The Solution of a Class of Third-Order Boundary Value Problems by the Reproducing Kernel Method

Abstract

This paper expands the application of reproducing kernel method to a class of third-order boundary value problems with mixed nonlinear boundary conditions. The analytical solution is represented in the form of series in the reproducing kernel space. The 𝑛-term approximation is obtained and is proved to converge to the analytical solution. The numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate that the method is simple and effective.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three layer beam, electromagnetic waves, or gravity driven flows. Third-order boundary value problems were discussed in many papers in recent years, for instance, see [1–6] and references therein. In [1–3], the authors used the spline functions to solve boundary value problems. In [4], the authors developed a second-order method for solving third-order three-point boundary value problems based on Padé approximant in a recurrence relation. In [5], the authors introduced Adomian decomposition method for multipoint boundary value problems (BVPs). In this paper, we use reproducing kernel to solve singular third-order boundary value problems with mixed boundary conditions. Recently, the reproducing kernel methods [7–10] emerge one after another. Using the reproducing kernel methods, the authors discussed two-point boundary value problems and periodic boundary value problems. For third-order boundary value problems with mixed nonlinear boundary conditions, however, this method has not yet been applied. In previous work, the reproducing kernel method cannot be used directly to solve third-order boundary value problems with nonlinear boundary conditions. Our work is to present a numerical algorithm for solving a class of singular third-order boundary value problems. By using this method, the analytical solution and approximate solution are given and uniformly converge to the exact solution and its corresponding derivatives. The algorithms are efficiently applied to solve some model problems.

Let us consider the following singular problems of third-order ordinary differential equations: 𝑢^′′′(𝑥)+𝑝₁(𝑥)𝑢^′′(𝑥)+𝑝₂(𝑥)𝑢^′(𝑥)+𝑝₃(𝑥)𝑢(𝑥)(𝑥)(),𝜆=𝐹,𝑥∈0,1_𝑖𝑢=𝑟_𝑖,(𝑖=1,2,3),(1.1) where 𝑝_𝑗(𝑥),𝑓(𝑥)∈𝐿²[0,1],(𝑗=1,2,3) are known functions. 𝜆_𝑖𝑢,(𝑖=1,2,3) are linear independence conditions of determining the solution. We assume that has a unique solution which belongs to 𝑊⁴₂[0,1], where 𝑊⁴₂[0,1] which is a reproducing kernel space is defined in the second section.

In order to solve , let 𝐿𝑢=𝑢^′′′(𝑥)+𝑝₁(𝑥)𝑢^′′(𝑥)+𝑝₂(𝑥)𝑢^′(𝑥)+𝑝₃(𝑥)𝑢(𝑥). It is easy to prove that 𝐿∶𝑊⁴₂[0,1]→𝐿²[0,1] is bounded linear operator.On the other hand, we assume that the conditions of determining the solution can be homogenized; after homogenization of these conditions, we put the conditions into the reproducing kernel space 𝑊⁴₂[0,1] constructed in the following section. Equation can be transformed into the following form in 𝑊⁴₂[0,1]: (𝐿𝑢)(𝑥)=𝐹(𝑥).(1.2) To solve problem (1.2), we give a space 𝑊⁴₂[0,1]={𝑢∣𝑢∈𝑊⁴₂[0,1]and𝜆_𝑖𝑢=0,𝑖=1,2,3}. The inner product in 𝑊⁴₂[0,1] is given by ⟨𝑢(𝑥),𝑣(𝑥)⟩=∑³_𝑖=0𝑢^(𝑖)(0)𝑣^(𝑖)(0)+∫¹₀𝑢⁽⁴⁾(𝑡)𝑣⁽⁴⁾(𝑡)𝑑𝑡. Like in [8], we can get the following reproducing kernel space.

