Advanced
A NUMERICAL ALGORITHM FOR SINGULAR MULTI-POINT BVPS USING THE REPRODUCING KERNEL METHOD
A NUMERICAL ALGORITHM FOR SINGULAR MULTI-POINT BVPS USING THE REPRODUCING KERNEL METHOD
The Pure and Applied Mathematics. 2014. Feb, 21(1): 51-60
Copyright © 2014, Korean Society of Mathematical Education
  • Received : October 17, 2013
  • Accepted : February 09, 2014
  • Published : February 28, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
YUNTAO JIA
SCHOOL OF SCIENCE, ZHUHAI CAMPUS, BEIJING INSTITUTE OF TECHNOLOGY, ZHUHAI GUANG- DONG ,519085Email address:jyt-002@126.com
YINGZHEN LIN
SCHOOL OF SCIENCE, ZHUHAI CAMPUS, BEIJING INSTITUTE OF TECHNOLOGY, ZHUHAI GUANG- DONG ,519085Email address:liliy5501@gmail.com

Abstract
In this paper, we construct a complex reproducing kernel space for singular multi-point BVPs, and skillfully obtain reproducing kernel expressions. Then, we transform the problem into an equivalent operator equation, and give a numerical algorithm to provide the approximate solution. The uniform convergence of this algorithm is proved, and complexity analysis is done. Lastly, we show the validity and feasibility of the numerical algorithm by two numerical examples.
Keywords
1. INTRODUCTION
Differential equations arise from various practical problems in mathematics and physics such as gas dynamics, nuclear physics, chemical reaction and geological prospecting etc. Multi-point BVPs can solve the contradictions in the process of actual research status efficiently and enhance their compatibility. Therefore, this problem has received a lot of attention of researchers. In [1 - 4] , the existence and uniqueness are investigated. In [5] , the author give the approximate solutions of a certain class of singular two-point (BVPs) by the Sinc-Galerkin method and homotopy-perturbation method. In [6] , the author solve two-point BVPs by variational iteration method. In [7] , the author present a method for solving a class of singular two-point BVPs based on cubic splines. For more information, please refer to [8 - 13] . Recently, in [14] , the author adopt differential transformation method to solve the two-point BVPs
PPT Slide
Lager Image
This method is able to provide an approximate solution of (1.1), while it has several disadvantages. This method is based on Taylor series which requires a high degree of smoothness. It has a local convergence region and the function f(x) are all polynomial functions.
Now, we present a new algorithm to make up the deficiencies in [14] . It can be extended to singular multi-point BVPs. We take the problem
PPT Slide
Lager Image
where p(x), q(x), f(x) are continuous complex functions on (0, 1] and x = 0 is the singular value point of p(x), c, δ are complex constants and c ∈ (0, 1), δ ≠ 1.
In this paper, we construct a complex reproducing kernel space
PPT Slide
Lager Image
[0, 1]. Then (1.2) can be transformed into an equivalent operator equation. Its approximate solution is provided. We also analyze the convergence and complexity of this algorithm. Finally, we give some numerical examples to verify the e®ectiveness of our algorithm.
2. SEVERAL COMPLEX REPRODUCING KERNEL SPACES
2 .1. The complex reproducing kernel space
PPT Slide
Lager Image
[0, 1]. We define the inner product space
PPT Slide
Lager Image
[0, 1] = { u(x) } u ″ is absolutely continuous complex function, u (3) L 2 [0, 1], u ′(0) = 0, u (1) = δu (c)}.
PPT Slide
Lager Image
Lemma 2.1. The space
PPT Slide
Lager Image
[0, 1] is a complex reproducing kernel space .
The proof can be found in [15] . Next, we give the reproducing kernel function Ry(x) of
PPT Slide
Lager Image
[0, 1]. For each y ∈ [0, 1] and each u(x)
PPT Slide
Lager Image
[0, 1], by applying (2.1), we have
PPT Slide
Lager Image
that is
PPT Slide
Lager Image
where
PPT Slide
Lager Image
We can obtain
PPT Slide
Lager Image
Now, we have
PPT Slide
Lager Image
Since Ry(x) C 2 , we get
PPT Slide
Lager Image
Similarly, if 0 ≤ y c , the function Ry(x) should satisfy the differential equation:
PPT Slide
Lager Image
and if c y ≤ 1, the function Ry(x) should satisfy the differential equation:
PPT Slide
Lager Image
If x y , (2.5) and (2.6) become
PPT Slide
Lager Image
Its characteristic equation is
λ6 = 0,
the characteristic roots are λ i = 0, ( i = 1, 2, ⋯, 6). So, we assume that
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
we get
PPT Slide
Lager Image
and
PPT Slide
Lager Image
In addition, we also need
PPT Slide
Lager Image
Combining(2.3), (2.4), (2.8) – (2.10) as well as y ∈ (0, c ) and y ∈ ( c , 1), we get 36 equations. We can find the undetermined coeffcients cij of (2.7) by solving the equations. If c = 1=2 and δ = 1 + 2 i , Ry(x) is the following expression
PPT Slide
Lager Image
2.2. The complex reproducing kernel space
PPT Slide
Lager Image
[0, 1].
