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.
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:
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.
Remark 1.1. We find that if
the problems are two-point BVPs; if
the problems are initial value problems; if
the problems are periodic BVPs; if
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
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).
It is clear that is the complete subspace of , so is a RKHS. If P, which is the orthogonal projection from to , is found, we can get the reproducing kernel of obviously. Under the assumptions of Section 2, note
P is self-conjugate. And
P is idempotent.
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
Put φi(x) = K1(xi, x), Ψi(x) = L*φi(x), where L* is the adjoint operator of L. Then
and the proof is complete.
The approximate solution of the (1.1) is
If (1.1) is linear, that is F(x, u(x)) = F(x), then the approximate solution of (1.1) can be obtained directly from (3.3). Else, the approximate process could be modified into the following form:
Next, the convergence of un(x) will be proved.
By Lemma 3.2, it is easy to obtain the following lemma.
Theorem 3.2. Suppose that ||un || is bounded in (3.3) and (1.1) has a unique solution. If is dense on [0, 1], then the n-term approximate solution un(x) derived from the above method converges to the analytical solution u(x) of (1.1).
Proof. First, we will prove the convergence of un (x).
From (3.4), we infer that
That means ||un+1|| ≥ ||un||. Due to the condition that ||un|| is bounded, ||un|| is convergent and there exists a constant ℓ such that
If m > n, then
In view of (um - um-1) ⊥ (um-1 - um-2) ⊥ ··· ⊥ (un+1 - un), it follows that
Secondly, we will prove that ū is the solution of (1.1).
Taking limits in (3.2), we get
If n = 1, then
If n = 2, then
It is clear that
Moreover, it is easy to see by induction that
That is, ū is the solution of (1.1).
The proof is complete.
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.
Example 4.1. Consider the following IBVPs:
Table 1. Numerical results for Example 4.1 (n = 21, N = 5)
Table 2. Numerical results for Example 4.1 (n = 51, N = 5)
Example 4.2. Consider the following IBVPs:
where f(x) = π cos(πx) - sin(πx)(x2 + (-1 + x) * x * sin2(π* x)). The true solution is u(x) = sin(πx) + 1. Using our method, take a3 = 1, b3 = c3 = 0, and N = 5, n = 21, 51, . The numerical results are given in Figures 1, 2, 3, and 4.
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.
The authors declare that they have no competing interests.
The work is supported by NSF of China under Grant Numbers 10971226.
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