Solving singular second-orderinitial/boundary value problems in reproducing kernel Hilbert space
© Gao et al; licensee Springer. 2012
Received: 13 January 2011
Accepted: 16 January 2012
Published: 16 January 2012
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.
where , for x ∈ [0, 1], p ≠ 0, p(x), q(x), r(x) ∈ 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.
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 , finite-element methods , Galerkin-wavelet methods , variational iteration method , spectral methods , finite difference methods , 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 and . Then we will construct a RKHS , in which every function satisfies the boundary condition of (1.1).
2.1. The RKHS
2.2. The RKHS
Inner space is defined as is absolutely continuous real valued functions, u"' ∈ L2[0, 1]}.
where a3, b3, c3 is random but satisfying that γ3 is linearly independent of γ1 and γ2.
and its corresponding reproducing kernel K2(t, s).
2.3. The RKHS
Inner space is defined as are absolutely continuous real valued functions, u"' ∈ L2[0, 1], and, a1 u(0) + b1 u'(0) + c1 u(1) = 0, a2 u(1) + b2u'(1) + c2u'(0) = 0}.
Theorem 2.1. Under the assumptions above, P is the orthogonal projection from to .
P is idempotent.
So P is the orthogonal projection from to .
The proof of the Theorem 2.1 is complete.
and the corresponding reproducing kernel K3(t, s) is given in Appendix 4.
3. The reproducing kernel method
In this section, the representation of analytical solution of (1.1) is given in the reproducing kernel space .
Note Lu = p(x)u"(x) + q(x)u'(x) + r(x)u(x) in (1.1). It is clear that is a bounded linear operator.
Lemma 3.1. Under the assumptions above, if is dense on [0, 1] then is the complete basis .
and the proof is complete.
Next, the convergence of u n (x) will be proved.
Lemma 3.2. There exists a constant M, satisfied , for all .
By Lemma 3.2, it is easy to obtain the following lemma.
Lemma 3.3. If , ||u n || is bounded, x n → y(n → ∞) and F(x, u(x)) is continuous, then .
Theorem 3.2. Suppose that ||u n || is bounded in (3.3) and (1.1) has a unique solution. If is dense on [0, 1], then the n-term approximate solution u n (x) derived from the above method converges to the analytical solution u(x) of (1.1).
Proof. First, we will prove the convergence of u n (x).
The completeness of shows that u n → ū as n → ∞ in the sense of .
Secondly, we will prove that ū is the solution of (1.1).
At the same time, . Clearly, u satisfies the boundary conditions of (1.1).
That is, ū is the solution of (1.1).
The proof is complete.
In fact, u n (x) is just the orthogonal projection of exact solution ū(x) onto the space .
4. Numerical example
In this section, some examples are studied to demonstrate the validity and applicability of the present method. We compute them and compare the results with the exact solution of each example.
Numerical results for Example 4.1 (n = 21, N = 5)
True solution u(x)
Approximate solution u11
Numerical results for Example 4.1 (n = 51, N = 5)
True solution u(x)
Approximate solution u11
Er Gao gives the main idea and proves the most of the theorems and propositions in the paper. He also takes part in the work of numerical experiment of the main results. Xinjian Zhang suggests some ideas for the prove of the main theorems. Songhe Song mainly accomplishes most part of the numerical experiments. All authors read and approved the final manuscript.
Appendix A: The reproducing kernel of
The work is supported by NSF of China under Grant Numbers 10971226.
