Numerical Algorithm for the Third-Order Partial Differential Equation with Three-Point Boundary Value Problem

Abstract

A numerical method based on the reproducing kernel theorem is presented for the numerical solution of a three-point boundary value problem with an integral condition. Using the reproducing property and the existence of orthogonal basis in a new reproducing kernel Hilbert space, we obtain a representation of exact solution in series form and its approximate solution by truncating the series. Moreover, the uniform convergency is proved and the effectiveness of the proposed method is illustrated with some examples.

1. Introduction

In this paper, we are concerned with the numerical solution of the following third-order partial differential equation with three-point boundary condition [1]: 𝜕³𝑢𝜕𝑡³+𝜕𝑎()𝜕𝑥𝑥,𝑡𝜕𝑢(),𝜕𝑥=𝑓𝑥,𝑡_1𝑐𝑢()[]𝑢()𝑥,𝑡𝑑𝑥=0,𝑡∈0,𝑇,0≤𝑐<1,𝑥,0=0,𝜕𝑢()𝜕𝜕𝑡𝑥,0=0,²𝑢𝜕𝑡²()[]𝑥,𝑇=0,𝑥∈0,1,𝑇>0, (1) where 𝑎(𝑥,𝑡) and its derivatives satisfy the conditions 0<𝑎₀<𝑎(𝑥,𝑡)<𝑎₁, |(𝑎(𝑥,𝑡))_𝑥|⩽𝑏, and 𝑓(𝑥,𝑡) is given smooth function in [0,1]×[0,𝑇].

Note that the third-order partial differential equations in (1) make a base of many mathematical models for dynamics of the soil moisture and subsoil waters [2], spreading of acoustic waves in a weakly heterogeneous environment [3]. Many physical phenomena and mechanical situations have been formulated into boundary value problems with integral boundary conditions [4, 5]. Later many works have appeared such as Ashyralyev and Aggez [6], Ashyralyev and Tetikoglua [7], Pulkina [8], and Ashyralyev and Gercek [9]. It should be noted that there are so much work devoted to the existence of solution for this type of boundary value problems where parabolic equations, hyperbolic equations, and mixed-type equations are considered [10, 11]. The proof of existence and uniqueness of solution especially for (1) has been studied by Latrous and Memou [1]. Recently, the reproducing kernel space method (RKSM) plays a crucial role in numerical solutions of differential and integral equations [12–20]. The main ideas of RKSM are based on the construction of reproducing kernel space (RKS). The reproducing kernel function can absorb all definite conditions. Then the numerical solution of definite problem is approximated by the reproducing kernel function. It is obvious that constructing a suitable reproducing kernel space and effectively calculating the reproducing kernel function expression become the key to apply RKSM.

However, due to the complex three-point value conditions with an integral condition in (1), the RKSM has not constructed suitable RKS to deal with the numerical solution. More precisely, the establishment of traditional RKS relies heavily on the two endpoints. Hence it can not be extended to three-point nonlocal boundary value problem which is based on intermediate point, especially with the integral boundary condition. Moreover, to the best of the authors’ knowledge, the numerical approximations of the problem equation (1) have not been studied before. Motivated by all the works above, we describe an improvement of the RKSM to find the numerical solution for (1). A new RKS is successfully established by some techniques like (3) and (7). Furthermore, other partial differential equations with multipoint boundary value conditions may be numerically solved using a similar process.

The outline of this paper is as follows. In the next section, new reproducing kernel spaces for solving problem (1) are constructed. Section 3 establishes a bounded linear operator and an orthogonal basis to use the RKSM. As a result, the approximate solution of the considered problem is obtained. In Section 4, some numerical results are given to demonstrate the accuracy of the present method. Also a conclusion is given in Section 5. Note that we have computed the numerical results using mathematic programming.

2. Constructive Method for the Reproducing Kernel Space 𝑊(Ω)

Definition 1 (see [12]). Let 𝐻 be a Hilbert function space on a set 𝑋. 𝐻 is called a reproducing kernel space if and only if, for any 𝑥∈𝑋, there exists a unique function 𝐾_𝑥(𝑦)∈𝐻, such that ⟨𝑓,𝐾_𝑥⟩=𝑓(𝑥) for any 𝑓∈𝐻. Meanwhile, 𝐾(𝑥,𝑦)=^Δ𝐾_𝑥(𝑦) is called a reproducing kernel.

