumptions of Theorem 2.5, Iterative method 2.1.2 converges with the rate of geometric progression and there hold the estimates uk − u ≤ C4,0 (b − a)4 pk , uk − u ≤ C4,1 (b − a)3 pk , ≤ C4,2 (b − a)2 pk , uk − u ≤ C4,3 (b − a)pk , uk − u where u is the exact solution of the problem (2.1.6) Ch 2.1 Consider the problem u(4) (x) = f (x, u(x), u (x), u (x), u (x)), u(a) = A1 , u(b) = B1 , u (a) = A2 , a < x < b, u (b) = B2 (2.1.12) Set v(x) = u(x) − P (x), where P (x) is the third degree polynomial satisfying the boundary conditions in this problem and denote F (x, v(x), v (x), v (x), v (x)) = f (x, v(x) + P (x), (v(x) + P (x)) , (v(x) + P (x)) , (v(x) + P (x)) ) Then, the problem (2.1.12) becomes v (4) (x) = F (x, v(x), v (x), v (x), v (x)), v(a) = v(b) = 0, v (a) = v (b) = a < x < b, Therefore, we can apply the results derived above to this problem 12 Theorem 2.8 Suppose that the function f is continuous and there exists constan M > such that |f (x, v0 , v1 , v2 , v3 )| ≤ M for all (x, v0 , v1 , v2 , v3 ) ∈ DM , where DM = (x, v0 , v1 , v2 , v3 ) | a ≤ x ≤ b, |vi | ≤ max |P (i) (x)| x∈[a,b] + C4,i (b − a)4−i M, i = 0, 1, 2, Then, the problem (2.1.12) has at least a solution We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis compared to the methods of R.P Agarwal (1984): Agarwal can only establish the existence of a solution of the problem or does not guarantee the existence of a solution of the problem meanwhile according to the proposed method, the problem has a unique solution or a unique positive solution 2.1.3 The case of nonlinear boundary conditions The thesis presents in detail the results of the work [A7] for the problem u(4) (x) = f (x, u, u ), u(0) = 0, u(L) = 0, < x < L, u (0) = g(u (0)), u (L) = h(u (L)) (2.1.13) Set u = v, u = w Then, the problem (2.1.13) is decomposed to the problems for w v u  x   w (x) = f x, v(t)dt, v(x) , < x < L, u (x) = w(x), < x < L,   w(0) = g(v(0)), u(0) = 0, w(L) = h(v(L)), u(L) = The solution u(x) from these problems depends on the function v Consequently, its derivative u also depends on v Therefore, we can represent this dependence by an operator T : C[0, L] → C[0, L] defined by T v = u Combining with u = v we get the operator equation v = T v, i.e., v is a fixed point of T To consider properties of the operator T, we introduce the space L S = v ∈ C[0, L], v(t)dt = We make the following assumptions on the given functions in the problem (2.1.13): there exist constants λf , λg , λh ≥ such that |f (x, u, v) − f (x, u, v)| ≤ λf max |u − u|, |v − v|, |g(u) − g(u)| ≤ λg |u − u|, |h(u) − h(u)| ≤ λh |u − u|, (2.1.14) for any u, u, v, v Applying Banach fixed point theorem for T, we establish the existence and uniqueness of a solution of the problem 13 Proposition 2.3 With assumption (2.1.14), the problem (2.1.13) has a unique solution if L L3 L q= λf max , + (λg + λh ) < (2.1.15) 16 2 The iterative method for solving the problem (2.1.13) is proposed as follows: Iterative method 2.1.3 (i) Given an initial approximation v0 (x), for example, v0 (x) = (ii) Knowing vk (x) (k = 0, 1, 2, ) solve consecutively two problems   w (x) = f x, x v (t)dt, v (x) , < x < L, uk (x) = wk (x), < x < L, k k k  wk (0) = g(vk (0)), wk (L) = h(vk (L)), uk (0) = uk (L) = (iii) Update vk+1 (x) = uk (x) Theorem 2.9 Under the assumptions (2.1.14), (2.1.15), Iterative method 2.1.3 converges with rate of geometric progression with the quotient q, and there hold the estimates uk − u ≤ qk v1 − v0 , 1−q uk − u ≤ L u −u , k where u is the exact solution of the original problem (2.1.13) For testing the convergence of the method, we