Theorem 1.1. The space 𝑊⁴₂[0,1] is a reproducing kernel space and its reproducing kernel is 𝐾(𝑥,𝑦), and 𝐾𝑥,𝑦=𝑅_𝑦(𝑥)−ℎ₁(𝑥)ℎ₁𝑦ℎ₁(𝑥)²−ℎ₂(𝑥)ℎ₂𝑦ℎ₂(𝑥)²−ℎ₃(𝑥)ℎ₃𝑦ℎ₃(𝑥)²,(1.3) where 𝑅_𝑦(𝑥)=5040−𝑦⁷+35𝑥³𝑦³4+𝑦−21𝑥²𝑦²−60+𝑦³+7𝑥𝑦720+𝑦⁵5040,𝑥<𝑦5040−𝑥⁷+5040𝑥𝑦+7𝑥⁶𝑦+1260𝑥²𝑦²−21𝑥⁵𝑦²+140𝑥³𝑦³+35𝑥⁴𝑦³5040,𝑦<𝑥,(1.4)ℎ₁(𝑥)=𝜆_1𝑦𝑅_𝑦(𝑥),ℎ₂(𝑥)=𝜆_2𝑦(𝑅(𝑥,𝑦)−(ℎ₁(𝑥)ℎ₁(𝑦)/‖ℎ₁(𝑥)‖²)),ℎ₃(𝑥)=𝜆_3𝑦(𝑅(𝑥,𝑦)−(ℎ₁(𝑥)ℎ₁(𝑦)/‖ℎ₁(𝑥)‖²)−(ℎ₂(𝑥)ℎ₂(𝑦)/‖ℎ₂(𝑥)‖²)),  the symbol 𝜆_𝑖𝑦 indicates that the operator 𝜆_𝑖 applies to the function of 𝑦.

2. The Reproducing Kernel Method

Let 𝜓_𝑖(𝑥)=(𝐿_𝑦𝐾_𝑥(𝑦))(𝑥_𝑖),𝑖=1,2,…, Practise Gram-Schmidt orthonomalization for {𝜓_𝑖(𝑥)}^∞_𝑖=1, we get 𝜓_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥),(2.1) where 𝛽_𝑖𝑘 are coefficients of Gram-Schmidt orthonormalization.

Theorem 2.1. If {𝑥_𝑖}^∞_𝑖=1 is distinct points dense in [0,1] and 𝐿⁻¹ is existent, then 𝑢(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘𝜓_𝑖(𝑥)(2.2) is an analytical solution of the problem (1.2).

Proof. Since {𝜓_𝑖(𝑥)}^∞_𝑖=1 is an orthonormal systems, 𝑢(𝑥) is expressed as 𝑢(𝑥)=^∞_𝑖=1𝑢(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑥),𝐿_𝑠𝐾_𝑥(𝑠)𝑥_𝑘𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿_𝑠𝑢(𝑥),𝐾_𝑥(𝑠)𝑥_𝑘𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿_𝑠𝑢(𝑠)𝑥_𝑘𝜓_𝑖(𝑥)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘𝜓_𝑖(𝑥).(2.3)

We denote the approximate solution of 𝑢_𝑛(𝑥) by 𝑢_𝑛(𝑥)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘𝜓_𝑖(𝑥).(2.4)

Theorem 2.2. Let 𝜀²_𝑛=‖𝑢(𝑥)−𝑢_𝑛(𝑥)‖² where 𝑢(𝑥),  𝑢_𝑛(𝑥) are given by (2.2) and (2.4), then the sequence of number 𝜀_𝑛(𝑥) is monotone decreasing and 𝜀_𝑛(𝑥)→0.

Proof. Because 𝜀²_𝑛=𝑢(𝑥)−𝑢_𝑛(𝑥)²=^∞_𝑖=𝑛+1𝑢(𝑥),𝜓_𝑖(𝑥)𝜓_𝑖(𝑥)=^∞_𝑖=𝑛+1𝑢(𝑥),𝜓_𝑖(𝑥)²,(2.5) clearly 𝜀_𝑛−1≥𝜀_𝑛 and consequently {𝜀_𝑛} is monotone decreasing in the sense of ‖⋅‖_𝑊⁴₂. By Theorem 2.1, we know ∑^∞_𝑖=1⟨𝑢(𝑥),𝜓_𝑖(𝑥)⟩𝜓_𝑖(𝑥) is convergent in the norm of ‖⋅‖_𝑊⁴₂, then we have 𝜀²_𝑛=𝑢(𝑥)−𝑢_𝑛(𝑥)²→0.(2.6) Hence, 𝜀_𝑛→0.

