Reproducing kernel method for singular fourth order four-point boundary value problems
Xiuying Li ∗Boying Wu
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, P.R.China.
Abstract
This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used di- rectly to solve singular fourth order four-point boundary value problems (BVPs), since there is no method of obtaining reproducing kernel satisfying four-point boundary conditions. The aim of this paper is to fill this gap. A method for obtaining reproducing kernel satisfying four-point boundary conditions is proposed so that reproducing kernel method can be used to solve singular fourth order four-point BVPs. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular fourth order four-point BVPs.
MSC: 34B10; 35G15; 34A25; 47B32.
Keywords: Reproducing kernel method; Singular; four-point boundary value problem.
1 Introduction
In this paper, we consider the following fourth order four-point boundary value problems:
⎧⎨
⎩
𝐿𝑢(𝑥)≡𝑎0(𝑥)𝑢′′′′(𝑥) +𝑎1(𝑥)𝑢′′′(𝑥) +𝑎2(𝑥)𝑢′′(𝑥) +𝑎3(𝑥)𝑢′(𝑥) +𝑎4(𝑥)𝑢(𝑥) =𝑓(𝑥),0< 𝑥 <1,
𝑢(0) = 0, 𝑢(𝛼) = 0, 𝑢(𝛽) = 0, 𝑢(1) = 0, (1.1)
where 𝛼 𝛽 ∈ (0,1), 𝑎𝑖(𝑥), 𝑓(𝑥)∈ 𝐶[0,1], 𝑖 = 0,1,2,3,4 and maybe 𝑎0(0) = 0 or𝑎0(1) = 0. That is, the equation may be singular at 𝑥 = 0,1. We consider 𝑢(0) = 0, 𝑢(𝛼) = 0, 𝑢(𝛽) = 0, 𝑢(1) = 0 since the boundary conditions 𝑢(0) =𝛾0, 𝑢(𝛼) =𝛾1, 𝑢(𝛽) =𝛾2, 𝑢(1) =𝛾3can be reduced to𝑢(0) = 0, 𝑢(𝛼) = 0, 𝑢(𝛽) = 0, 𝑢(1) = 0.
Multi-point boundary value problems arise in a variety of applied mathematics and physics. Two-point boundary value problems have been extensively studied in the literature. Also, the existence and multiplicity of solutions of fourth order four-point boundary value problems have been studied by many authors, see [1-8] and the references
∗ E-mail: [email protected](X.Y. Li); [email protected](B.Y. Wu).
therein. However, research for methods for solving singular fourth order four-point boundary value problems has proceeded very slowly. Geng [9] proposed a method for a class of second order three-point BVPs by convert the original problem into an equivalent integro-differential equation. In this paper, we will apply reproducing kernel method (RKM) presented by Cui, Geng, et al [10-18] to solve singular fourth order four-point boundary value problem (1.1).
The rest of the paper is organized as follows. In the next section, the reproducing kernel satisfying four-point boundary conditions is constructed. The RKM is applied to (1.1) in section 3. The numerical examples are presented in section 4. Section 5 ends this paper with a brief conclusion.
2 Construction of RK satisfying four-point boundary conditions
By using the methods in [9-18], it is impossible to obtain reproducing kernel (RK) satisfying four-point boundary conditions of (1.1). In this section, we will make great efforts to fill this gap.
First, we construct a reproducing kernel space𝑊5[0,1] in which every function satisfies𝑢(0) =𝑢(1) = 0.
𝑊5[0,1] is defined as 𝑊5[0,1] = {𝑢(𝑥) ∣ 𝑢(𝑥), 𝑢′(𝑥),𝑢′′(𝑥),𝑢′′′(𝑥), 𝑢′′′′(𝑥) are absolutely continuous real value functions,𝑢(5)(𝑥)∈𝐿2[0,1], 𝑢(0) = 0, 𝑢(1) = 0}. The inner product and norm in 𝑊5[0,1] are given, respectively, by
(𝑢(𝑦), 𝑣(𝑦))𝑊5=𝑢(0)𝑣(0) +𝑢(1)𝑣(1) +𝑢′(0)𝑣′(0) +𝑢′(1)𝑣′(1) +𝑢′′(0)𝑣′′(0) +
∫ 1
0 𝑢(5)𝑣(5)𝑑𝑦 and
∥𝑢∥𝑊5=√
(𝑢, 𝑢)𝑊5, 𝑢, 𝑣∈𝑊5[0,1].