perform some experiments for the case of the known exact solutions and also for the case of the unknown exact solutions 2.2 2.2.1 The boundary value problem for the nonlocal nonlinear fourth order differential equation The case of boundary conditions of simply supported type The thesis presents in detail the results of the work [A2] for the problem L (4) |u (s)|2 ds u (x) u (x) − M = f (x, u(x), u (x), u (x), u (x)), < x < L, u(0) = u(L) = 0, u (0) = u (L) = 2.2.1.1 (2.2.1) The existence and uniqueness of a solution For function ϕ(x) ∈ C[0, L], consider the nonlinear operator A : C[0, L] → C[0, L] defined by (Aϕ)(x) = M ( u 2 )u (x) + f (x, u(x), u (x), u (x), u (x)), 14 (2.2.2) where is the norm in L2 [0, L], u(x) is a solution of the problem u(4) (x) = ϕ(x), < x < L, u(0) = u(L) = 0, u (0) = u (L) = (2.2.3) Proposition 2.4 A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x) is a solution of the operator equation ϕ = Aϕ if and only if the function u(x) determined from the boundary value problem (2.2.3) satisfies the problem (2.2.1) By setting v(x) = u (x), the problem (2.2.3) is decomposed to the problems v (x) = ϕ(x), < x < L, v(0) = v(L) = 0, u (x) = v(x), < x < L, u(0) = u(L) = Then the operator A is represented in the form (Aϕ)(x) := M ( y 22 )v(x) + f (x, u(x), y(x), v(x), z(x)), y(x) = u (x), z(x) = v (x) For any number R > 0, we define the set DR := (x, u, y, v, z) | ≤ x ≤ L, |u| ≤ L3 R L2 R LR 5L4 R , |y| ≤ , |v| ≤ , |z| ≤ 384 24 Let B[O, R] denote the closed ball in the space C[0, L] Lemma 2.2 If there are constants R > 0, ≤ m ≤ , λM , K1 , K2 , K3 , K4 ≥ L such that mL2 , |M (s)| ≤ m, |f (x, u, y, v, z)| ≤ R − R2 L7 for all (x, u, y, v, z) ∈ DR and ≤ s ≤ , then, the operator A maps B[O, R] 576 into itself If, in addition, |M (s2 ) − M (s1 )| ≤ λM |s2 − s1 |, |f (x, u2 , y2 , v2 , z2 ) − f (x, u1 , y1 , v1 , z1 )| ≤ K1 |u2 − u1 | + K2 |y2 − y1 | + K3 |v2 − v1 | + K4 |z2 − z1 |, R L7 (i = 1, 2) and for all (x, ui , yi , vi , zi ) ∈ DR , ≤ si ≤ 576 q = K1 5L4 L3 L2 L mL2 λM R2 L9 + K2 + K3 + K4 + + such that |f (t, u)| ≤ A, ∀(t, u) ∈ [0, L] × [−L2 R, L2 R], |g(u)| ≤ B, ∀u ∈ [−L2 R, L2 R], |M (s)| ≤ m, Then, if L2 A ∀s ∈ [0, L3 R ] + LB ≤ R(1 − mL2 ), the problem (2.2.4) has at least a solution Theorem 2.13 Suppose that the assumptions of Theorem 2.12 hold Further assume that there exist constants λf , λg , λM > such that |f (x, u) − f (x, v)| ≤ λf |u − v|, |g(u) − g(v)| ≤ λg |u − v|, ∀(x, u), (x, v) ∈ [0, L] × [−L2 R, L2 R], ∀u, v ∈ [−L2 R, L2 R], ∀u, v ∈ [0, L3 R ] |M (u) − M (v)| ≤ λM |u − v|, Then, if q = 4L5 R λM + unique solution 2.2.2.2 L4 λf + L3 λg + 2mL2 < 1, the problem (2.2.4) has a Iterative method and numerical examples The iterative method for solving the problem (2.2.4) is proposed as follows: Iterative method 2.2.2 i) Given an initial approximation u0 (x), example, u0 (x) = 0, in [0, L] ii) Knowing uk (x) (k = 0, 1, 2, ) solve consecutively the final value problem vk (x) = f (x, uk (x)), vk (L) = −M uk 2 < x < L, uk (L), uk+1 (x) = vk (x) + M uk uk+1 (0) = uk+1 (0) = 17 2 vk (L) = g(uk (L)), uk (x), < x < L, Theorem 2.14 Under the assumptions of Theorem 2.13, Iterative method 2.2.2 converges with rate of geometric progression with the quotient q and there hold the estimates uk − u ∞ ≤ L uk − u ∞ ≤ L2 uk − u ∞ ≤ L2 qk u − u0 1−q ∞, where u is the exact solution of the original problem (2.2.4) We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis compared to the methods of T.F Ma (2003): According to the proposed method, the problem has a unique solution meanwhile Ma’s method can only establish