Theorem 2.3 (convergence analysis). If 𝑢(𝑥), 𝑢_𝑛(𝑥) are given by (2.2) and (2.4), then 𝑢_𝑛(𝑥) and 𝑢_𝑛^(𝑘)(𝑥) uniformly convergent to 𝑢(𝑥) and 𝑢^(𝑘)(𝑥), where 𝑘=0,1,2,3.

Proof. For any 𝑥∈[0,1],𝑘=0,1,2,3𝑢_𝑛^(𝑘)(𝑥)−𝑢^(𝑘)(𝑥)=𝑢_𝑛(𝑡)(𝑡),𝜕−𝑢^𝑘𝐾(𝑥,𝑡)𝜕𝑥^𝑘≤𝑢_𝑛(𝑡)(𝑡⋅𝜕−𝑢)^𝑘𝐾(𝑥,𝑡)𝜕𝑥^𝑘,(2.7) then there exists 𝐶_𝑘>0 such that, |𝑢_𝑛^(𝑘)(𝑥)−𝑢^(𝑘)(𝑥)|≤𝐶_𝑘‖𝑢_𝑛(𝑡)−𝑢(𝑡)‖=𝐶_𝑘𝜀_𝑛→0.

Theorem 2.4. If {𝑥_𝑘}^∞_𝑘=1 is distinct points dense in [0,1] and 𝑢(𝑥),𝑢_𝑛(𝑥) are given by (2.2) and (2.4), then 𝐿𝑢(𝑥_𝑘)=𝐿𝑢_𝑛(𝑥_𝑘).

Proof. We may set projective operator 𝑃_𝑛∶𝑊⁴₂[0,1]→{∑^𝑛_𝑚=1𝑐_𝑚𝜓_𝑚(𝑥),𝑐_𝑚∈𝑅}. Hence, 𝐿𝑢_𝑛𝑥_𝑘=𝑢_𝑛(𝜉),𝐿_𝑥𝑘𝐾_𝑥𝑘(𝜉)=𝑢_𝑛(𝜉),𝜓_𝑘(𝜉)=𝑃_𝑛𝑢(𝜉),𝜓_𝑘(𝜉=𝑢(𝜉)),𝑃_𝑛𝜓_𝑘(𝜉)=𝑢(𝜉),𝜓_𝑘(𝜉)=𝑢(𝜉),𝐿_𝑥𝑘𝐾_𝑥𝑘(𝜉)=𝐿_𝑥𝑘𝑢(𝜉),𝐾_𝑥𝑘(𝜉)=𝐿_𝑥𝑘𝑢𝑥_𝑘=𝐿𝑢𝑥_𝑘.(2.8)

Theorem 2.5 (error estimate). If {𝑥_𝑘}^∞_𝑘=1 is distinct points dense in [0,1] and 𝑢(𝑥), 𝑢_𝑛(𝑥) are given by (2.2) and (2.4), then |𝑢(𝑥)−𝑢_𝑛(𝑥)|<(𝑀/𝑛), where 𝑀=‖∑^∞_𝑖=𝑛+1∑^𝑖_𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘)𝜓_𝑖(𝑥)‖⋅‖𝜕𝐾_𝜂(𝜉)/𝜕𝜂‖.

