Approximate Solutions to Three-Point Boundary Value Problems with Two-Space Integral Condition for Parabolic Equations

Abstract

We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.

1. Introduction

Nonclassical boundary value problems with nonlocal boundary conditions arise naturally in various engineering models and physical phenomena, for example, chemical engineering, thermoelasticity, underground water flow, and population dynamics [1–4]. The importance of boundary value problems with integral boundary conditions has been pointed out by Samarskiĭ [5].

Boundary value problems for parabolic equations with an integral boundary condition are investigated in the literature for the development, analysis, and implementation of accurate methods [6–11]. Integral boundary conditions of models emerged in previous literatures can be summed up as^𝑏_𝑎𝑢(𝑥,𝑡)𝑑𝑥=ℎ(𝑡),𝑢(𝑥,𝑡)∈[𝑎,𝑏]×[0,𝑇]. (1.1) However, Marhoune [12] studied the parabolic equation with a generalized integral boundary condition (1.2)–(1.4). This model is more universal, and it extended usual integral boundary conditions. The form is as follows:𝜕𝑢(𝑥,𝑡)(𝑡)𝜕𝜕𝑡−𝑎²𝑢(𝑥,𝑡)𝜕𝑥²=𝑓(𝑥,𝑡),(𝑥,𝑡)∈(0,1)×(0,𝑇), (1.2) subject to the initial-boundary value conditions𝑢()(𝑥)(),𝑢()()(),𝑥,0=𝜑,𝑥∈0,10,𝑡=𝑢1,𝑡,𝑡∈0,𝑇 (1.3) and the integral condition^𝛼₀𝑢(𝑥,𝑡)𝑑𝑥+¹_𝛽𝑢(𝑥,𝑡)𝑑𝑥=0,0<𝛼<𝛽<1,𝛼+𝛽=1, (1.4) with the function 𝑎(𝑡),  and its derivatives are bounded on the interval [0,𝑇]: 0<𝑎₀<𝑎(𝑡)<𝑎₁, 0<𝑎₂<𝑎^′(𝑡)<𝑎₃. In the following, we may assume 𝜑(𝑥)=0  because it can be got from homogeneous boundary conditions. The existence and uniqueness of the solution for (1.2)–(1.4) have been proved in [12].

A practical model is typed by various definite conditions under different environment. Investigation about the definite conditions is the key problem to the model. Due to condition (1.4), it is difficult to construct reproducing kernel space, so nobody gives the algorithm for the above problem by applying reproducing kernel theory. In this paper, the author successfully constructs a novel reproducing kernel space which includes boundary conditions (1.3)-(1.4) and gains the expression of the reproducing kernel skillfully. Meanwhile, we provide a simple algorithm for solving (1.2)–(1.4). Based on the orthogonal basis in the reproducing kernel space, the exact solution is given by the form of series. Meanwhile, using a similar process, it is possible to solve other linear ordinary differential equations, partial differential equations with the same boundary value conditions.

2. Constructive Method for the Reproducing Kernel Space 𝐻₀(Ω)

𝐻(Ω)  and 𝐻₀(Ω)(Ω=[0,1]×[0,𝑇])  are inner product spaces and 𝐻₀(Ω)⊂𝐻(Ω), and  they are defined in the following.