the existence of a solution or cannot ensure the existence of a solution CONCLUSION OF CHAPTER In this chapter, we investigate the unique solvability and iterative method for five boundary value problems for local or nonlocal nonlinear fourth order differential equations with different boundary conditions: The case of boundary conditions of simply supported type, combined boundary conditions, Dirichlet boundary condition, nonlinear boundary conditions By using the reduction of these problems to the operator equations for the function to be sought or for an intermediate function, we prove that under some assumptions, which are easy to verify, the operator is contractive Then, the uniqueness of a solution is established, and the iterative method for solving the problem converges We also give some examples for illustrating the applicability of the obtained theoretical results, including examples showing the advantages of the method in the thesis compared with the methods of other authors 18 Chapter The existence and uniqueness of a solution and the iterative method for solving boundary value problems for nonlinear fourth order partial equations Continuing the development of the techniques in Chapter 2, in Chapter 3, we also obtain the results of the existence and uniqueness of a solution, and the convergence of iterative methods for solving two boundary value problems for a nonlinear biharmonic equation and a nonlinear biharmonic equation of Kirchhoff type The results of this chapter are presented in articles [A1], [A4] in the list of works of the author related to the thesis 3.1 The nonlinear boundary value problem for the biharmonic equation The thesis presents in detail the results of the work [A5] for the problem ∆2 u = f (x, u, ∆u), x ∈ Ω, u = 0, ∆u = 0, x ∈ Γ, (3.1.1) where Ω is a connected bounded domain in R2 with a smooth (or piecewise smooth) boundary Γ 3.1.1 The existence and uniqueness of a solution For function ϕ(x) ∈ C(Ω), consider the nonlinear operation A : C(Ω) → C(Ω) defined by (Aϕ)(x) = f (x, u(x), ∆u(x)), (3.1.2) where u(x) is a solution of the problem ∆2 u = ϕ(x), x ∈ Ω, u = ∆u = 0, x ∈ Γ (3.1.3) Proposition 3.1 A function ϕ(x) is a solution of the operator equation Aϕ = ϕ, i.e., ϕ(x) is a fixed point of the operator A defined by (3.1.2)-(3.1.3) if and only if the function u(x) being the solution of the boundary value problem (3.1.3) solves the problem (3.1.1) 19 Lemma 3.1 Suppose that Ω is a connected bounded domain in RK (K ≥ 2) with a smooth boundary (or smooth of each piece) Γ Then, for the solution of the problem −∆u = f (x), x ∈ Ω, u = 0, x ∈ Γ, there holds the estimate R u ≤ CΩ f , where u = maxx∈Ω¯ |u(x)|, CΩ = and R is the radius of the circle containing the domain Ω If Ω is the unit square then u ≤ f For each positive number M denote DM = {(x, u, v)| x ∈ Ω, |u| ≤ CΩ2 M, |v| ≤ CΩ M } Theorem 3.1 Assume that there exist numbers M, K1 , K2 ≥ such that |f (x, u, v)| ≤ M, ∀(x, u, v) ∈ DM , |f (x, u2 , v2 ) − f (x, u1 , v1 )| ≤ K1 |u2 − u1 | + K2 |v2 − v1 |, ∀(x, ui , vi ) ∈ DM , i = 1, q := (K2 + CΩ K1 )CΩ < ¯ and u ≤ C M Then the problem (3.1.1) has a unique solution u(x) ∈ C(Ω) Ω Denote + = {(x, u, v)| x ∈ Ω, ≤ u ≤ CΩ2 M, −CΩ M ≤ v ≤ 0} DM Theorem 3.2 (Positivity of solution) Assume that there exist numbers M, K1 , K2 ≥ such that + ≤ f (x, u, v) ≤ M, ∀(x, u, v) ∈ DM + |f (x, u2 , v2 ) − f (x, u1 , v1 )| ≤ K1 |u2 − u1 | + K2 |v2 − v1 |, ∀(x, ui , vi ) ∈ DM , i = 1, 2, q := (K2 + CΩ K1 )CΩ < ¯ and Then the problem (3.1.1) possesses a unique positive solution u(x) ∈ C(Ω) ≤ u(x) ≤ CΩ2 M 3.1.2 Solution method