By [9-16], it is easy to obtain the following theorem.
Theorem 2.1. The space 𝑊5[0,1]is a reproducing kernel Hilbert space. That is, there exists 𝑘(𝑥, 𝑦)∈𝑊5[0,1], for any 𝑢(𝑦)∈𝑊5[0,1] and each fixed 𝑥∈[0,1], 𝑦 ∈[0,1], such that (𝑢(𝑦), 𝑘(𝑥, 𝑦))𝑊5 =𝑢(𝑥). The reproducing kernel 𝑘(𝑥, 𝑦)can be denoted by
𝑘(𝑥, 𝑦) =
⎧⎨
⎩
ℎ(𝑥, 𝑦), 𝑦≤𝑥,
ℎ(𝑦, 𝑥), 𝑦 > 𝑥, (2.1)
where ℎ(𝑥, 𝑦) = 3628801 [𝑦((3𝑥4−4𝑥3+ 1)𝑦8−9(𝑥−1)2𝑥(2𝑥+ 1)𝑦7+ 36(𝑥−1)2𝑥2𝑦6+ 3𝑥(𝑥8−6𝑥7+ 12𝑥6−42𝑥4+ 635100𝑥3−907225𝑥2+ 30240𝑥+ 241920)𝑦3−𝑥(4𝑥8−27𝑥7+ 72𝑥6−84𝑥5+ 2721675𝑥3−3991720𝑥2+ 181440𝑥+ 1088640)𝑦2+ 90720(𝑥−1)2𝑥2𝑦+ 362880(𝑥−1)2𝑥(2𝑥+ 1))].
Next, we construct a reproducing kernel space𝑊𝛼𝛽5 [0,1] in which every function satisfies 𝑢(0) =𝑢(𝛼) =𝑢(𝛽) = 𝑢(1) = 0.
𝑊𝛼𝛽5 [0,1] is defined as𝑊𝛼𝛽5 [0,1] ={𝑢(𝑥)∣𝑢(𝑥)∈𝑊5[0,1], 𝑢(𝛼) = 0, 𝑢(𝛽) = 0}.
Clearly,𝑊𝛼𝛽5 [0,1] is a closed subspace of𝑊5[0,1].
Theorem 2.2. If 𝑘(𝛼, 𝛼)𝑘(𝛽, 𝛽)−𝑘2(𝛼, 𝛽)∕= 0, then the reproducing kernel 𝑘𝛼𝛽 of 𝑊𝛼𝛽5 [0,1]is given by 𝑘𝛼𝛽(𝑥, 𝑦) =𝑘(𝑥, 𝑦) +𝑘(𝑥, 𝛽)𝑘(𝛼, 𝑦)𝑘(𝛼, 𝛽) +𝑘(𝑥, 𝛼)𝑘(𝛽, 𝑦)𝑘(𝛼, 𝛽)−𝑘(𝑥, 𝛽)𝑘(𝛽, 𝑦)𝑘(𝛼, 𝛼)−𝑘(𝑥, 𝛼)𝑘(𝛼, 𝑦)𝑘(𝛽, 𝛽)
𝑘(𝛼, 𝛼)𝑘(𝛽, 𝛽)−𝑘2(𝛼, 𝛽) (2.2)
Proof. It is easy to see that𝑘𝛼𝛽(𝑥, 𝛼) =𝑘𝛼𝛽(𝑥, 𝛽) = 0, and therefore𝑘𝛼𝛽(𝑥, 𝑦)∈𝑊𝛼𝛽5 [0,1].
For∀ 𝑢(𝑦)∈𝑊𝛼𝛽5 [0,1], obviously,𝑢(𝛼) = 0, 𝑢(𝛽) = 0, it follows that
(𝑢(𝑦), 𝑘𝛼𝛽(𝑥, 𝑦))𝑊5= (𝑢(𝑦), 𝑘(𝑥, 𝑦))𝑊5 =𝑢(𝑥).
That is,𝑘𝛼𝛽(𝑥, 𝑦) is of ”reproducing property”. Thus,𝑘𝛼𝛽(𝑥, 𝑦) is the reproducing kernel of𝑊𝛼𝛽5 [0,1] and the proof is complete.
