Báo cáo toán học: " Solving singular second-order initial/boundary value problems in reproducing kernel Hilbert space" pptx

Solving singular second-order initial/boundary value problems in reproducing kernel Hilbert space Boundary Value Problems 2012, 2012:3 doi:10.1186/1687-2770-2012-3 Er Gao gao.nudter@gmai

This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted

PDF and full text (HTML) versions will be made available soon

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

kernel Hilbert space

Boundary Value Problems 2012, 2012:3 doi:10.1186/1687-2770-2012-3

Er Gao (gao.nudter@gmail.com)Songhe Song (shsong31@gmail.com)Xinjian Zhang (xjz_20075@163.com)

ISSN 1687-2770

Article type Research

Submission date 13 January 2011

Acceptance date 16 January 2012

Publication date 16 January 2012

Article URL http://www.boundaryvalueproblems.com/content/2012/1/3

This peer-reviewed article was published immediately upon acceptance It can be downloaded,

printed and distributed freely for any purposes (see copyright notice below)

For information about publishing your research in Boundary Value Problems go to

http://www.boundaryvalueproblems.com/authors/instructions/

For information about other SpringerOpen publications go to

http://www.springeropen.comBoundary Value Problems

© 2012 Gao et al ; licensee Springer.

Solving singular second-order initial/boundary value problems in

reproducing kernel Hilbert space

Er Gao*, Songhe Song** and Xinjian Zhang***

Department of Mathematics and Systems Science, College of Science,National University of Defense Technology, Changsha 410073, China

sin-Mathematics Subject Classification (2000) 35A24, 46E20, 47B32

1 Introduction

Initial and boundary value problems of ordinary differential equationsplay an important role in many fields Various applications of boundary tophysical, biological, chemical, and other branches of applied mathematicsare well documented in the literature The main idea of this paper is topresent a new algorithm for computing the solutions of singular second-orderinitial/boundary value problems (IBVPs) of the form:

the problems are anti-periodic BVPs

Such problems have been investigated in many researches Specially, theexistence 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 methodsare 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 tions in numerical analysis, differential equation, probability and statistics,machine learning and precessing image Recently, using the reproducing ker-nel method, Cui and Geng [12, 13, 14, 15, 16] have make much effort to solvesome special boundary value problems

applica-According to our method, which is presented in this paper, some ducing kernel Hilbert spaces have been presented in the first step And in thesecond step, the homogeneous IBVPs is deal with in the RKHS Finally, oneanalytic approximation of the solutions of the second-order BVPs is given byreproducing kernel method under the assumption that the solution to (1.1)

contin-uous real valued functions, 𝑢′ ∈ 𝐿2[0, 1]} The inner product in 𝑊1

2[0, 1] isgiven by

and the norm∥𝑢∥𝑊1 is denoted by ∥𝑢∥𝑊1 = √(𝑢, 𝑢)𝑊1 From [17][18],

𝑊21[0, 1] is a reproducing kernel Hilbert space and the reproducing kernelis

and its corresponding reproducing kernel 𝐾2(𝑡, 𝑠).

2.3 The RKHS 𝐻23[0, 1]

Inner space 𝐻3

2[0, 1] is defined as 𝐻3

2[0, 1] = {𝑢(𝑥)∣𝑢, 𝑢′, 𝑢′′ are absolutelycontinuous real valued functions, 𝑢′′′ ∈ 𝐿2[0, 1], and 𝑎1𝑢(0)+𝑏1𝑢′(0)+𝑐1𝑢(1) =

0, 𝑎2𝑢(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 𝑊3

The proof of the Theorem 2.1 is complete.

Now, 𝐻23[0, 1] is a RKHS if endowed the inner product with the innerproduct below

3 The reproducing kernel method

In this section, the representation of analytical solution of (1.1) is given

in the reproducing kernel space 𝐻23[0, 1]

Note 𝐿𝑢 = 𝑝(𝑥)𝑢′′(𝑥) + 𝑞(𝑥)𝑢′(𝑥) + 𝑟(𝑥)𝑢(𝑥) in (1.1) It is clear that

𝐿 : 𝐻23[0, 1] → 𝑊21[0, 1] is a bounded linear operator

Put 𝜑𝑖(𝑥) = 𝐾1(𝑥𝑖, 𝑥), Ψ𝑖(𝑥) = 𝐿∗𝜑𝑖(𝑥), where 𝐿∗ is the adjoint operator

2[0, 1]

The orthogonal system {Ψ𝑖(𝑥)}∞𝑖=1 of 𝐻3

2[0, 1] can be derived from Gram–Schmidt orthogonalization process of {Ψ𝑖(𝑥)}∞𝑖=1, and

Proof From Lemma 3.1, {Ψ𝑖(𝑥)}∞𝑖=1is the complete system of 𝐻3

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 processcould be modified into the following form:

Next, the convergence of 𝑢𝑛(𝑥) will be proved

Lemma 3.2 There exists a constant 𝑀 , satisfied ∣𝑢(𝑥)∣ ≤ 𝑀 ∥𝑢∥𝐻3, for all𝑢(𝑥) ∈ 𝐻23[0, 1]