and numerical examples Consider the following iterative process for finding fixed point ϕ of the operator A and simultaneously for finding the solution u of the original boundary value problem: Iterative method 3.1.1 Given an initial approximation ϕ0 ∈ B[O, M ], for example, ϕ0 (x) = f (x, 0, 0), x ∈ Ω 20 (3.1.4) Knowing ϕk Ω (k = 0, 1, ) solve sequentially two Poisson problems ∆vk = ϕk , vk = 0, x ∈ Ω, x ∈ Γ, ∆uk = vk , uk = 0, x ∈ Ω, x ∈ Γ (3.1.5) Update ϕk+1 = f (x, uk , vk ) (3.1.6) Theorem 3.3 Suppose that the assumptions of Theorem 3.1 (or Theorem 3.2) hold Then Iterative method 3.1.1 converges and there holds the estimate qk ||uk − u|| ≤ CΩ ϕ1 − ϕ0 , (1 − q) where u is the exact solution of the problem (3.1.1) Theorem 3.4 (Monotony) Assume that all the conditions of Theorem 3.1 (or Theorem 3.2) are satisfied In addition, we assume that the function f (x, u, v) is (1) (2) increasing in u and decreasing in v for any (x, u, v) ∈ DM Then, if ϕ0 , ϕ0 ∈ (2) (1) B[O, M ] are initial approximations and ϕ0 (x) ≤ ϕ0 (x) for any x ∈ Ω then the (1) (2) sequences {uk }, {uk } generated by the iterative process (3.1.4)-(3.1.6) satisfy the relation (2) (1) uk (x) ≤ uk (x), k = 0, 1, ; x ∈ Ω Corollary 3.1 Denote ϕmin = (x,u,v)∈DM f (x, u, v), ϕmax = max f (x, u, v) (x,u,v)∈DM Under the assumptions of Theorem 3.4, if starting from ϕ0 = ϕmin we obtain the increasing sequence {uk (x)}, inversely, starting from ϕ0 = ϕmax we obtain the decreasing sequence {uk (x)}, both of them converge to the exact solution u(x) of the problem and uk (x) ≤ u(x) ≤ uk (x) To numerically realize the above iterative method, we use difference schemes of second and fourth order of accuracy for solving second order boundary value problems (3.1.5) at each iteration The numerical examples show the advantage of the proposed method compared to the methods in Y.M Wang (2007), Y An, R Liu (2008), S Hu, L Wang (2014) on the convergence rate or conclusions about the uniqueness of the solution 3.2 The nonlinear boundary value problem for the biharmonic equation of Kirchhoff type The thesis presents in detail the results of the work [A1] for the problem ∆2 u = M |∇u|2 dx ∆u + f (x, u), Ω u = 0, ∆u = 0, x ∈ Ω, (3.2.1) x ∈ Γ, where Ω is a connected bounded domain in RK (K ≥ 2) with a smooth (or piecewise smooth) boundary Γ 21 3.2.1 The existence and uniqueness of a solution For function ϕ(x) ∈ C(Ω), consider the nonlinear operator A : C(Ω) → C(Ω) defined by |∇u|2 dx ∆u + f (x, u), (Aϕ)(x) = M (3.2.2) Ω where u(x) is a solution of the problem ∆2 u = ϕ(x), x ∈ Ω, u = ∆u = 0, x ∈ Γ (3.2.3) Proposition 3.2 A function ϕ(x) is a fixed point of the operator A, i.e., ϕ(x) is a solution of the operator equationAϕ = ϕ, if and only if the function u(x) determined from the boundary value problem (3.2.3) satisfies the problem (3.2.1) For any number R > and the coefficient CΩ defined by Lemma 3.1, we define the set DR = (x, u) | x ∈ Ω; |u| ≤ CΩ2 R Theorem 3.5 Assume that there exist constants R, λf , m, λM > 0, m ≤ 1/CΩ such that |M (s)| ≤ m, |M (s1 ) − M (s2 )| ≤ λM |s1 − s2 |, |f (x, u)| ≤ R(1 − mCΩ ), |f (x, u1 ) − f (x, u2 )| ≤ λf |u1 − u2 |, for all (x, u), (x, ui ) ∈ DR (i = 1, 2); ≤ s, s1 , s2 ≤ CΩ3 R2 SΩ , q = λf CΩ2 + mCΩ + 2λM R2 CΩ4 SΩ < 1, where SΩ is the measure of the domain Ω Then the problem (3.2.1) has a unique solution u(x) ∈ C(Ω) which satisfies the estimates u ≤ CΩ2 R, ∆u ≤ CΩ R Remark 3.1 As seen from Theorem 3.5 for the existence and uniqueness of solution of the problem (3.2.1) we require the conditions on the function f (x, u) and M (s) only in bounded domains Due to