Iterative method for solving two - Point boundary value problems for fourth order differential equations and systems

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 〈α, β〉, 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 L[y(x)] ≡ p0(x)d ny dxn + p1(x) dn−1y dxn−1 + ...+ pn(x)y = 0, (1.3.1) Mi(y(a), y(b)) ≡ n−1∑ k=0 ( αik dky(a) dxk + βik 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 − lim x→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 − lim x→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)d ny 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∑ k=0 ( αik dky(a) dxk + βik dky(b) dxk ) = 0, i = 1, ...n. (1.3.4) 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 y(x) = ∫ 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)). 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), Minho´s 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), u′′(x)), 0 < x < 1, u(0) = u(1) = u′′(0) = u′′(1) = 0. (2.1.1) 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, u′′). 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{ v′′ϕ = ϕ(x), 0 < x < 1, vϕ(0) = vϕ(1) = 0, (2.1.4) { u′′ϕ = vϕ(x), 0 < x < 1, uϕ(0) = uϕ(1) = 0. (2.1.5) 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) ‖v‖ ≤ 1 8 ‖ϕ‖, ‖u‖ ≤ 1 64 ‖ϕ‖, (2.1.7) where ‖.‖ is the maximum norm in C[0, 1]. (ii) If ϕ(x) ≥ 0 in [0, 1] then −‖ϕ‖/8 ≤ v(x) ≤ 0 end 0 ≤ u(x) ≤ ‖ϕ‖/64 in [0, 1]. For each number M > 0 denotes DM = { (x, u, v) | 0 ≤ x ≤ 1, |u| ≤ M 64 , |v| ≤ M 8 } , (2.1.9) 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) q := 1 64 (L1 + 8L2) < 1. (2.1.12) Then the problem (2.1.1) has a unique solution u(x) ∈ C[0, 1], satisfying the estimate ‖u‖ ≤M/64. Consider a particular case of Theorem 2.1. Denote D+M = { (x, u, v) | x ∈ [0, 1], 0 ≤ u ≤ M 64 ,−M 8 ≤ v ≤ 0 } . (2.1.16) 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) ∈ D+M . (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) ∈ D+M , i = 1, 2. (iii) q := 1 64 (L1 + 8L2) < 1. (2.1.19) 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{ v′′k = ϕk(x), 0 < x < 1, vk(0) = vk(1) = 0, (2.1.21) { u′′k = 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 ||uk − u|| ≤ q k 64(1− q)||ϕ1 − ϕ0||, (2.1.24) 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 u and decreasing in v for any (x, u, v) ∈ DM . Then, if ϕ(1)0 , ϕ(2)0 ∈ B[O,M ] are initial approximations and ϕ (1) 0 (x) ≤ ϕ(2)0 (x) for any x ∈ [0, 1] then the sequences u (1) k , u (2) k generated by the iterative process satisfy the property u (1) k (x) ≤ u(2)k (x), k = 0, 1, ...; x ∈ [0, 1]. Theorem 2.4. Denote ϕmin = min (x,u,v)∈DM f(x, u, v), ϕmax = max (x,u,v)∈DM f(x, u, v). 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 = ‖uk − uk−1‖ ≤ 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) = −5u′′ − (u+ 1)2 + sin2 pix+ 1, u(0) = u(1) = u′′(0) = u′′(1) = 0. In this example f(x, u, v) = −5v − (u+ 1)2 + sin2 pix+ 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 = sinpix. The sequences of approximations αn and βn in Bai (2004) are generated by solving the equation of the form u(4) + 5u′′ + 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), u′(x), u′′(x), u′′′(x)), 0 < x < 1, u(0) = u(1) = u′′(0) = u′′(1) = 0. (2.2.1) 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 DM = { (x, u, y, v, z) | 0 ≤ x ≤ 1, |u| ≤ M 64 , |y| ≤ M 16 , |v| ≤ M 8 , |z| ≤ M 2 } , (2.2.9) D+M = { (x, u, y, v, z)| 0 ≤ x ≤ 1; 0 ≤ u ≤ M 64 ; |y| ≤ M 16 ; −M 8 ≤ v ≤ 0; |z| ≤ M 2 } , (2.2.22) 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), u′(x), u′′(x), u′′′(x)), 0 < x < 1, u(0) = u′(0) = u′′(1) = u′′′(1) = 0, (2.2.30) 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) = u ′ ϕ(x), zϕ(x) = v ′ ϕ(x). (2.2.35) Here vϕ(x), uϕ(x) respectively are the solutions of the sequence of problems{ v′′ϕ(x) = ϕ(x), 0 < x < 1, vϕ(1) = v ′ ϕ(1) = 0, (2.2.36){ u′′ϕ(x) = vϕ(x), 0 < x < 1, uϕ(0) = u ′ ϕ(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 DM = {(x, u, y, v, z)| 0 ≤ x ≤ 1, |u| ≤ M 8 , |y| ≤ M 6 , |v| ≤ M 2 , |z| ≤M}, (2.2.39) 14 D+M = { (x, u, y, v, z)| 0 ≤ x ≤ 1; 0 ≤ u ≤ M 8 ; 0 ≤ y ≤ M 6 ; 0 ≤ v ≤ M 2 ;−M ≤ z ≤ 0 } , (2.2.57) 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{ v′′k = ϕk(x), 0 < x < 1, vk(1) = v ′ k(1) = 0, (2.2.60) { u′′k = vk(x), 0 < x < 1, uk(0) = u ′ k(0) = 0. (2.2.61) 3. Update ϕk+1 = f(x, uk, u ′ k, vk, v ′ k). (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 ωh = {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(4)(x) = f(x, u(x), v(x), u′′(x), v′′(x)), 0 < x < 1, v(4)(x) = h(x, u(x), v(x), u′′(x), v′′(x)), 0 < x < 1, (3.1.1) with boundary conditions{ u(0) = u(1) = u′′(0) = u′′(1) = 0, v(0) = v(1) = v′′(0) = v′′(1) = 0. (3.1.2) where f, h : [0, 1] × R+ × R+ × R− × R− → R+ are continuous functions and u′′, v′′ 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. To investigate the problem (3.1.1)-(3.1.2), vi w = ( ϕ ψ ) , ϕ, ψ ∈ C[0, 1], consider the operator equation w = Tw, (3.1.9) where T is defined by Tw = ( (Aw)(x) (Bw)(x) ) = ( f(x, uϕ(x), vψ(x), rϕ(x), zψ(x)) h(x, uϕ(x), vψ(x), rϕ(x), zψ(x)) ) . (3.1.10) Here rϕ(x), uϕ(x), zψ(x), vψ(x) respectively are the solutions of the sequence of problems { r′′ϕ(x) = ϕ(x), 0 < x < 1, rϕ(0) = rϕ(1) = 0, (3.1.11){ u′′ϕ(x) = rϕ(x), 0 < x < 1, uϕ(0) = uϕ(1) = 0. (3.1.12){ z′′ψ(x) = ψ(x), 0 < x < 1, zψ(0) = zψ(1) = 0, (3.1.13){ v′′ψ(x) = zψ(x), 0 < x < 1, vψ(0) = vψ(1) = 0. (3.1.14) Proposition 3.1.