Since 𝐾(𝑥,𝑦)=𝐾_𝑥(𝑦)=⟨𝐾_𝑥,𝐾_𝑦⟩=⟨𝐾_𝑦,𝐾_𝑥⟩=𝐾_𝑦(𝑥)=𝐾(𝑦,𝑥), one has the following property.

Lemma 2. A reproducing kernel function of real reproducing kernel space is symmetric.

Definition 3. 𝑊₁[0,1]={𝑢(𝑥)∣𝑢^′′(𝑥) is an absolutely continuous real value function in [0,1], 𝑢⁽³⁾(𝑥)∈𝐿²[0,1], 𝑢(0)=0, and ∫¹_𝑐𝑢(𝑥)𝑑𝑥=0, 0<𝑐<1}. The inner product is given by ⟨𝑢(𝑥),𝑣(𝑥)⟩₁=𝑢^′(0)𝑣^′(0)+𝑢^′′(0)𝑣^′′(0)+¹₀𝑢⁽³⁾(𝑥)𝑣⁽³⁾(𝑥)𝑑𝑥. (2)

Theorem 4. 𝑊₁[0,1] is a reproducing kernel space. Moreover the reproducing kernel 𝑅(𝑥,𝑦) can be denoted by 𝑅=𝑟𝑥,𝑦₁𝑥,𝑦=𝑎₁+𝑎₂𝑥+𝑎₃𝑥²+⋯+𝑎₆𝑥⁵𝑟,𝑥<𝑦<𝑐,₂𝑥,𝑦=𝑏₁+𝑏₂𝑥+𝑏₃𝑥²+⋯+𝑏₆𝑥⁵+𝜆₁𝑥⁶𝑟6!,𝑐<𝑥<𝑦,₃𝑥,𝑦=𝑐₁+𝑐₂𝑥+𝑐₃𝑥²+⋯+𝑐₆𝑥⁵𝑟,𝑦<𝑐<𝑥,₄𝑥,𝑦=𝑑₁+𝑑₂𝑥+𝑑₃𝑥²+⋯+𝑑₆𝑥⁵𝑟,𝑦<𝑥<𝑐,₅𝑥,𝑦=𝑒₁+𝑒₂𝑥+𝑒₃𝑥²+⋯+𝑒₆𝑥⁵+𝜆₁𝑥⁶𝑟6!,𝑐<𝑦<𝑥,₆𝑥,𝑦=𝑓₁+𝑓₂𝑥+𝑓₃𝑥²+⋯+𝑓₆𝑥⁵+𝜆₂𝑥⁶6!,𝑥<𝑐<𝑦. (3)

Proof. 𝑊₁[0,1] is a RKS which is a generalization of [12, Theorem 1.3.1,1.3.2] with essentially the same proofs. Let 𝑅(𝑥,𝑦) be the reproducing kernel function of 𝑊₁[0,1]. In view of (2), 𝑢(0)=0, and ∫¹_𝑐𝑢(𝑦)𝑑𝑦=0, we have the following equality using the integration by parts: 𝑢(𝑥),𝑅𝑥,𝑦₁=𝑢^′(0)𝜕_𝑥𝑅0,𝑦+𝑢^′′(0)𝜕²_𝑥𝑅+0,𝑦₁₀𝑢⁽³⁾(𝑥)𝜕³_𝑥𝑅𝑥,𝑦𝑑𝑥=𝑢^′(0)𝜕_𝑥𝑅0,𝑦+𝑢^′′(0)𝜕²_𝑥𝑅+0,𝑦₁₀𝑢³(𝑥)𝜕³_𝑥𝑅𝑥,𝑦𝑑𝑥+𝜆¹_𝑐𝑢(𝑥)𝑑𝑥=𝑢^′(0)𝜕_𝑥𝑅0,𝑦+𝜕⁴_𝑥𝑅0,𝑦+𝑢^′′(0)𝜕²_𝑥𝑅0,𝑦−𝜕³_𝑥𝑅(1)𝜕0,𝑦+𝑢⁵_𝑥𝑅1,𝑦−𝑢^′(1)𝜕⁴_𝑥𝑅1,𝑦+𝑢^′′(1)𝜕³_𝑥𝑅1,𝑦−^𝑐₀𝑢(𝑥)𝜕⁶_𝑥𝑅−𝑥,𝑦𝑑𝑥_1𝑐𝑢(𝑥)𝜕⁶_𝑥𝑅𝑥,𝑦−𝜆𝑑𝑥. (4) Here 𝜆 is an arbitrary function of 𝑦. In order to obtain reproducing property, namely, 𝑢(𝑥),𝑅𝑥,𝑦₁=𝑢𝑦, (5) let 𝜕_𝑥𝑅0,𝑦+𝜕⁴_𝑥𝑅𝜕0,𝑦=0,_2𝑥𝑅0,𝑦−𝜕³_𝑥𝑅𝜕0,𝑦=0,_𝑖𝑥𝑅1,𝑦=0,(𝑖=3,4,5),𝜕⁶_𝑥𝑅,𝜕𝑥,𝑦=−𝛿𝑥−𝑦𝑦<𝑐,𝑥<𝑐_6𝑥𝑅𝑥,𝑦−𝜆₂,𝜕=0𝑦>𝑐,𝑥<𝑐_6𝑥𝑅𝑥,𝑦−𝜆₁,𝜕=−𝛿𝑥−𝑦𝑦>𝑐,𝑥>𝑐_6𝑥𝑅𝑥,𝑦=0𝑦<𝑐,𝑥>𝑐. (6)(7)
Since the eigenvalues of (7) are all zero and sixfold, the general solutions of (7) have the form of (3). Next, we need to establish 38 equations for calculating the coefficients which are functions on 𝑦. It is obvious that 4 equations can be obtained from boundary value conditions and (6) give 10 equations. According to (7), we have 12 equations. Finally, 12 equations follow from the continuity at 𝑐.