this the assumptions on the growth of these functions at infinity, which are needed in F Wang, Y An (2012) for the case when M is a function and in other works for the case M = const are freed This is an advantage of our result over the results of others Moreover, the conditions of Theorem 3.5 are simple and are easy to be verified as will be seen from the numerical examples Remark 3.2 In Theorem 3.5, if M (s) = m = const then λM = and by q = λf CΩ2 + mCΩ < Thus, the assumptions of the theorem are reduced to the boundedness and the satisfaction of Lipschitz condition of f (x, u) in the domain DR These conditions obviously are not complicated as in Y An, R Liu (2008), R Pei (2010), S Hu, L Wang (2014) 22 Remark 3.3 In the case if the right-hand side function f = f (u), the conditions for f in Theorem 3.5 become the boundedness and the Lipschitz condition for f (u) in the domain DR = u; |u| ≤ CΩ2 R 3.2.2 Solution method and numerical examples The iterative method for solving the problem (3.1.1) is proposed as follows: Iterative method 3.2.1 i) Given ϕ0 ∈ B[O, R], for example, ϕ0 (x) = f (x, 0), x ∈ Ω ii) Knowing ϕk (k = 0, 1, 2, ) solve successively two second order problems iii) Update ∆vk = ϕk , vk = 0, x ∈ Ω, x ∈ Γ, ϕk+1 (x) = M ∆uk = vk , uk = 0, Ω |∇uk | dx x ∈ Ω, x ∈ Γ (3.2.4) vk + f (x, uk ) Theorem 3.6 Under the assumptions of Theorem 3.5, Iterative method 3.2.1 converges and there holds the estimate CΩ2 q k uk − u ≤ ϕ1 − ϕ0 , 1−q where u is the exact solution of the problem (3.2.1) In order to numerically realize the above iterative process we use the difference scheme of fourth order accuracy for solving (3.2.4), and formulas of fourth order accuracy for approximating ∇u In order to test the convergence of the proposed iterative method we perform some experiments for the case of known exact solutions and also for the case of unknown exact solutions of the problem (3.2.1) in unit square Some examples show the advantage of the proposed method compared to the method in F Wang, Y An (2012) in the conclusion of the existence and uniqueness of solution of the problem CONCLUSION OF CHAPTER In this chapter, we investigate the unique solvability and iterative method for two boundary value problems for nonlinear biharmonic equation and nonlinear biharmonic equation of Kirchhoff type - For both problems, under some easily verified conditions, we establish the existence and uniqueness of solution Especially, for the boundary problem for the biharmonic equation, we also consider the positive property of the solution - We propose iterative methods for solving these problems and prove the convergence of the iterative process Especially, for the boundary problem for the biharmonic equation, we also show the monotonicity of the approximation sequences 23 - We give some examples for illustrating the applicability of the obtained theoretical results including examples which the existence or uniqueness is not guaranteed by other authors CONCLUSIONS By using the reduction of the original problem to the operator equations for the function to be sought or for an intermediate function and applying the fixed point theorem to this operator, the thesis proposes a simple but very effective method to investigate the unique solvability and iterative methods for solving the problems The main results of the thesis include: We establish the existence and uniqueness of solution and propose iterative methods for solving some boundary value problems for local or nonlocal nonlinear fourth order ordinary differential equations with different boundary conditions