Definition 2.1. The inner product space 𝐻(Ω) is defined by𝐻(Ω)=𝑢(𝑥,𝑡)∣𝜕³_𝑥2𝑡𝑢()()𝑢()()𝑥,𝑡isabsolutelycontinuous,𝑢𝑥,0=0,0,𝑡=𝑢1,𝑡,𝜕⁵_𝑥3𝑡2𝑢(𝑥,𝑡)∈𝐿²(Ω). (2.1)
The inner product of 𝐻(Ω) is defined by ⟨𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)⟩_H=¹_𝑗=0¹_𝑖=0𝜕_𝑡^𝑗+𝑖_𝑗𝑥𝑖𝑢(0,0)𝜕_𝑡^𝑗+𝑖_𝑗𝑥𝑖𝑣(0,0)+¹₀𝜕_𝑡^𝑗+3_𝑗𝑥3𝑢(𝑥,0)𝜕_𝑡^𝑗+3_𝑗𝑥3𝑣()+𝑥,0𝑑𝑥_1𝑖=0𝑇0𝜕_𝑡^2+𝑖_2𝑥𝑖𝑢(0,𝑡)𝜕_𝑡^2+𝑖_2𝑥𝑖𝑣(0,𝑡)𝑑𝑡+^𝑇₀¹₀𝜕⁵_𝑡2𝑥3𝑢(𝑥,𝑡)𝜕⁵_𝑡2𝑥3𝑣(𝑥,𝑡)𝑑𝑥𝑑𝑡. (2.2) And it possesses associated norm ‖⋅‖.

Lemma 2.2. Inner space 𝐻(Ω) is a Hilbert reproducing kernel space. Its reproducing kernel function 𝐾𝑥,𝑦,𝑡,𝑠=𝑅𝑥,𝑦𝐺(𝑡,𝑠), (2.3) and for any 𝑢(𝑥,𝑡)∈𝐻(Ω), 𝑢(𝑥,𝑡)=𝑢𝑦,𝑠,𝐾𝑥,𝑦,𝑡,𝑠_𝐻, (2.4) where 𝑅=𝑟𝑥,𝑦^𝑦(𝑥)𝑟,𝑥≤𝑦,_𝑥𝑦𝐺()=(),𝑦<𝑥,𝑡,𝑠𝑠𝑡2+𝑡2−𝑡³6(),𝑡≤𝑠,𝑡𝑠2+𝑠2−𝑠³6,𝑡>𝑠, (2.5) where 𝑟_𝑦(𝑥)=1()𝑥1872018720+𝑥𝑦−15−𝑥³2+𝑦−120𝑦18+𝑦−6𝑦−156𝑥⁴−10𝑥𝑦(3+𝑥)𝑦6+𝑦−4𝑦−120. (2.6)

This proof can be found in [13–17].

Clearly, one has(1)𝐺()(),(2)[]()(3)[].𝑡,𝑠=𝐺𝑠,𝑡,𝑅𝑥,𝑦=𝑅𝑦,𝑥∀𝑠∈0,𝑇,𝐺0,𝑠=0,∀𝑦∈0,1,𝑅0,𝑦=𝑅1,𝑦 (2.7)

Lemma 2.3. Fix an 𝑠∈[0,𝑇], 𝐺(𝑡,𝑠)(∫^𝛼₀𝑅(𝑥,𝑦)𝑑𝑦+∫¹_𝛽𝑅(𝑥,𝑦)𝑑𝑦)∈𝐻(Ω).

Definition 2.4. The subspace 𝐻₀(Ω) is defined by 𝐻₀(Ω)=𝑢(𝑥,𝑡)∈𝐻(Ω),^𝛼₀𝑢(𝑥,𝑡)𝑑𝑥+¹_𝛽𝑢(𝑥,𝑡)𝑑𝑥=0. (2.8)
Obviously, 𝐻₀(Ω) is a closed subspace of the reproducing kernel 𝐻(Ω).  It is very important to obtain the representation of reproducing kernel in 𝐻₀(Ω),  which is the base of our algorithm. Therefore, our work begins with some lemmas to provide constructive method for reproducing kernel in 𝐻₀(Ω).

Lemma 2.5. Fix an 𝑠∈[0,𝑇], 𝐺(𝑡,𝑠)(∫^𝛼₀𝑅(𝑥,𝑦)𝑑𝑦+∫¹_𝛽𝑅(𝑥,𝑦)𝑑𝑦)≢0.