Proof. For every given 𝑥∈[0,1], there is always 𝑥_𝑖∈{𝑥_𝑘}^∞_𝑘=1 satisfying 𝑥_𝑖<𝑥 and 𝑥−𝑥_𝑖=1/𝑛. By Theorem 2.5 and 𝑥_𝑖∈{𝑥_𝑘}^∞_𝑘=1 implying 𝐿𝑢(𝑥_𝑖)=𝐿𝑢_𝑛(𝑥_𝑖), so we obtain 𝐿𝑢(𝑥)−𝐿𝑢_𝑛(𝑥)=𝐿𝑢(𝑥)−𝐿𝑢𝑥_𝑖−𝐿𝑢_𝑛(𝑥)−𝐿𝑢_𝑛𝑥_𝑖.(2.9) For application reproducing kernel property, we have 𝑢(𝑥)=𝑢(𝜉),𝐾_𝑥(𝜉),𝐿𝑢(𝑥)=𝑢(𝜉),𝐿𝐾_𝑥(𝜉).(2.10) We also have 𝐿𝑢(𝑥)−𝐿𝑢_𝑛(𝑥)=𝐿𝑢(𝑥)−𝐿𝑢𝑥_𝑖−𝐿𝑢_𝑛(𝑥)−𝐿𝑢_𝑛𝑥_𝑖=𝑢(𝜉),𝐿𝐾_𝑥(𝜉)−𝐿𝐾_𝑥𝑖(𝜉)−𝑢_𝑛(𝜉),𝐿𝐾_𝑥(𝜉)−𝐿𝐾_𝑥𝑖(𝜉=𝑢(𝜉))−𝑢_𝑛(𝜉),𝐿𝐾_𝑥(𝜉)−𝐿𝐾_𝑥𝑖(𝜉).(2.11) Moreover, 𝑢(𝑥)−𝑢_𝑛(𝑥)=𝐿⁻¹𝐿𝑢(𝑥)−𝐿𝑢_𝑛(𝑥≤𝑢(𝜉))−𝑢_𝑛(𝜉),𝐿⁻¹𝐿𝐾_𝑥(𝜉)−𝐿⁻¹𝐿𝐾_𝑥𝑖(𝜉=𝑢(𝜉))−𝑢_𝑛(𝜉),𝐾_𝑥(𝜉)−𝐾_𝑥𝑖(𝜉≤)𝑢−𝑢_𝑛𝐾_𝑥(𝜉)−𝐾_𝑥𝑖(𝜉).(2.12) It is noted that we take norm of ‖𝐾_𝑥(𝜉)−𝐾_𝑥𝑖(𝜉)‖ for variable 𝜉. The function 𝑘_𝑥(𝜉) is derived on 𝑥 in the interval of [0,1], so we have 𝐾_𝑥(𝜉)−𝐾_𝑥𝑖(𝜉)=(𝜕𝐾_𝜂(𝜉)/𝜕𝜂)(𝑥−𝑥_𝑖). Hence, 𝑢(𝑥)−𝑢_𝑛(𝑥)≤𝑢−𝑢_𝑛𝜕𝐾_𝜂(𝜉)𝜕𝜂𝑥−𝑥_𝑖=𝑢−𝑢_𝑛𝜕𝐾_𝜂(𝜉)𝜕𝜂𝑥−𝑥_𝑖=^∞𝑖_{𝑖=𝑛+1𝑘=1}𝛽_𝑖𝑘𝐹𝑥_𝑘𝜓_𝑖(𝑥)𝜕𝐾_𝜂(𝜉)𝜕𝜂𝑥−𝑥_𝑖≤𝑀𝑛.(2.13)

3. Numerical Experiment

For showing the effectiveness of our method, we consider the following problems.

Example 3.1 (see [2, 3]). Considering the following third-order boundary values problem 𝑢^′′′(𝑥)−𝑥𝑢(𝑥)=𝑥³−2𝑥²−5𝑥−3𝑒^𝑥𝑢(0)(1),0≤𝑥≤1,=𝑢=0,𝑢^′(0)=1,(3.1) where the exact solution is 𝑢_𝑇(𝑥)=𝑥(1−𝑥)𝑒^𝑥. By the homogeneous process of the boundary condition, let 𝑣(𝑥)=𝑢(𝑥)−𝑥(1−𝑥), problem can be transformed into the equivalent form 𝑣^′′′(𝑥)−𝑥𝑣(𝑥)=(1−𝑥)𝑥²+𝑥³−2𝑥²−5𝑥−3𝑒^𝑥𝑣(0)(1),0≤𝑥≤1,=𝑣=0,𝑣^′(0)=0.(3.2) The numerical results are presented in Tables 1, 2, and 3.