Proof For all 𝑥 ∈ [0, 1] and 𝑢 ∈ 𝐻23[0, 1], there are

∣𝑢(𝑥)∣ = ∣(𝑢(⋅), 𝐾3(⋅, 𝑥))∣ ≤ ∥𝐾3(⋅, 𝑥)∥𝐻3 ⋅ ∥𝑢∥𝐻3

Since 𝐾3(⋅, 𝑥) ∈ 𝐻3

2[0, 1], note

𝑀 = max

𝑥∈[0,1]∥𝐾3(⋅, 𝑥)∥𝐻3.That is, ∣𝑢(𝑥)∣ ≤ 𝑀 ∥𝑢∥𝐻3

By Lemma 3.2, it is easy to obtain the following lemma

Lemma 3.3 If 𝑢𝑛 −→ ¯∥⋅∥ 𝑢(𝑛 → ∞), ∥𝑢𝑛∥ is bounded, 𝑥𝑛 → 𝑦(𝑛 → ∞) and

𝐹 (𝑥, 𝑢(𝑥)) is continuous, then 𝐹 (𝑥𝑛, 𝑢𝑛−1(𝑥𝑛)) → 𝐹 (𝑦, ¯𝑢(𝑦))

Theorem 3.2 Suppose that ∥𝑢𝑛∥ is bounded in (3.3) and (1.1) has a uniquesolution If {𝑥𝑖}∞

𝑖=1 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

∥𝑢𝑛∥ is convergent and there exists a constant ℓ such that

In view of (𝑢𝑚− 𝑢𝑚−1) ⊥ (𝑢𝑚−1− 𝑢𝑚−2) ⊥ ⋅ ⋅ ⋅ ⊥ (𝑢𝑛+1− 𝑢𝑛), it follows that

Secondly, we will prove that ¯𝑢 is the solution of (1.1)

Taking limits in (3.2), we get

¯𝑢(𝑥) =

Moreover, it is easy to see by induction that

(𝐿¯𝑢)(𝑥𝑗) = 𝐹 (𝑥𝑗, 𝑢𝑗−1(𝑥𝑗)), 𝑗 = 1, 2, (3.6)Since {𝑥𝑖}∞

𝑖=1 is dense on [0, 1], for all 𝑌 ∈ [0, 1], there exists a subsequence{𝑥𝑛𝑗}∞

That is, ¯𝑢 is the solution of (1.1)

The proof is complete

In fact, 𝑢𝑛(𝑥) is just the orthogonal projection of exact solution ¯𝑢(𝑥) ontothe space Span{Ψ𝑖}𝑛

𝑖=1

4 Numerical example

In this section, some examples are studied to demonstrate the validityand applicability of the present method We compute them and compare theresults with the exact solution of each example

Example 4.1 Consider the following IBVPs:

𝑢(1) + 𝑢′(1) + 𝑢′(0) = 0,

where 𝑓 (𝑥) = 10𝑥𝑒10(𝑥−𝑥 ) + 40𝑒10(𝑥−𝑥 ) (1−2𝑥)(𝑥−𝑥 )+ 𝑥2(1 − 𝑥)(𝑒20(𝑥−𝑥 ) +20𝑒10(𝑥−𝑥2)2(1 − 2𝑥)2− 40𝑒10(𝑥−𝑥 2 ) 2

(𝑥 − 𝑥2) + 400𝑒10(𝑥−𝑥2)2(1 − 2𝑥)2(𝑥 − 𝑥2)2).The exact solution is 𝑢(𝑥) = 𝑒10(𝑥−𝑥 2 ) 2

− 1 Using our method, take 𝑎3 =

1, 𝑏3 = 𝑐3 = 0 and 𝑛 = 21, 51, 𝑁 = 5, 𝑥𝑖 = 𝑛−1𝑖−1 The numerical results aregiven in Tables 1 and 2

Example 4.2 Consider the following IBVPs:

1, 2, 3 and 4

Contributions

Er Gao gives the main idea and proves the most of the theorems andpropositions in the paper He also takes part in the work of numerical exper-iment of the main results Xinjian Zhang suggests some ideas for the prove

of the main theorems Songhe Song mainly accomplishes most part of thenumerical experiments All authors read and approved the final manuscript

Competing interests

The authors declare that they have no competing interests

The work is supported by NSF of China under Grant Numbers 10971226

The reproducing kernel of 𝐻3

Λ7 = (6𝑎22(𝑎1𝑠(−2𝑐3+ 𝑏3𝑠) + 𝑏1(2𝑐3− 𝑎3𝑠2)) + 3𝑎2(2𝑐1𝑐2(−2𝑐3+ 𝑎3𝑠2)+ 𝑏2(−𝑏3𝑐1+ 20𝑏1𝑐3 + 6𝑐1𝑐3+ 𝑎3𝑐1𝑠 − 20𝑎1𝑐3𝑠 − 10𝑐1𝑐3𝑠 − 10𝑎3𝑏1𝑠2+ 10𝑎1𝑏3𝑠2− 3𝑎3𝑐1𝑠2+ 5𝑏3𝑐1𝑠2)) + 5𝑏2(3𝑐1𝑐2(−2𝑐3+ 𝑎3𝑠2)