Proof. Otherwise, for all 𝑢(𝑥,𝑡)∈𝐻(Ω), by (2.4), ^𝛼₀𝑢(𝑥,𝑡)𝑑𝑥+¹_𝛽𝑢(𝑥,𝑡)𝑑𝑥=^𝛼₀𝑢𝑦,𝑠,𝑅𝑥,𝑦𝐺(𝑡,𝑠)_𝐻𝑑𝑥+¹_𝛽𝑢𝑦,𝑠,𝑅𝑥,𝑦𝐺(𝑡,𝑠)_𝐻=𝑢()𝑑𝑥𝑦,𝑠,𝐺𝑡,𝑠^𝛼₀𝑅𝑥,𝑦𝑑𝑥+¹_𝛽𝑅𝑥,𝑦𝑑𝑥_𝐻≡0, (2.9) then 𝑢(𝑥,𝑡)∈𝐻₀(Ω), which is contradictory.

Lemma 2.6. Fix an 𝑠∈[0,𝑇], 𝐺(𝑡,𝑠)(∫^𝛼₀𝑅(𝑥,𝑦)𝑑𝑦+∫¹_𝛽𝑅(𝑥,𝑦)𝑑𝑦)∉𝐻₀(Ω), namely, ∫^𝛼₀[∫^𝛼₀𝑅𝑑𝑦+∫¹_𝛽𝑅𝑑𝑦]𝑑𝑥+∫¹_𝛽[∫^𝛼₀𝑅𝑑𝑦+∫¹_𝛽𝑅𝑑𝑦]𝑑𝑥≠0.

Proof. Otherwise, forall𝑢(𝑥,𝑡)∈𝐻₀(Ω)⊂𝐻(Ω),  then 0=^𝛼₀𝑢(𝑥,𝑡)𝑑𝑥+¹_𝛽𝑢()=𝑥,𝑡𝑑𝑥_𝛼0𝑢𝑦,𝑠,𝑅𝑥,𝑦𝐺(𝑡,𝑠)_𝐻𝑑𝑥+¹_𝛽𝑢𝑦,𝑠,𝑅𝑥,𝑦𝐺(𝑡,𝑠)_𝐻=𝑢(𝑑𝑥𝑦,𝑠,𝐺𝑡,𝑠)^𝛼₀𝑅𝑥,𝑦𝑑𝑥+¹_𝛽𝑅𝑥,𝑦𝑑𝑥_𝐻, (2.10) which implies that 𝐺(𝑡,𝑠)(∫^𝛼₀𝑅(𝑥,𝑦)+∫¹_𝛽𝑅(𝑥,𝑦))≡0, and this contradicts with Lemma 2.5.

Consider a function𝐾₀()∫𝑥,𝑦,𝑡,𝑠=𝐾𝑥,𝑦,𝑡,𝑠+𝐺𝑡,𝑠^𝛼0𝑅𝑑𝑦+∫¹_𝛽𝑅𝑑𝑦∫¹_𝛽𝑅𝑑𝑥−∫^𝛼₀𝑅𝑑𝑥∫^𝛼₀∫^𝛼₀𝑅𝑑𝑦+∫¹_𝛽𝑅𝑑𝑦𝑑𝑥+∫¹_𝛽∫^𝛼₀𝑅𝑑𝑦+∫¹_𝛽𝑅𝑑𝑦𝑑𝑥, (2.11) and one can check carefully that 𝐾₀(𝑥,𝑦,𝑡,𝑠)∈𝐻₀(Ω)  is the reproducing kernel of 𝐻₀(Ω).

3. An Orthogonal Basis of 𝐻₀(Ω)

Let an operator 𝕃∶𝐻₀(Ω)→𝐿²(Ω),  putting(𝕃𝑢)(𝑥,𝑡)=𝜕_𝑡𝑢(𝑥,𝑡)−𝑎(𝑡)𝜕²_𝑥2𝑢(𝑥,𝑡), (3.1) rewriting (1.2)–(1.4) as()(),𝑢()()(),𝕃𝑢)(𝑥,𝑡=𝑓𝑥,𝑡𝑥,0=0,𝑢0,𝑡=𝑢1,𝑡_𝛼0𝑢(𝑥,𝑡)𝑑𝑥+¹_𝛽𝑢(𝑥,𝑡)𝑑𝑥=0. (3.2)

Lemma 3.1. 𝕃 is a bounded linear operator.