PPT Slide
Lager Image
[0, 1] = { u(x) | u is absolutely continuous complex function, u ′ ∈ L 2 [0, 1]}.
The inner product is given by
PPT Slide
Lager Image
It is easy to prove that
PPT Slide
Lager Image
[0, 1] is a complex reproducing kernel space and its reproducing kernel is
PPT Slide
Lager Image
3. A SOLUTION OF (1.2)
In this section, we investigate how to obtain approximate solutions of (1.2). First, we transform (1.2) into an equivalent operator equation (3.1). Then we give its approximate solution. Also, the convergence and complexity analysis are provided.
3.1. Equivalent operator equation. The equation (1.2) can be transformed into the following form:
where α is constant and satisfy
PPT Slide
Lager Image
(b ≠ 0).
Define linear operator
PPT Slide
Lager Image
:
PPT Slide
Lager Image
[0,1] →
PPT Slide
Lager Image
[0, 1] by
Obviously, operator
PPT Slide
Lager Image
is bounded. The equation (1.2) can be converted into an equivalent operator equation:
PPT Slide
Lager Image
where f1(x) = xaf(x) .
3.2. The numerical solution for operator equation (3.1). We choose a countable dense subset
PPT Slide
Lager Image
⊂ (0, 1] and define 𝜓 i(x) as
Theorem 3.1. The function system
PPT Slide
Lager Image
is a complete system in the space
PPT Slide
Lager Image
[0, 1].
Proof . For an arbitrary i , we have,
0 = ⟨u(x), 𝜓i(x)⟩ = ⟨u(x), (Rx(·))(xi)⟩ = (⟨u(x),Rx(·)⟩)(xi) = (u(·))(xi) = (u)(xi).
Note that
PPT Slide
Lager Image
is dense in [0, 1], so (
PPT Slide
Lager Image
u )( x ) = 0. By the existence of
PPT Slide
Lager Image
−1 , it follows that u ≡ 0. Therefore,
PPT Slide
Lager Image
is a complete system in
PPT Slide
Lager Image
[0, 1]. □
Furthermore, we obtain an orthogonal system
PPT Slide
Lager Image
of
PPT Slide
Lager Image
[0, 1] derived from Gram-Schmidt orthonormalization process from
PPT Slide
Lager Image
:
Theorem 3.2. If
PPT Slide
Lager Image
is dense on [0, 1], then the solution of (1.2) is
Proof . We expand u(x) into a Fourier series as follows
Now, we can get the approximate solution un(x) by truncating the nth – term of the exact solution u(x) ,
PPT Slide
Lager Image
3.3. Theoretical analysis for our algorithm.
Theorem 3.3. An approximate solution un(x) is uniform convergence to u(x) on [0, 1]. Moreover ,
PPT Slide
Lager Image
,
PPT Slide
Lager Image
are both uniform convergence to u′(x) and u″(x) on [0, 1] .
Proof . Note that
un(x) = ⟨un, Rx⟩, u(x) = ⟨u,Rx,
and
PPT Slide
Lager Image
By applying Schwarz s inequality and the boundedness of
PPT Slide
Lager Image
( i = 0, 1, 2), we have
PPT Slide
Lager Image
So
PPT Slide
Lager Image
Theorem 3.4. The time complexity of the algorithm is O ( n 3 ).
Proof . There are three steps to calculate the approximate solution un(x) of (1.2) .
(1) Assume the number of multiplications required is C in one calculation of the inner product ⟨ φii ⟩, then the total number of multiplications required is n ( n + 1) C =2 in calculation of all inner products.
(2) Orthogonalization of the system
PPT Slide
Lager Image
needs 3 layers of nested loops, that is, the number of multiplication is
(3) The number of multiplication is n 2 when calculating un(x) using (3.2). To sum up, the total number of multiplication is
4. NUMERICAL EXAMPLES
In this section, some numerical examples are studied to demonstrate the accuracy of the present algorithm. Results obtained by this algorithm are compared with the exact solution of each example and are shown to be in good agreement with the exact solution.
