1. Introduction

2. The Problem with Periodic Boundary Conditions

Let’s examine the problem given below;

(1)

(2)

(3)

(4)

(5)

where Fourier Coefficients: Then we obtain the solution:

(6)

3. Analytical solution of the Problem

Let's assume the following rules for the functions used in the problem: where

Definition 1. 

is called Banach (B) norm.

Proof. 

As iteration for the problem

Applying Cauchy, Bessel, Hölder inequalities, Lipschitz condition, we have

(7)

From the theorem ${\mathrm{T}}^{(0)}(\tau )\in B$.

Applying for the step N,

(8)

We get ${\mathrm{T}}^{(\mathrm{N})}(\tau )\in B$since ${\mathrm{T}}^{(\mathrm{N}+1)}(\tau )\in B.$

Let us show that, ${\mathrm{T}}^{(\mathrm{N}+1)}(\tau )$is converged for $\mathrm{N}\to \infty$. Applying Cauchy, Bessel, Hölder inequalities, Lipschitz condition, we have

(9)

where let .

Applying for the step N,

(10)

We obtain ${\mathrm{T}}^{(\mathrm{N}+1)}(\tau )\to {\mathrm{T}}^{(\mathrm{N})}(\tau )$.

Let us show that, ${\mathrm{T}}^{(\mathrm{N}+1)}(\tau )\to \mathrm{T}(\tau )$is converged for $\mathrm{N}\to \infty$.

Applying Cauchy, Bessel, Hölder inequalities, Lipschitz condition, we have

(11)

From the Gronwall’s inequality,

(12)

From Eq. 12, we get ${\mathrm{T}}^{(\mathrm{N}+1)}(\tau )\to \mathrm{T}(\tau )\mathrm{.}$

To show the uniqueness, we get two solutions the problem $\left\{T(\tau ),U(\tau )\right\}$.

Applying Cauchy, Bessel, Hölder inequalities, Lipschitz condition, we have From the Gronwall’s inequality,

(13)

We get $T(\tau )=U(\tau )\mathrm{.}$

4. Stability of Solution

Proof. 

Applying Cauchy, Bessel, Hölder inequalities and Lipschitz condition, we have From the Gronwall’s inequality,

(14)

From Eq. 14, for $\theta \to \overline{\theta }$. Then $T\to \overline{T}$.

5. Numerical Method for Problem

In this section, we use implicit finite-difference approximation of the discretized problem (1)-(4);

(15)

(16)

(17)

(18)

where the computational domain $\left[0,\pi \right]\times \left[0,\pi \right]\times \left[0,\tau \right]$ is discretized as follows:

Where $\Delta \alpha =\pi /\mathrm{n}\alpha$, $\Delta \beta =\pi /\mathrm{n}\beta$ and $\Delta \tau =\tau /\mathrm{n}$ are the space α direction, space β direction and time steps, respectively. nx, ny and n is are three positive integers.

In order to define periodic boundary condition (Eq. 17 and 18) for implicit finite difference scheme, one-dimensional schematic figure with numerical meshes is represented in Figure 2. Point-1 and point-nα are boundary points of one-dimensional solution domain. When we determine spatial dicretization with finite difference scheme, the used finite difference formulations are defined as;

(19)

where ${\mathrm{r}}_{\alpha }=\Delta \tau /\Delta {\alpha }^2$, and the coefficient matrix can be rewrittwen for one-dimensional solution domain.

(20)

This one-dimensional formulation and coefficient matrix can be extended in two-dimensional solution domain. Two-dimensional schematic figure with numerical meshes (9 finite difference meshes) is represented in Figure 3. Of course, matrix is constructed for 9 finite difference meshes, therefore the matrix size is 9×9. Point-1, 2 and 3 are periodic with 7, 8 and 9, respectively, at β-direction. Point-1, 4 and 7 are periodic with 3, 6 and 9, respectively, at α direction. Coefficient matrix can be written as;

(21)

where $r_{\beta }=\Delta \tau /\Delta {\beta }^2$. If large number of meshes are used, the matrix can be extended in a same way. For example, if we have 20 meshes both α and β directions, our matrix size will be 400×400, total element of matrix is 160 000, and our matrix can be constructed in a same manner to define points of periodic boundary condition and inner (no periodic boundary condition points).

Example 1. 

Consider the heat diffision problem with heat source (1)-(4):

7. Conclusions

Nomenclature

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