Proof. Noting that 𝜕_𝑡𝑢(𝑥,𝑡)=𝑢𝑦,𝑠,𝜕_𝑡𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0≤𝑀₁‖𝑢‖_𝐻0,𝜕²_𝑥2𝑢(𝑥,𝑡)=𝑢𝑦,𝑠,𝜕²_𝑥2𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0≤𝑀₂‖𝑢‖_𝐻0, (3.3) where 𝑀₁,𝑀₂ are positive real numbers, ‖𝕃𝑢‖²_𝐿2=^𝑇₀¹₀((𝕃𝑢)(𝑥,𝑡))²𝑑𝑥𝑑𝑡=^𝑇₀¹₀𝜕_𝑡𝑢(𝑥,𝑡)−𝑎(𝑡)𝜕²_𝑥2𝑢(𝑥,𝑡)²≤𝑑𝑥𝑑𝑡_𝑇010𝜕_𝑡𝑢²+𝑎²(𝑡)𝜕²_𝑥2𝑢²+2|𝑎(𝑡)|𝜕_𝑡𝑢𝜕²_𝑥2𝑢𝑑𝑥𝑑𝑡. (3.4) Combining with the bounded function 𝑎(𝑡), it holds that 𝕃 is a bounded linear operator.

We will choose and fix a countable dense subset 𝑆={(𝑥₁,𝑡₁),(𝑥₂,𝑡₂),…}⊂Ω   and define 𝜓_𝑖(𝑥,𝑡) by𝜓_𝑖(𝑥,𝑡)=^def𝕃_(𝑦,𝑠)𝐾₀𝑥,𝑦,𝑡,𝑠𝑥_𝑖,𝑡_𝑖. (3.5)

Lemma 3.2. Consider the following:  𝜓_𝑖(𝑥,𝑡)∈𝐻₀(Ω),𝑖=1,2,…,  (see [11]).

Lemma 3.3. The function system {𝜓_𝑖(𝑥,𝑡)}^∞_𝑖=1 is a complete system of the space 𝐻₀(Ω).

Proof. For every 𝑖, we have 0=𝑢(𝑥,𝑡),𝜓_𝑖(𝑥,𝑡)_𝐻0=𝑢(𝑥,𝑡),𝕃_(𝑦,𝑠)𝐾₀𝑥,𝑦,𝑡,𝑠𝑥_𝑖,𝑡_𝑖𝐻0=𝕃_(𝑦,𝑠)𝑢(𝑥,𝑡),𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0𝑥_𝑖,𝑡_𝑖=𝕃_(𝑦,𝑠)𝑢𝑦,𝑠𝑥_𝑖,𝑡_𝑖=(𝕃𝑢)𝑥_𝑖,𝑡_𝑖, (3.6) which shows that (𝕃𝑢)(𝑥,𝑡)=0  due to the denseness of 𝑆. 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  of 𝐻₀(Ω), such that𝜓_𝑖(𝑥,𝑡)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑥,𝑡), (3.7) where 𝛽_𝑖𝑘 are orthogonal coefficients.

4. Numerical Algorithm

In this section, it is explained how to deduce the exact solution from the orthogonal basis {𝜓_𝑖(𝑥,𝑡)}^∞_𝑖=1  of 𝐻₀(Ω).

