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

and Applied Analysis 3 The inner product ofH Ω is defined by 〈u x, t , v x, t 〉H 1 ∑ j 0 [ 1 ∑ i 0 ∂ j i tjxi u 0, 0 ∂ i tjxi v 0, 0 ∫1 0 ∂ j 3 tjx3 u x, 0 ∂ 3 tjx3 v x, 0 dx ]


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 Samarskiȋ 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 a u 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: subject to the initial-boundary value conditions with the function a t , and its derivatives are bounded on the interval 0, T : 0 < a 0 < a t < a 1 , 0 < a 2 < a t < a 3 . 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.

Constructive Method for the Reproducing Kernel Space H 0 Ω
H Ω and H 0 Ω Ω 0, 1 × 0, T are inner product spaces and H 0 Ω ⊂ H Ω , and they are defined in the following.

2.1
Abstract and Applied Analysis 3 The inner product of H Ω is defined by

2.2
And it possesses associated norm · .

2.6
This proof can be found in 13-17 .
Clearly, one has Obviously, H 0 Ω is a closed subspace of the reproducing kernel H Ω . It is very important to obtain the representation of reproducing kernel in H 0 Ω , which is the base of our algorithm. Therefore, our work begins with some lemmas to provide constructive method for reproducing kernel in H 0 Ω .
Proof. Otherwise, for all u x, t ∈ H Ω , by 2.4 ,  then u x, t ∈ H 0 Ω , which is contradictory.

An Orthogonal Basis of H 0 Ω
Let an operator L : Proof. Noting that where M 1 , M 2 are positive real numbers,

3.4
Combining with the bounded function a t , it holds that L is a bounded linear operator.
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 .

Abstract and Applied Analysis
Applying Gram-Schmidt process, we obtain an orthogonal basis { ψ i x, t } ∞ i 1 of H 0 Ω , such that where β ik are orthogonal coefficients.

Numerical Algorithm
In this section, it is explained how to deduce the exact solution from the orthogonal basis Proof. The exact solution u x, t can be expanded to a Fourier series in terms of normal orthogonal basis

4.2
We obtain the n-truncation approximate solution of 3.2 , which is n-truncation Fourier series of the exact solution u x, t in 3.2 , so u n x, t → u x, t , as n → ∞.

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.

5.1
The exact solution is u x, t e t − 1 7 − 36x 36x 3 . The numerical results are collected in Table 1.   8 Abstract and Applied Analysis

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.