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.
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