3 Application of RKM to (1.1)
In (1.1), it is clear that 𝐿 : 𝑊𝛼𝛽5 [0,1] → 𝑊21[0,1] is a bounded linear operator. Put 𝜑𝑖(𝑥) = 𝑘(𝑥𝑖, 𝑥) and 𝜓𝑖(𝑥) = 𝐿∗𝜑𝑖(𝑥) where𝑘(𝑥𝑖, 𝑥) is the RK of𝑊21[0,1],𝐿∗ is the adjoint operator of𝐿. The orthonormal system{𝜓𝑖(𝑥)}∞𝑖=1of 𝑊𝛼𝛽5 [0,1] can be derived from Gram-Schmidt orthogonalization process of{𝜓𝑖(𝑥)}∞𝑖=1,
𝜓𝑖(𝑥) =
∑𝑖 𝑘=1
𝛽𝑖𝑘𝜓𝑘(𝑥),(𝛽𝑖𝑖 >0, 𝑖= 1,2, ...). (3.3)
By RKM presented in [9-17], we have the following theorem.
Theorem 3.1. For (1.1), if {𝑥𝑖}∞𝑖=1 is dense on [0,1], then {𝜓𝑖(𝑥)}∞𝑖=1 is the complete system of 𝑊𝛼𝛽5 [0,1] and 𝜓𝑖(𝑥) =𝐿𝑦𝑘𝛼𝛽(𝑥, 𝑦)∣𝑦=𝑥𝑖.
Theorem 3.2. If {𝑥𝑖}∞𝑖=1 is dense on [0,1]and the solution of(1.1) is unique, then the solution of(1.1) is 𝑢(𝑥) =
∑∞ 𝑖=1
∑𝑖 𝑘=1
𝛽𝑖𝑘𝑓(𝑥𝑘)𝜓𝑖(𝑥). (3.4)
Now, the approximate solution 𝑢𝑛 can be obtained by taking finitely many terms in the series representation of 𝑢(𝑥) and
𝑢𝑛(𝑥) =∑𝑛
𝑖=1
∑𝑖 𝑘=1
𝛽𝑖𝑘𝑓(𝑥𝑘)𝜓𝑖(𝑥).
4 Numerical examples
In this section, two numerical examples are studied to demonstrate the accuracy of the present method. The examples are computed using Mathematica 5.0. Results obtained by the method are compared with the analytical solution of each example and are found to be in good agreement with each other.
Example 4.1
Consider the following singular fourth order four-point boundary value problem
⎧⎨
⎩
𝑥4(1−𝑥)𝑢′′′′(𝑥) +𝑒2𝑥2𝑢′′′(𝑥) + 2𝑒𝑥sin√
𝑥𝑢′′(𝑥) + 2𝑢′(𝑥) +𝑥𝑢(𝑥) =𝑓(𝑥), 0≤𝑥≤1, 𝑢(0) = 0, 𝑢(12) = sinh12, 𝑢(14) = sinh14, 𝑢(1) = sinh 1,
where 𝑓(𝑥) = (1−𝑥) sinh𝑥𝑥4+ sinh𝑥𝑥+12𝑒𝑥/2cosh𝑥+ 2 cosh𝑥+ 2𝑒𝑥sin√
𝑥sinh𝑥. The exact solution is given by 𝑢(𝑥) = sinh𝑥. Using our method, taking𝑥𝑖= 𝑛−1𝑖−1, 𝑖= 1,2,⋅ ⋅ ⋅, 𝑛, 𝑛= 6, 11, 51, the absolute errors ∣𝑢𝑛(𝑥)−𝑢(𝑥)∣ between the approximate solution and exact solution are given in Figure 1.
Example 4.2
Consider the following singular fourth order four-point boundary value problem
⎧⎨
⎩
sin𝑥(𝑒𝑥−1)2𝑢′′′′(𝑥) +𝑒𝑥2𝑢′′′(𝑥) + 2 sin√
𝑥𝑢′′(𝑥) + sinh𝑥𝑢′(𝑥) +𝑥𝑢(𝑥) =𝑓(𝑥), 0≤𝑥≤1, 𝑢(0) = 0, 𝑢(12) = sinh12, , 𝑢(14) = sinh14, 𝑢(1) = sinh 1,
where 𝑓(𝑥) = (𝑒𝑥−1)2sin2𝑥+𝑥sin𝑥−2 sin√𝑥sin𝑥−𝑒𝑥/2cos𝑥+ cos𝑥sinh𝑥. The exact solution is given by 𝑢(𝑥) = sin𝑥. Using our method, taking 𝑥𝑖 = 𝑛−1𝑖−1, 𝑖= 1,2,⋅ ⋅ ⋅, 𝑛, 𝑛 = 6, 11, 51, the absolute errors ∣𝑢𝑛(𝑥)−𝑢(𝑥)∣ between the approximate solution and exact solution are given in Figure 2.