For 𝑐=1/2, the concrete expression of 𝑅(𝑥,𝑦) is given by Lemma 2𝑟₁=𝑦𝑥,𝑦₅+𝑥120²𝑦²12𝑦+3+𝑥𝑦−𝑥𝑦⁴−247𝑥𝑦95941801080+210𝑥+70𝑥²−45𝑥³+12𝑥⁴×1080+210𝑦+70𝑦²−45𝑦³+12𝑦⁴;𝑟₂=𝑦𝑥,𝑦₅+𝑥120²𝑦²12𝑦+3+𝑥𝑦−𝑥𝑦⁴+247𝑦1535068801−17292𝑥−3300𝑥²−1280𝑥³+960𝑥⁴−384𝑥⁵+64𝑥⁶×1080+210𝑦+70𝑦²−45𝑦³+12𝑦⁴;𝑟₃=𝑦𝑥,𝑦₅+𝑥120²𝑦²12𝑦+3+𝑥𝑦−𝑥𝑦⁴−72424561100801−17292𝑥−3300𝑥²−1280𝑥³+960𝑥⁴−384𝑥⁵+64𝑥⁶×1−17292𝑦−3300𝑦²−1280𝑦³+960𝑦⁴−384𝑦⁵+64𝑦⁶;𝑟₄𝑥,𝑦=𝑟₁𝑦,𝑥;𝑟₅𝑥,𝑦=𝑟₂;𝑟𝑦,𝑥₆𝑥,𝑦=𝑟₃𝑦,𝑥. (8)

Similar to Theorem 4, we show another RKS: 𝑊₂[0,𝑇]={𝑢(𝑡)∣𝑢⁽³⁾(𝑡) is an absolutely continuous real value function in [0,𝑇], 𝑢⁽⁴⁾(𝑡)∈𝐿²[0,𝑇], and 𝑢(0)=𝑢^′(0)=𝑢^′′(𝑇)=0}. The inner product is given by ⟨𝑢(𝑡),𝑣(𝑡)⟩₂=𝑢⁽³⁾(0)𝑣⁽³⁾(0)+^𝑇₀𝑢⁽⁴⁾(𝑡)𝑣⁽⁴⁾(𝑡)𝑑𝑡 (9) and we calculate the reproducing kernel function is 𝑄(𝑡,𝑠) as the following form: 𝑞₁()=1𝑡,𝑠𝑡5040²7𝑠𝑡⁴−𝑡⁵+140𝑠²𝑡(𝑠−3𝑇)+35𝑠²𝑡²(𝑠−3𝑇)+21𝑠²𝑠³−5𝑠(4+𝑠)𝑇+60𝑇²+20𝑇³,𝑞𝑡≤𝑠,₂()=1𝑡,𝑠𝑠5040²7𝑠⁴𝑡−𝑠⁵+21𝑡⁵+35𝑡³𝑠(𝑠+4)−12𝑇−105𝑡⁴𝑇+105𝑡²𝑇4𝑇²+12𝑇−𝑠(𝑠+4),𝑠≤𝑡. (10)