Example 1. Consider equation
where f(x) =
PPT Slide
Lager Image
[(8+4 i )−(16−8 i )
PPT Slide
Lager Image
−8 ie −(1−5 i ) ex +(3−2 i ) x −((2−4i) + (4 + 8 i )
PPT Slide
Lager Image
− 4 e ) x 2 +
PPT Slide
Lager Image
. Its exact solution is
PPT Slide
Lager Image
Applying our algorithm and taking the number of nodes as n=50 and 100, the absolute errors of real part (a.e.Re) and the absolute errors of imaginary part (a.e.Im) are shown in Table 1 . It shows that the approximate solution is getting more and more accurate as n increases.
The absolute errors for Example 1
PPT Slide
Lager Image
The absolute errors for Example 1
Example 2. Consider equation
where
The function f(x) is continuous at x =
PPT Slide
Lager Image
, but not differentiable. So the method of [14] is invalid for example 2. While using our algorithm, we choose 100 points in (0, 1]. The numerical results |
PPT Slide
Lager Image
u 100 f | are given in the following Table 2 .
Numerical results |u100−f| for Example 2
PPT Slide
Lager Image
Numerical results |u100f| for Example 2
5. CONCLUSION
In this paper, we present a new numerical algorithm in complex reproducing kernel space for singular multi-point BVPs. We give the rigorous theoretical analysis, the uniform convergence of the approximate solution. The numerical examples show that by using this algorithm we obtain better solution and fix the deficiencies of [14] .
View Fulltext  
Wong F. , Lian W. 1996 Positice solution for singular boundary value problems Comput. Math. Appl. 32 (9) 41 - 49
Kelevedjiev P. 2002 Existence of positice solution to a singular second order boundary value problem Nonlinear Anal. 50 1107 - 1118    DOI : 10.1016/S0362-546X(01)00803-3
Xu X. , Ma J. 2004 A note on singular nonlinear boundary value problems J. Math. Anal. Appl. 293 108 - 124    DOI : 10.1016/j.jmaa.2003.12.017
Liu Y. , Yu H. 2005 Existence and uniqueness of positice solution for singular boundary value problem Comput. Math. Appl. 50 133 - 143    DOI : 10.1016/j.camwa.2005.01.022
Al-Khaled K. 2007 Theory and computation in singular boundary value problems Chaos Solitons Fractals. 33 678 - 684    DOI : 10.1016/j.chaos.2006.01.047
Junfeng L.U. 2007 Variational iteration method for solving two-point boundary value problems J. Comput. Appl. Math. 207 92 - 101    DOI : 10.1016/j.cam.2006.07.014
Ravi Kanth A.S.V. , Reddy Y.N. 2005 Cubic spline for a class of two-point boundary value problems Appl. Math. Comput. 207 733 - 740
Graef John R. , Kong Lingju 2008 Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems Nonlinear Anal. 68 1529 - 1552    DOI : 10.1016/j.na.2006.12.037
Kong Qingkai , Wang Min 2010 Positive solutions of even order system periodic boundary value problems Nonlinear Anal. 72 1778 - 1791    DOI : 10.1016/j.na.2009.09.019
Ikram A 2006 Nonpolynomial spline approach to the solution of a system of second-order boundary-value problems Appl. Math. Comput. 173 1208 - 1218    DOI : 10.1016/j.amc.2005.04.064
Aziz Imran 2010 The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets Math. Comput. Modelling. 52 1577 - 1590    DOI : 10.1016/j.mcm.2010.06.023
Caballero J. , Harjani J. , Sadarangani K. 2011 Positive solutions for a class of singular fractional boundary value problems Comput. Math. Appl. 62 1325 - 1332    DOI : 10.1016/j.camwa.2011.04.013
He Tieshan , Yang Fengjian , Chen Chuanyong , Peng Shiguo 2011 Existence and multi-plicity of positive solutions for nonlinear boundary value problems with a parameter Comput. Math. Appl. 61 3355 - 3363    DOI : 10.1016/j.camwa.2011.04.039
Ravi Kanth A.S.V. , Aruna K. 2008 Solution of singular two-point boundary value problems using differential transformation method Phys. Lett. A. 372 4671 - 4673    DOI : 10.1016/j.physleta.2008.05.019
Wu Boying , Lin Yingzhen 2012 Application of the reproducing kernel space Science Press