5 Conclusion
In this paper, we apply RKM to singular fourth order four-point boundary value problems and obtain approximate solutions with a high degree of accuracy. Therefore, RKM is an accurate and reliable analytical technique for singular fourth order four-point boundary value problems
Acknowledgments
The authors would like to express their thanks to unknown referees for their careful reading and helpful comments.
References
[1] S.H. Chen, W. Ni, C.P. Wang, Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Applied Mathematics Letters 19(2006)161-168.
[2] I.Y. Karaca, Fourth-order four-point boundary value problem on time scales, Applied Mathematics Letters, 21(2008)1057- 1063.
[3] X.G. Zhang, L.S. Liu, Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator, Journal of Mathematical Analysis and Applications, 336(2007)1414-1423.
[4] C.Z. Bai, D.D. Yang, H.B. Zhu, Existence of solutions for fourth order differential equation with four-point boundary
[5] D.X. Ma, X.Z. Yang, Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 223(2009)543-551.
[6] Q. Zhang, S.H. Chen, J.H. Lv, Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 196(2006)387-393.
[7] Y.L. Zhong, S.H. Chen, C.P. Wang, Existence results for a fourth-order ordinary differential equation with a four-point boundary condition, Applied Mathematics Letters 21(2008)465-470.
[8] R. P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinary differential equations with singu- larities, J. Math. Anal. Appl. 297(2004)131-151.
[9] F.Z. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2009.08.002.
[10] F.Z. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Applied Mathematics and Computation 213 (2009) 163-169.
[11] F.Z. Geng, M.G. Cui, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied Mathematics and Computation 192 (2007)389-398.
[12] F.Z. Geng, M.G. Cui, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society 45(3)(2008)77-87.
[13] F.Z. Geng, M.G. Cui, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications 327(2007)1167-1181.
[14] F.Z. Geng, M.G. Cui, Analytical Approximation to Solutions of Singularly Perturbed Boundary Value Problems, Bulletin of the Malaysian Mathematical Sciences Society (accepted).
[15] M.G. Cui, F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, Journal of Compu- tational and Applied Mathematics 205(2007)6-15.
[16] M.G. Cui, F.Z. Geng, A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied Mathematics and Computation 188(2007)1389-1401.
[17] M.G. Cui, Z. Chen, The exact solution of nonlinear age-structured population model, Nonlinear Analysis: Real World Applications 8(2007)1096-1112.
[18] C.L. Li, M.G. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied Mathematics and Computation 143(2-3)(2003)393-399.
List of figure captions
Figure 1: Absolute errors for Example 4.1 (n=6, 11, n=51).
Figure 2: Absolute errors for Example 4.2 (n=6, 11, n=51).
0.2 0.4 0.6 0.8 1 x 5´10-7
1´10-6 1.5´10-6 2´10-6
Absolute errors
0.2 0.4 0.6 0.8 1 x
1´10-7 2´10-7 3´10-7 4´10-7
Absolute errors
0.2 0.4 0.6 0.8 1 x
2´10-9 4´10-9 6´10-9 8´10-9 1´10-8
Absolute errors
Figure 1: Absolute errors for Example 4.1 (n=6, 11, n=51) ( The left: 𝑛= 6; the middle: 𝑛= 11; the right: 𝑛= 51 )
0.2 0.4 0.6 0.8 1 x
2´10-7 4´10-7 6´10-7 8´10-7 1´10-6 1.2´10-6 1.4´10-6
Absolute errors
0.2 0.4 0.6 0.8 1 x
5´10-9 1´10-8 1.5´10-8 2´10-8
Absolute errors
0.2 0.4 0.6 0.8 1 x
2´10-10 4´10-10 6´10-10 8´10-10
Absolute errors
Figure 2: Absolute errors for Example 4.2 (n=6, 11, n=51) ( The left: 𝑛= 6; the middle: 𝑛= 11; the right: 𝑛= 51 )