Definition 5. Let Ω=[0,1]×[0,𝑇], and 𝑊(Ω)={𝑢(𝑥,𝑡)∣(𝜕⁵𝑢/𝜕𝑥²𝜕𝑡³) is a completely continuous real value function in Ω, 𝑢(0,𝑡)=𝑢(𝑥,0)=(𝜕𝑢/𝜕𝑡)(𝑥,0)=(𝜕²𝑢/𝜕𝑡²)(𝑥,𝑇)=0, ∫¹_𝑐𝑢(𝑥,𝑡)𝑑𝑥=0, and (𝜕⁷𝑢/𝜕𝑥³𝜕𝑡⁴)∈𝐿²(Ω)}. The inner product in 𝑊(Ω) is given by ⟨𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)⟩_𝑤=²_𝑖=1^𝑇₀𝜕⁴𝜕𝑡⁴𝜕^𝑖𝜕𝑥^𝑖𝑢()𝜕0,𝑡⁴𝜕𝑡⁴𝜕^𝑖𝜕𝑥^𝑖𝑣(+𝜕0,𝑡)𝑑𝑡₃𝜕𝑡³𝑢(),𝜕𝑥,0³𝜕𝑡³𝑣(𝑥,0)₁+_Ω𝜕³𝜕𝑥³𝜕⁴𝜕𝑡⁴𝑢()𝜕𝑥,𝑡³𝜕𝑥³𝜕⁴𝜕𝑡⁴𝑣(𝑥,𝑡)𝑑𝑥𝑑𝑡. (11)

Theorem 6. The space 𝑊(Ω) is a reproducing kernel space and its reproducing kernel is 𝐾(𝑥,𝑡),𝑦,𝑠=𝑅𝑥,𝑦𝑄(𝑡,𝑠), (12) where 𝑅(𝑥,𝑦) and 𝑄(𝑡,𝑠) are reproducing kernel functions of 𝑊₁[0,1] and 𝑊₂[0,𝑇], respectively.

Proof. We need to prove that 𝐾((𝑥,𝑡,𝑦,𝑠)) satisfies the reproducing property; namely, 𝑢(𝑥,𝑡),𝐾𝑥,𝑡,𝑦,𝑠_𝑤=𝑢(𝑥,𝑡),𝑅𝑥,𝑦𝑄(𝑡,𝑠)_𝑤=²_𝑖=1^𝑇₀𝜕⁴𝜕𝑡⁴𝜕^𝑖𝜕𝑥^𝑖𝑢()𝜕0,𝑡⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠^𝑖𝜕𝑥^𝑖𝑅+𝜕0,𝑦𝑑𝑡₃𝜕𝑡³𝑢()𝜕𝑥,0,𝑅𝑥,𝑦³𝜕𝑡³𝑄(0,𝑠)₁+_Ω𝜕³𝜕𝑥³𝜕⁴𝜕𝑡⁴𝑢()𝜕𝑥,𝑡³𝜕𝑥³𝑅𝜕𝑥,𝑦⁴𝜕𝑡⁴𝑄(=𝑡,𝑠)𝑑𝑥𝑑𝑡_𝑇02𝑖=1𝜕⁴𝜕𝑡⁴𝜕^𝑖𝜕𝑥^𝑖𝑢()𝜕0,𝑡⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠^𝑖𝜕𝑥^𝑖𝑅+𝜕0,𝑦𝑑𝑡₃𝜕𝑡³𝑄()𝜕0,𝑠³𝜕𝑡³𝑢+𝑦,0_𝑇010𝜕³𝜕𝑥³𝜕⁴𝜕𝑡⁴𝑢()𝜕𝑥,𝑡³𝜕𝑥³𝑅𝜕𝑥,𝑦⁴𝜕𝑡⁴𝑄(=𝑡,𝑠)𝑑𝑥𝑑𝑡_𝑇02𝑖=1𝜕⁴𝜕𝑡⁴𝜕^𝑖𝜕𝑥^𝑖𝑢()𝜕0,𝑡⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠^𝑖𝜕𝑥^𝑖𝑅+0,𝑦₁₀𝜕³𝜕𝑥³𝜕⁴𝜕𝑡⁴𝑢()𝜕𝑥,𝑡³𝜕𝑥³𝑅×𝜕𝑥,𝑦₄𝜕𝑡⁴𝑄(+𝜕𝑡,𝑠)𝑑𝑥𝑑𝑡₃𝜕𝑡³𝑄()𝜕0,𝑠³𝜕𝑡³𝑢=𝑦,0_𝑇0𝜕⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠⁴𝜕𝑡⁴²_𝑖=1𝜕^𝑖𝜕𝑥^𝑖𝑢()𝜕0,𝑡^𝑖𝜕𝑥^𝑖𝑅+0,𝑦₁₀𝜕³𝜕𝑥³𝑢()×𝜕𝑥,𝑡₃𝜕𝑥³𝑅+𝜕𝑥,𝑦𝑑𝑥𝑑𝑡₃𝜕𝑡³𝑄()𝜕0,𝑠³𝜕𝑡³𝑢=𝜕𝑦,0₃𝜕𝑡³𝑄()𝜕0,𝑠³𝜕𝑡³𝑢+𝑦,0_𝑇0𝜕⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠⁴𝜕𝑡⁴𝑢(𝑥,𝑡),𝑅𝑥,𝑦₁=𝜕𝑑𝑡₃𝜕𝑡³𝑄()𝜕0,𝑠³𝜕𝑡³𝑢𝑦,0+^𝑇₀𝜕⁴𝜕𝑡⁴𝑄()𝜕𝑡,𝑠⁴𝜕𝑡⁴𝑢=𝑢(𝑦,𝑡𝑑𝑡𝑦,𝑡,𝑄𝑡,𝑠)₂=𝑢𝑦,𝑠. (13)