In addition, for the problems with Dirichlet boundary condition and combined boundary conditions, we also consider the positivity of the solution For the boundary problem for the biharmonic equation and the biharmonic equation of Kirchhoff type, we establish the existence and uniqueness of the solution, propose the iterative method for finding solution In addition, for the problem for the biharmonic equation, we also consider the positivity of the solution We give some examples for illustrating the applicability of the obtained theoretical results, including examples of advantages of the method in the thesis compared to the methods of other authors We perform experiments for illustrating the effectiveness of iterative methods FUTURE RESEARCH Study nonlinear boundary problems for higher order ordinary differential equations with other boundary conditions Study nonlinear boundary problems for fourth order and higher order partial differential equations with other boundary conditions and the complex righthand side Study the nonlinear boundary problem for the fourth and higher order differential equations systems with more complex boundary conditions 24 THE LIST OF WORKS OF THE AUTHOR RELATED TO THE THESIS [A1] Quang A Dang, Thanh Huong Nguyen (2018), Existence results and iterative method for solving a nonlinear biharmonic equation of Kirchhoff type, Computers and Mathematics with Applications, 76, pp 11-22 (SCI) [A2] Dang Quang A, Nguyen Thanh Huong (2018), The unique solvability and approximation of BVP for a nonlinear fourth order Kirchhoff type equation, East Asian Journal on Applied Mathematics, 8(2), pp 323-335 (SCIE) [A3] Quang A Dang, Thanh Huong Nguyen (2019), Solving the Dirichlet problem for fully fourth order nonlinear differential equation, Afrika Matematika, 30, pp 623-641 (ESCI, SCOPUS) [A4] Dang Quang A, Nguyen Thanh Huong (2018), Existence results and numerical method for a fourth order nonlinear problem, International Journal of Applied and Computational Mathematics, 4:148, DOI 10.1007/s40819-0180584-9 (SCOPUS) [A5] Dang Quang A, Truong Ha Hai, Nguyen Thanh Huong, Ngo Thi Kim Quy (2017), Solving a nonlinear biharmonic boundary value problem, Journal of Computer Science and Cybernetics, 33(4), pp 309-324 [A6] Dang Quang A, Nguyen Thanh Huong (2016), Existence results and iterative methods for solving a nonlocal fourth order boundary value problem, Journal of Mathematical Applications, 14(2), pp 63-78 [A7] Quang A Dang, Nguyen Thanh Huong (2013), Iterative method for solving a beam equation with nonlinear boundary conditions, Advances in Numerical Analysis, Volume 2013, Article ID 470258, pages [A8] Vu Vinh Quang, Nguyen Thanh Huong (2017), Difference schemes for solving boundary value problems for high order linear and nonlinear differential equation, Proceeding of the 10th National Conference on Fundamental and Applied Information Technology Research (FAIR’10), Hanoi, pp 358-368 25 ... of Science and Technology – Vietnam Academy of Science and Technology Supervisor 1: Prof Dr Dang Quang A Supervisor 2: Dr Vu Vinh Quang Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be... Quang A, Pham Ky Anh, Nguyen Dong Anh, Nguyen Huu Cong, Nguyen Van Dao, Le Luong Tai The existence, the uniqueness, the positivity of solutions and the iterative method for solving some boundary... difference scheme of fourth order accuracy for solving this problem as follows  c1  c v − (−25v0 + 48v1 − 36v2 + 16v3 − 3v4 ) = F0 , 0   12h vi 1 − 2vi + vi+ 1 = Fi , i = 1, 2, , N − 1,    d v +
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