Solving singular second-order initial/boundary value problems in reproducing kernel Hilbert space

Gao et al. Boundary Value Problems 2012, 2012:3 http://www.boundaryvalueproblems.com/content/2012/1/3 RESEARCH Open Access Solving singular second-orderinitial/boundary value problems in reproducing kernel Hilbert space Er Gao*, Songhe Song and Xinjian Zhang * Correspondence: gao. Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha 410073, China Abstract In this paper, we presents a reproducing kernel method for computing singular second-order initial/boundary value problems (IBVPs). This method could deal with much more general IBVPs than the ones could do, which are given by the previous researchers. According to our work, in the first step, the analytical solution of IBVPs is represented in the RKHS which we constructs. Then, the analytic approximation is exhibited in this RKHS. Finally, the n-term approximation is proved to converge to the analytical solution. Some numerical examples are displayed to demonstrate the validity and applicability of the present method. The results obtained by using the method indicate the method is simple and effective. Mathematics Subject Classification (2000) 35A24, 46E20, 47B32. 1. Introduction Initial and boundary value problems of ordinary differential equations play an important role in many fields. Various applications of boundary to physical, biological, chemical, and other branches of applied mathematics are well documented in the literature. The main idea of this paper is to present a new algorithm for computing the solutions of singular second-order initial/boundary value problems (IBVPs) of the form: ⎧ ⎨ p(x)u (x) + q(x)u (x) + r(x)u(x) = F(x, u), a1 u(0) + b1 u (0) + c1 u(1) = 0, ⎩ a2 u(1) + b2 u (1) + c2 u (0) = 0, (1:1) where u(x) ∈ W23 [0, 1] , for x Î [0, 1], p ≠ 0, p(x), q(x), r(x) Î C[0, 1]. a1, b1,c1, a2, b2, c2 arc real constants and satisfy that a1 u(0) + b1 u’(0) + c1 u (1) and a2 u(1) + b2u’(1) + c2u’(0) are linear independent. F(x, u) is continuous. Remark 1.1. We find that if b1 = c1 = b2 = c2 = 0, a1 = 0, a2 = 0, (1:2) c2 = 0, (1:3) the problems are two-point BVPs; if b1 = c1 = a2 = b2 = 0, a1 = 0, © 2012 Gao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gao et al. Boundary Value Problems 2012, 2012:3 http://www.boundaryvalueproblems.com/content/2012/1/3 Page 2 of 11 the problems are initial value problems; if b1 = a2 = 0, a1 = c1 = 0, b2 = c2 = 0, (1:4) the problems are periodic BVPs; if b1 = a2 = 0, a1 = −c1 = 0, b2 = −c2 = 0, (1:5) the problems are anti-periodic BVPs. Such problems have been investigated in many researches. Specially, the existence and uniqueness of the solution of (1.1) have been discussed in [1-5]. And in recent years, there are also a large number of special-purpose methods are proposed to provide accurate numerical solutions of the special form of (1.1), such as collocation methods [6], finite-element methods [7], Galerkin-wavelet methods [8], variational iteration method [9], spectral methods [10], finite difference methods [11], etc. On the other hands, reproducing kernel theory has important applications in numerical analysis, differential equation, probability and statistics, machine learning and precessing image. Recently, using the reproducing kernel method, Cui and Geng [12-16] have make much effort to solve some special boundary value problems. According to our method, which is presented in this paper, some reproducing kernel Hilbert spaces have been presented in the first step. And in the second step, the homogeneous IBVPs is deal with in the RKHS. Finally, one analytic approximation of the solutions of the second-order BVPs is given by reproducing kernel method under the assumption that the solution to (1.1) is unique. 