- Erbe LH, Wang HY: On the existence of positive solutions of ordinary differential equations. Proc Am Math Soc 1994, 120(3):743–748. 10.1090/S0002-9939-1994-1204373-9MATHMathSciNetView ArticleGoogle Scholar
- Kaufmann ER, Kosmatov N: A second-order singular boundary value problem. Comput Math Appl 2004, 47: 1317–1326. 10.1016/S0898-1221(04)90125-3MATHMathSciNetView ArticleGoogle Scholar
- Yang F-H: Necessary and sufficient condition for the existence of positive solution to a class of singular second-order boundary value problems. Chin J Eng Math 2008, 25(2):281–287.MATHGoogle Scholar
- Zhang X-G: Positive solutions of nonresonance semipositive singular Dirichlet boundary value problems. Nonlinear Anal 2008, 68: 97–108. 10.1016/j.na.2006.10.034MATHMathSciNetView ArticleGoogle Scholar
- Ma R-Y, Ma H-L: Positive solutions for nonlinear discrete periodic boundary value problems. Comput Math Appl 2010, 59: 136–141. 10.1016/j.camwa.2009.07.071MATHMathSciNetView ArticleGoogle Scholar
- Russell RD, Shampine LF: A collocation method for boundary value problems. Numer Math 1972, 19: 1–28. 10.1007/BF01395926MATHMathSciNetView ArticleGoogle Scholar
- Stynes M, O'Riordan E: A uniformly accurate finite-element method for a singular-perturbation problem in conservative form. SIAM J Nu-mer Anal 1986, 23: 369–375. 10.1137/0723024MATHMathSciNetView ArticleGoogle Scholar
- Xu JC, Shann WC: Galerkin-wavelet methods for two-point boundary value problems. Numer Math 1992, 63: 123–144. 10.1007/BF01385851MATHMathSciNetView ArticleGoogle Scholar
- He J-H: Variational iteration method--a kind of non-linear analytical technique: some examples. Nonlinear Mech 1999, 34: 699–708. 10.1016/S0020-7462(98)00048-1MATHView ArticleGoogle Scholar
- Capizzano SS: Spectral behavior of matrix sequences and discretized boundary value problems. Linear Algebra Appl 2001, 337: 37–78. 10.1016/S0024-3795(01)00335-4MATHMathSciNetView ArticleGoogle Scholar
- Ilicasu FO, Schultz DH: High-order finite-difference techniques for linear singular perturbation boundary value problems. Comput Math Appl 2004, 47: 391–417. 10.1016/S0898-1221(04)90033-8MATHMathSciNetView ArticleGoogle Scholar
- Cui MG, Geng F-Z: A computational method for solving one-dimensional variable-coefficient Burgers equation. Appl Math Comput 2007, 188: 1389–1401. 10.1016/j.amc.2006.11.005MATHMathSciNetView ArticleGoogle Scholar
- Cui MG, Chen Z: The exact solution of nonlinear age-structured population model. Nonlinear Anal Real World Appl 2007, 8: 1096–1112. 10.1016/j.nonrwa.2006.06.004MATHMathSciNetView ArticleGoogle Scholar
- Geng FZ: Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Appl Math Comput 2009, 215: 2095–2102. 10.1016/j.amc.2009.08.002MATHMathSciNetView ArticleGoogle Scholar
- Geng F-Z, Cui M-G: Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl Math Comput 2007, 192: 389–398. 10.1016/j.amc.2007.03.016MATHMathSciNetView ArticleGoogle Scholar
- Jiang W, Cui M-G, Lin Y-Z: Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method. Com-mun Nonlinear Sci Numer Simulat 2010, 15: 1754–1758. 10.1016/j.cnsns.2009.07.022MATHMathSciNetView ArticleGoogle Scholar
- Zhang XJ, Long H:Computating reproducing kernels for (I). Math Numer Sin 2008, 30(3):295–304. (in Chinese)MATHMathSciNetGoogle Scholar
- Zhang XJ, Lu S-R:Computating reproducing kernels for (II). Math Numer Sin 2008, 30(4):361–368. (in Chinese)MATHMathSciNetView ArticleGoogle Scholar
- Long H, Zhang X-J: Construction and calculation of reproducing kernel determined by various linear differential operators. Appl Math Comput 2009, 215: 759–766. 10.1016/j.amc.2009.05.063MATHMathSciNetView ArticleGoogle Scholar