3. The Numerical Method

A subspace in 𝐿²(Ω) is defined by 𝐿₀(Ω)={𝑢(𝑥,𝑡)∣𝑢(𝑥,𝑡)∈𝐿²(Ω),𝑢(𝑥,𝑡)iscontinuousinΩ}. (14) Then we define a linear operator ℒ∶𝑊(Ω)→𝐿₀(Ω): ℒ𝑢(𝑥,𝑡)=^Δ𝜕³𝑢𝜕𝑡³+𝜕𝑎()𝜕𝑥𝑥,𝑡𝜕𝑢.𝜕𝑥 (15)

Lemma 7. ℒ is an invertible bounded linear operator.

Proof. Consider the following ‖ℒ𝑢‖²_𝐿0=_Ω((ℒ𝑢)(𝑥,𝑡))²=𝑑𝑥𝑑𝑡_Ω𝜕³𝑢𝜕𝑡³+𝜕𝑎()𝜕𝑥𝑥,𝑡𝜕𝑢𝜕𝑥²𝑑𝑥𝑑𝑡≤𝐶_Ω𝜕³𝑢𝜕𝑡³²+2𝜕𝑢𝜕𝜕𝑥³𝑢𝜕𝑡³+𝜕²𝑢𝜕𝑡²²𝑑𝑥𝑑𝑡. (16) Due to the definition of  𝑊(Ω), we get ‖𝜕^𝑖_𝑥𝑅(𝑥,𝑦)𝜕^𝑗_𝑡𝑄(𝑡,𝑠)‖_𝑤≤𝑀, 𝑖=0,1,2 and 𝑗=0,1,2,3; then 𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝑢(𝑥,𝑡)=𝑢𝑦,𝑠,𝜕^𝑖_𝑥𝑅𝑥,𝑦𝜕^𝑗_𝑡𝑄(𝑡,𝑠)_𝑤≤𝜕^𝑖_𝑥𝑅𝑥,𝑦𝜕^𝑗_𝑡𝑄(𝑡,𝑠)_𝑤‖𝑢‖_𝑤≤𝑀‖𝑢‖_𝑤; (17) it follows that ‖ℒ𝑢‖²_𝐿0≤𝑀²‖𝑢‖²_𝑤. (18) Therefore (1) is turned into the following operator equation: (ℒ𝑢)(𝑥,𝑡)=𝑓(𝑥,𝑡). (19) Since (19) has a unique solution [1], it indicates ℒ is invertible. The proof is complete.