2. Some RKHS In this section, we will introduce the RKHS W21 [0, 1] and W23 [0, 1] . Then we will construct a RKHS H23 [0, 1] , in which every function satisfies the boundary condition of (1.1). 2.1. The RKHS W21 [0, 1] Inner space W21 [0, 1] is defined as W21 [0, 1] = {u(x)|u is absolutely continuous real valued functions, u’ Î L2[0, 1]}. The inner product in W21 [0, 1] is given by 1 (f , h)W21 = f (0)h(0) + f  (t)h (t)dt, f , h ∈ W21 [0, 1] (2:1) 0 and the norm ||u||W21 is denoted by ||u||W21 =  (u, u)W21 . From [17,18], W21 [0, 1] is a reproducing kernel Hilbert space and the reproducing kernel is K1 (t, s) = 1 + min{t, s} (2:2) 2.2. The RKHS W23 [0, 1] Inner space W23 [0, 1] is defined as W23 [0, 1] = {u(x)|u, u , u is absolutely continuous real valued functions, u"’ Î L2[0, 1]}. From [15,17-19], it is clear that W23 [0, 1] become a reproducing kernel Hilbert space if we endow it with suitable inner product. Gao et al. Boundary Value Problems 2012, 2012:3 http://www.boundaryvalueproblems.com/content/2012/1/3 Page 3 of 11 Zhang and Lu [18] and Long and Zhang [19] give us a clue to relate the inner product with the boundary conditions (1.1). Set L = D3, and ⎧ ⎨ γ1 f = a1 f (0) + b1 f  (0) + c1 f (1), (2:3) γ2 f = a2 f (1) + b2 f  (1) + c2 f  (0), ⎩ γ3 f = a3 f (0) + b3 f  (0) + c3 f  (0), where a3, b3, c3 is random but satisfying that g3 is linearly independent of g1 and g2. It is easy to know that g1, g2, g3 are linearly independent in Ker L. Then from [18,19], it is easy to know one of the inner products of W23 [0, 1] (f , h)W23 = 3  1 γi f γi h + i=1 f  (t)h (t)dt, f , h ∈ W23 [0, 1] (2:4) 0 and its corresponding reproducing kernel K2(t, s). 2.3. The RKHS H23 [0, 1] Inner space H23 [0, 1] is defined as H23 [0, 1] = {u(x)|u, u , u are absolutely continuous real valued functions, u"’ Î L2[0, 1], and, a1 u(0) + b1 u’(0) + c1 u(1) = 0, a2 u(1) + b2u’(1) + c2u’(0) = 0}. It is clear that H23 [0, 1] is the complete subspace of W23 [0, 1] , so H23 [0, 1] is a RKHS. If P, which is the orthogonal projection from W23 [0, 1] to H23 [0, 1] , is found, we can get the reproducing kernel of H23 [0, 1] obviously. Under the assumptions of Section 2, note 1 Pf (t) = (γ3 f )e3 (t) + G(t, τ ) · f  (τ )dτ , ∀f ∈ W23 [0, 1] (2:5) 0 Theorem 2.1. Under the assumptions above, P is the orthogonal projection from H23 [0, 1] to H23 [0, 1] . Proof. For all f ∈ W23 [0, 1] , We have (γ1 (Pf ))(t) = (γ2 (Pf ))(t) = 0 That means Pf ∈ H23 [0, 1] . At the same time, for any f , h ∈ W23 [0, 1] ⎛ (Pf , h) = ⎝(γ3 f )e3 (t) + 1 0 1 = (γ3 f )(γ3 h) + ⎞ G(t, τ ) · Lf (τ ) dτ , h⎠ ⎛ ⎝L 0 1 ⎞ G(t, τ ) · Lf (τ )dτ ⎠ · Lh(t) dt 0 1 Lf (t) · Lh(t) dt = (γ3 f )(γ3 h) + ⎛ 0 (f , Ph) = ⎝f , (γ3 h)e3 (t) + 1 ⎞ G(t, τ ) · Lh(τ ) dτ ⎠ 0 1 1 Lf (t) · L = (γ3 f )(γ3 h) + 0 G(t, τ ) · Lh(τ ) dτ dt 0 1 Lf (t) · Lh(t) dt = (γ3 f )(γ3 h) + 0 Gao et al. Boundary Value Pro (...truncated)