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
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
Inner space is defined as is absolutely continuous real valued functions, u' ∈ L2[0, 1]}. The inner product in is given by
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}.
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
Theorem 2.1. Under the assumptions above, P is the orthogonal projection from to .
Proof. For all , We have
That means . At the same time, for any
P is self-conjugate. And
P is idempotent.
So P is the orthogonal projection from to .
The proof of the Theorem 2.1 is complete.
Now, is a RKHS if endowed the inner product with the inner product below
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.
Put φi(x) = K1(xi, x), Ψi(x) = L*φi(x), where L* is the adjoint operator of L. Then
Lemma 3.1. Under the assumptions above, if is dense on [0, 1] then is the complete basis .
The orthogonal system of can be derived from Gram-Schmidt orthogonalization process of , and
Theorem 3.1. If is dense on [0, 1] and the solution of (1.1) is unique, the solution can be expressed in the form
Proof. From Lemma 3.1, is the complete system of . Hence we have
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.
Lemma 3.2. There exists a constant M, satisfied , for all .
Proof. For all x ∈ [0, 1] and , there are
Since , note
By Lemma 3.2, it is easy to obtain the following lemma.
Lemma 3.3. If , ||un|| is bounded, xn → y(n → ∞) and F(x, u(x)) is continuous, then .
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
The orthonormality of yield 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
The completeness of shows that un → ū as n → ∞ in the sense of .
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
Since is dense on [0, 1], for all Y ∈ [0, 1], there exists a subsequence such that
It is easy to see that . Let j → ∞, by the continuity of F(x, u(x)) and Lemma 3.3, we have
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, un(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.
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.
Appendix A: The reproducing kernel of
The reproducing kernel of is
The work is supported by NSF of China under Grant Numbers 10971226.
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