Theorem 4.1. The exact solution of (3.2) can be expressed by 𝑢(𝑥,𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢𝑥_𝑘,𝑡_𝑘𝜓_𝑖(𝑥,𝑡). (4.1)

Proof. The exact solution 𝑢(𝑥,𝑡)  can be expanded to a Fourier series in terms of normal orthogonal basis 𝜓_𝑖(𝑥,𝑡) in 𝐻₀(Ω), 𝑢(𝑥,𝑡)=^∞_𝑖=1𝑢(𝑥,𝑡),𝜓_𝑖(𝑥,𝑡)_𝐻0𝜓_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑢(𝑥,𝑡),𝜓_𝑘(𝑥,𝑡)_𝐻0𝜓_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑢(𝑥,𝑡),𝕃_(𝑦,𝑠)𝐾₀𝑥,𝑦,𝑡,𝑠𝑥_𝑘,𝑡_𝑘𝐻0𝜓_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝕃_(𝑦,𝑠)𝑢(𝑥,𝑡),𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0𝑥_𝑘,𝑡_𝑘𝜓_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘(𝕃𝑢)𝑥_𝑘,𝑡_𝑘𝜓_𝑖()=𝑥,𝑡_{∞𝑖𝑖=1𝑘=1}𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑡_𝑘𝜓_𝑖(𝑥,𝑡). (4.2)

We obtain the 𝑛-truncation approximate solution of (3.2),𝑢_𝑛(𝑥,𝑡)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑥_𝑘,𝑡_𝑘𝜓_𝑖(𝑥,𝑡), (4.3) which is 𝑛-truncation Fourier series of the exact solution 𝑢(𝑥,𝑡) in (3.2), so 𝑢_𝑛(𝑥,𝑡)−−−→^{‖⋅‖𝐻0}𝑢(𝑥,𝑡),  as 𝑛→∞.

Theorem 4.2. Consider the following:  |𝑢_𝑛(𝑥,𝑡)−𝑢(𝑥,𝑡)|→0,  as  𝑛→∞.

Proof. Since 𝐾₀(𝑥,𝑥,𝑡,𝑡)  is continuous on Ω, it follows that 𝐾₀(𝑥,𝑥,𝑡,𝑡)≤𝐶, for all (𝑥,𝑡)∈Ω,  where 𝐶 is a constant. When ‖𝑢_𝑛−𝑢‖_𝐻0→0, 𝑢_𝑛(𝑥,𝑡)−𝑢(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−𝑢(𝑥,𝑡),𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0≤𝑢_𝑛−𝑢_𝐻0𝐾₀𝑥,𝑦,𝑡,𝑠_𝐻0=𝐾₀(𝑥,𝑥,𝑡,𝑡)𝑢_𝑛−𝑢_𝐻0⟶0. (4.4)

5. Numerical Example

In this section, a numerical example is studied to demonstrate the accuracy of the present algorithm. The example is computed by Mathematica 5.0. Results obtained by the algorithm are compared with the analytical solution and are found to be in good agreement.

Example 5.1. Consider a three-point boundary value problem of parabolic equations with two-space integral condition 𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡−sin𝑡²𝑢(𝑥,𝑡)𝜕𝑥²=𝑒^𝑡7−36𝑥+36𝑥³−216𝑒^𝑡(𝑡),𝑢()(),𝑢()(),−1𝑥sin𝑥,0=0,𝑥∈0,10,𝑡=𝑢1,𝑡₀^1/3𝑢(𝑥,𝑡)𝑑𝑥+¹_2/3𝑢(𝑥,𝑡)𝑑𝑥=0,𝑡∈(0,1). (5.1) The exact solution is 𝑢(𝑥,𝑡)=(𝑒^𝑡−1)(7−36𝑥+36𝑥³). The numerical results are collected in Table 1.

Table 1 

Numerical results.

6. Conclusion

In this paper, we construct a reproducing kernel space by new method, in which each function satisfies boundary value conditions of considered problems. In this space, a numerical algorithm is presented for solving a class of parabolic equations with two-space integral boundary condition. Exact solution with series form is given. Approximate solution obtained by present algorithm converges to exact solution uniformly.

Acknowledgments

The authors appreciate the constructive comments and suggestions provided from the kind referees and editor. This work was supported by Youth Foundation of Heilongjiang Province under Grant no. QC2010036, Fundamental Research Funds for the Central Universities under Grant no. HIT.NSRIF.2009050, and Academic Foundation for Youth of Harbin Normal University (11KXQ-04) and also by 10KXQ-05.