For the reproducing kernel function of 𝑊(Ω), we choose a countable dense subset 𝑆=𝑥₁,𝑡₁,𝑥₂,𝑡₂,…⊂Ω (20) and define Ψ_𝑖(𝑥,𝑡)=ℒ_(𝑦,𝑠)𝐾(𝑥,𝑡),𝑦,𝑠_{(𝑦,𝑠)=(𝑥𝑖,𝑡𝑖)},𝑖=1,2,…, (21) where ℒ_(𝑦,𝑠) denotes operator ℒ which acts on (𝑦,𝑠).

Lemma 8. The function system {Ψ_𝑖(𝑥,𝑡)}^∞_𝑖=1 is a complete system in 𝑊(Ω).

Proof. It follows that Ψ_𝑖(𝑥,𝑡)∈𝑊(Ω) from the definition of 𝑊(Ω) and Ψ_𝑖(𝑥,𝑡). Next we will see it is complete; that is, if ⟨𝑢(𝑥,𝑡),Ψ_𝑖(𝑥,𝑡)⟩_𝑤=0, then we can get 𝑢(𝑥,𝑡)≡0. For every 𝑖, it holds 0=𝑢(𝑥,𝑡),Ψ_𝑖(𝑥,𝑡)_𝑤=𝑢(𝑥,𝑡),ℒ_(𝑦,𝑠)𝐾(𝑥,𝑡),𝑥_𝑖,𝑡_𝑖𝑤=ℒ_(𝑦,𝑠)𝑢(𝑥,𝑡),𝐾(𝑥,𝑡),𝑥_𝑖,𝑡_𝑖𝑤=ℒ𝑢𝑥_𝑖,𝑡_𝑖. (22) Note that (𝑥_𝑖,𝑡_𝑖)∈𝑆 is a countable dense subset in Ω; hence, ℒ𝑢(𝑥,𝑡)=0. It follows that 𝑢(𝑥,𝑡)≡0 from the existence of ℒ⁻¹.

Applying Gram-Schmidt process, we obtain an orthogonal basis {Ψ_𝑖(𝑥,𝑡)}^∞_𝑖=1 in 𝑊(Ω): Ψ_𝑖(𝑥,𝑡)=^𝑖_𝑘=1𝛽_𝑖𝑘Ψ_𝑘(𝑥,𝑡),𝑖=1,2,…. (23)

Theorem 9. If 𝑢(𝑥,𝑡) is the solution of (19), then the approximate solution can be formed by 𝑢_𝑚(𝑥,𝑡)=^𝑚𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑡_𝑘Ψ_𝑖(𝑥,𝑡). (24)

Proof. From 𝑢(𝑥,𝑡)∈𝑊(Ω) and (21), it holds that 𝑢(𝑥,𝑡)=^∞_𝑖=1𝑢(𝑥,𝑡),Ψ_𝑖(𝑥,𝑡)_𝑤Ψ_𝑖()=𝑥,𝑡_∞𝑖=1𝑢(𝑥,𝑡),^𝑖_𝑘=1𝛽_𝑖𝑘Ψ_𝑘(𝑥,𝑡)_𝑤Ψ_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑢(𝑥,𝑡),Ψ_𝑘(𝑥,𝑡)_𝑤Ψ_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑢(),ℒ𝑥,𝑡_(𝑦,𝑠)𝐾(𝑥,𝑡),𝑦,𝑠_{(𝑦,𝑠)=(𝑥𝑘,𝑡𝑘)𝑤}×Ψ_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘ℒ_(𝑦,𝑠)𝑢(),𝐾(),𝑥,𝑡𝑥,𝑡𝑦,𝑠_{𝑤(𝑦,𝑠)=(𝑥𝑘,𝑡𝑘)}×Ψ_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘ℒ_(𝑦,𝑠)𝑢𝑥_𝑘,𝑡_𝑘Ψ_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑡_𝑘Ψ_𝑖(𝑥,𝑡). (25) The approximate solution of (19) is 𝑚-truncation of Fourier series about the exact solution 𝑢(𝑥,𝑡) in (28), so 𝑢_𝑚(𝑥,𝑡)→𝑢(𝑥,𝑡) in 𝑊(Ω) as 𝑚→∞.