Table 1 

The absolute error of Example 3.1.

Table 2 

The numerical results of Example 3.1.

Table 3 

The numerical results of Example 3.1.

Example 3.2 (see [11, 14]). Considering the following third-order obstacle problems: 𝑢^′′′(𝑥)=10,0≤𝑥≤4,141≤𝑥≤4,𝑢(𝑥)−1,3𝜋4≤𝑥≤1,(3.3) where 𝑢(0)=𝑢^′(0)=𝑢^′(1)=0, the exact solution is 𝑢_𝑇(𝑥)=12𝑎₁𝑥²1,0≤𝑥≤4,1+𝑎₂𝑒^𝑥+𝑒^−𝑥/2𝑎₃√cos3𝑥2+𝑎₄√sin3𝑥2,143≤𝑥≤412𝑎₅𝑥(𝑥−2)+𝑎₆,34≤𝑥≤1,(3.4) where 𝑎₁=0.24391096222648,𝑎₂=−0.17847234452746,𝑎₃𝑎=−0.81893573565615,₄=−0.30266818001866,𝑎₅=−0.24213890868443,𝑎₆=−0.06537630092111.(3.5) The numerical results are presented in Table 4.

Table 4 

The observed maximum errors of Example 3.2.

Example 3.3. Considering the following boundary value problems with nonclassical condition: 𝑢^′′′(𝑥)+1𝑥√𝑢1−𝑥^′′(𝑥)+1𝑥𝑥()𝑢1−𝑥^′(𝑥)+1𝑥²√𝑢(𝑥)(𝑥)𝑢(0)1−𝑥=𝑓,0<𝑥<1,=0,𝑢^′(0)=¹₀𝑢(𝑥)𝑑𝑥,2𝑢(1)=𝑢^′(1).(3.6) We determine 𝑓(𝑥) to get the true solution, given by 𝑢_𝑇(𝑥)=𝑥𝑒^𝑥. The numerical results are presented in Table 5.

Table 5 

The numerical results of Example 3.3.

Example 3.4. Considering the following singular third-order three points boundary value problems with nonlinear condition 𝑢^′′′(𝑥)+𝑝𝑥^𝛼𝑢^′′(𝑥)−𝑘²𝑥^𝛼(1−𝑥)^𝛽𝑢^′(𝑥)+𝑞sin(𝑥)𝑥^𝛼(1−𝑥)^𝛽𝑢(𝑥)(𝑥)𝑢=𝑓,0<𝑥<1,_′(1)=0,2𝑢^′(0)=𝑢^′′(0)²1,𝑢2=0.(3.7) We determine 𝑓(𝑥) to get the true solution, given by 𝑢_𝑇(𝑥)=𝑟(𝑘(2𝑥−1)−2sinh(𝑘𝑥)+2(cosh(𝑘𝑥))tanh(𝑘/2))/2𝑘³. The numerical results are presented in Table 6.

Table 6 

The numerical results of Example 3.4, 𝑝=𝑞=𝛽=1,𝛼=2,𝑘=2,𝑛=20.

Example 3.5. Considering the following boundary value problems with nonlinear condition: 𝑢^′′′(𝑥)−1√𝑢1+𝑥^′(𝑥)(𝑥)(𝑥),𝑢(0)+2𝑢=𝑓=𝑢^′(0)=0,𝑢^′(1)²=2¹₀𝑢(𝑥)𝑑𝑥.(3.8) We determine 𝑓(𝑥) to get the true solution, given by 𝑢_𝑇(𝑥)=(1/2)𝑥³. The numerical results are presented in Table 7.

Table 7 

The numerical results of Example 3.5.

4. Conclusions and Remarks

In this work, we present an algorithm for solving third-order mixed boundary value problems (BVPs) based on the reproducing kernel method. The method can be generalized to get reproducing kernel of problem with linear conditions. All computations are performed by the Mathematica 7.0 software package.

Acknowledgments

This paper is supported by the Natural Science Foundation of Inner Mongolia (no. 2009MS0103) and the project of Inner Mongolia University of Technology (no. ZS201036).