+ 𝑏2(−2𝑏3𝑐1+ 16𝑏1𝑐3+ 10𝑐1𝑐3+ 2𝑎3𝑐1𝑠 − 16𝑎1𝑐3𝑠 − 16𝑐1𝑐3𝑠 − 8𝑎3𝑏1𝑠2+ 8𝑎1𝑏3𝑠2− 5𝑎3𝑐1𝑠2+ 8𝑏3𝑐1𝑠2)))(2𝑏1𝑐3+ 2𝑐1𝑐3+ 𝑎3𝑐1𝑡 − 2𝑎1𝑐3𝑡

− 2𝑐1𝑐3𝑡 − 𝑎3𝑏1𝑡2 − 𝑎3𝑐1𝑡2+ 𝑏3(𝑎1𝑡2+ 𝑐1(−1 + 𝑡2)))

References

[1] Erbe, L.H., Wang, H.-Y.: On the existence of positive solutions of dinary differential equations Proc Am Math Soc 120(3), 743–748(1994)

or-[2] Kaufmann, E.R., Kosmatov, N.: A second-order singular boundaryvalue problem Comput Math Appl 47, 1317–1326 (2004)

[3] Yang, F.-H.: Necessary and sufficient condition for the existence of tive solution to a class of singular second-order boundary value problems.Chin J Eng Math 25(2), 281–287 (2008)

posi-[4] Zhang, X.-G.: Positive solutions of nonresonance semipositive singularDirichlet boundary value problems Nonlinear Anal 68, 97–108 (2008)[5] Ma, R.-Y., Ma, H.-L.: Positive solutions for nonlinear discrete periodicboundary value problems Comput Math Appl 59, 136–141 (2010)[6] Russell, R.D., Shampine, L.F.: A collocation method for boundary valueproblems Numer Math 19, 1–28 (1972)

[7] Stynes, M., O’Riordan, E.: A uniformly accurate finite-element methodfor a singular-perturbation problem in conservative form SIAM J Nu-mer Anal 23, 369–375 (1986)

boundary value problems Numer Math 63, 123–144 (1992)

[9] He, J.-H.: Variational iteration method—a kind of non-linear analyticaltechnique: some examples Nonlinear Mech 34, 699–708 (1999)

[10] Capizzano, S.S.: Spectral behavior of matrix sequences and discretizedboundary value problems Linear Algebra Appl 337, 37–78 (2001)[11] Ilicasu, F.O., Schultz, D.H.: High-order finite-difference techniques forlinear singular perturbation boundary value problems Comput Math.Appl 47, 391–417 (2004)

[12] Cui, M.-G., Geng, F.-Z.: A computational method for solving dimensional variable-coefficient Burgers equation Appl Math Comput

one-188, 1389–1401 (2007)

[13] Cui, M.-G., Chen, Z.: The exact solution of nonlinear age-structuredpopulation model Nonlinear Anal Real World Appl 8, 1096–1112(2007)

[14] Geng, F.-Z.: Solving singular second order three-point boundary valueproblems using reproducing kernel Hilbert space method Appl Math.Comput 215, 2095–2102 (2009)

[15] Geng, F.-Z., Cui, M.-G.: Solving singular nonlinear second-order odic boundary value problems in the reproducing kernel space Appl.Math Comput 192, 389–398 (2007)

peri-[16] Jiang, W., Cui, M.-G., Lin, Y.-Z.: Anti-periodic solutions for type equations via the reproducing kernel Hilbert space method Com-mun Nonlinear Sci Numer Simulat 15, 1754–1758 (2010)

Rayleigh-[17] Zhang, X.-J., Long, H.: Computating reproducing kernels for 𝑊𝑚

2 [𝑎, 𝑏](I) Math Numer Sin 30(3), 295–304 (2008) (in Chinese)

[18] Zhang, X.-J., Lu, S.-R.: Computating reproducing kernels for 𝑊𝑚

2 [𝑎, 𝑏](II) Math Numer Sin 30(4), 361–368 (2008) (in Chinese)

[19] Long, H., Zhang, X.-J.: Construction and calculation of reproducingkernel determined by various linear differential operators Appl Math.Comput 215, 759–766 (2009)

Figure 4

The relative error of Example 4.2 (𝑛 = 51, 𝑁 = 5)

Tables

Table 1

Numerical results for Example 4.1 (𝑛 = 21, 𝑁 = 5)

𝑥 True solution 𝑢(𝑥) Approximate solution 𝑢 11 Absolute error Relative error

0.0020

0.0025

0.0015

0.0020

0.0025

0.2 0.4 0.6 0.8 1.0 0.0003

0.0004

0.0005