Theorem 10. The approximate solution 𝑢_𝑚(𝑥,𝑡) and its derivatives uniformly converge to exact solution 𝑢(𝑥,𝑡) and its derivatives, respectively.

Proof. By the properties of 𝐾((𝑥,𝑡),(𝑦,𝑠)), we know that there exist positive real numbers 𝑀=𝑀_(𝑖,𝑗) (𝑖=0,1,2 and 𝑗=0,1,2,3), such that 𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝐾(𝑥,𝑡),𝑦,𝑠_𝑤≤𝑀. (26) Therefore, as 𝑚→∞ we have 𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝑢_𝑚(𝑥,𝑡)−𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝑢(=𝜕𝑥,𝑡)_{𝑥𝑖+𝑗𝑖𝑡𝑗}𝑢_𝑚()()=𝜕𝑥,𝑡−𝑢𝑥,𝑡_{𝑥𝑖+𝑗𝑖𝑡𝑗}𝑢_𝑚𝑦,𝑠−𝑢𝑦,𝑠,𝐾(𝑥,𝑡),𝑦,𝑠_𝑤=𝑢_𝑚𝑦,𝑠−𝑢𝑦,𝑠,𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝐾(𝑥,𝑡),𝑦,𝑠_𝑤≤𝑢_𝑚−𝑢_𝑤𝜕_𝑥^𝑖+𝑗_𝑖𝑡𝑗𝐾(𝑥,𝑡),𝑦,𝑠_𝑤≤𝑀𝑢_𝑚−𝑢_𝑤. (27)

4. Numerical Simulation and Comparison

In this section we will give some numerical examples of multipoint boundary value problem that show the exactness and usefulness of our presented process.

Example 11. This problem corresponds to (1) with 𝑎(𝑥,𝑡)=(𝑡+1)sin𝑥 and 𝑓(𝑥,𝑡)=6𝑥(5−8𝑥²)−𝑡²(𝑡−6)(𝑡+1)((24𝑥²−5)cos𝑥+48𝑥sin𝑥). The exact solution is 𝑢(𝑥)=(8𝑥³−5𝑥)(−𝑡³+6𝑡²). The numerical results are given in Table 1 for 𝑚=20. Here we take 𝑇=2 and 𝑐=1/2.

Table 1 

Numerical results for Example 11.

Example 12. Consider the following three-point nonlocal elliptic-parabolic problem in [9]: 𝜕²𝑢𝜕𝑡²+𝜕()𝜕𝑥1+𝑥𝜕𝑢𝑒𝜕𝑥=−𝑡sin𝑥+^−𝑡+𝑡(cos𝑥−𝑥sin𝑥)0<𝑡<1,0<𝑥<𝜋,𝜕𝑢+𝜕𝜕𝑡)𝜕𝑥(1+𝑥𝜕𝑢=𝜕𝑥−2𝑒^−𝑡+𝑒+1−𝑡sin𝑥_−𝑡),𝑢()=1+𝑡(cos𝑥−𝑥sin𝑥−1<𝑡<0,0<𝑥<𝜋,1,𝑥2𝑢()+1−1,𝑥2𝑢−12+𝑒,𝑥₋₁−𝑒2−12𝑒^1/2+74𝑢()()sin𝑥,0≤𝑥≤𝜋,𝑡,0=𝑢𝑡,𝜋=0,1≤𝑡≤−1. (28)

The exact solution of this problem is 𝑢(𝑡,𝑥)=(𝑒^−𝑡+𝑡)sin𝑥. In terms of (24), we calculate the approximate solution 𝑢_𝑚(𝑥,𝑡) for 𝑚=30,60,90. Comparing the maximum absolute error by our method with finite difference methods, Table 2 shows that our method has better accuracy.

Table 2 

Maximum absolute error for Example 12.

5. Conclusion

In summary, a new numerical algorithm is provided to solve three-point boundary value problems in a very favorable reproducing kernel space. Using the good properties of reproducing kernel space such as reproducing property and existence of orthogonal basis, we obtain the series pattern approximate solution through operator equation. Numerical results show that the present method is an accurate and reliable analytical technique.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors appreciate the constructive comments and suggestions provided from the kind referees and editor. This work was supported by Academic Foundation for Youth of Harbin Normal University (KGB201226), science research funds for overseas returned Chinese scholars of Heilongjiang Province (No. LC2013C01), and Science Research Foundation of Heilongjiang Province (No. A201411).