(5)

Now using (3), (4) and (5), we get (Refer the work of Mittal and Ghalaut in [2] )

From (1) consider only the approximation of the sum of the first and the third terms, that is, the sum of

and

(6)

where

Again from (1) consider only the approximation of the sum of the first and the fourth terms, that is, the sum of

and, and we get

(7)

Once again from (1) consider only the approximation of the sum of the second and the fourth terms, that is, the sum of and; to get

(8)

Again taking the approximation of the term by

(9)

Equation (9) implying that

(10)

Now letting and adding (6), (7), (8) and twice of (10), we get

(11)

Now choose and consider the following terms in (11)

(12)

Again we can write the term in (12) as

(13)

Using (12), (13), and multiplying both sides of (11) by and rearranging and simplifying further, we get

(14)

where

The system of equations in (14) is a linear sparse system, and thereby when solving we save both work and storage compared with a general system of equations. Such savings are basically true of finite difference methods: they yield sparse systems because each equation involves only few variables.

To solve equation (14), consider first in the direction, next in the direction and lastly in the direction, and thus (14) can be written in matrix form as

(15)

where

and it has blocks and each is of order.

,

are of order.

For the domain

,

For the domain,

and are the same as in the domain.

Here in, the matrices and are circulant matrices of order; and

, and

such that each represents a known boundary values of and values of, and

, and

Thus, we write (15) as

(16)

3. Extended Hockney’s Method

Observe that matrices and are real symmetric matrices and hence their eigenvalues and eigenvectors can easily be obtained as

For

, ,

,

and for

, ,

,

Let be an eigenvector of and corresponding to the eigenvalue and; and matrix be a modal matrix of and, such that

The modal matrix Q is defined by

, for; where 

  for

Let be a matrix of order; thus satisfy since.

Since and are symmetric matrices, we have

where

where

where

Let

, (17)

where

,;

and

Pre-multiplying equation (16) by and applying (17), we get

(18)

Now from each equation of (18) we collect the first equations and put them as one group of equation

(19)

Now put in equation (19) and collect the entire first set of equations, for and to get

and                       (20a)

Again consider the second equations by putting, and get

and                      (20b)

Continuing in this manner and finally considering the last equations for, we obtain

and                       (20c)

All these set of equations (20a)-(20c) are tri-diagonal ones and hence we solve for by using Thomas algorithm. With the help of (17) again we get all and this solves (14) as desired. By doing this we generally reduce the number of computations and computational time.

4. Numerical Results

In order to test the efficiency and adaptability of the proposed method, computational experiments are done on some selected problems that may arise in practice, for which the analytical solutions of are known to us. The computed solutions are found for all grid points for any values of and. Here results are reported at some randomly taken mesh points in terms of the absolute maximum error from Table 1 to 7.

Example 1. Consider with the boundary conditions,

, and

The analytical solution is and the computed results of this example are shown in table 1.

Example 2. Consider with the boundary conditions

,

, , and

The analytical solution is and the computed results of this example are shown in table 2.

Example 3. Consider with the boundary conditions

, ,

 

 

,

The analytical solution is and the computed results of this example are shown in table 3.

Example 4. Consider with the boundary conditions

, , and

The analytical solution is and the computed results of this example are shown in table 4.

Example 5 Consider, where with the boundary conditions

,

The analytical solution is and the computed results of this example are shown in Table 5.

 

This example was considered by M.C. Lai [1] as a test problem and our results are better than their results in terms of accuracy. For instance, for (8,16,16) the maximum absolute error in their result is 9.1438e-004 and while ours is 3.28689e-004.

Example 6 Consider, where with the boundary conditions

,; and

The analytical solution is and the computed results are shown in table 6.

Example 5.7 Consider where with the boundary conditions

,;

The analytical solution is and the computed results of this example are shown in table 7.

 

5. Conclusions

In this work, we have transformed the three dimensional Poisson’s equation in cylindrical coordinates system into a system of algebraic linear equations using its equivalent fourth-order finite difference approximation scheme. The resulting large number of algebraic equation is, then, systematically arranged in order to get a block matrix. By extending Hockney’s method to three dimensions, we reduced the obtained matrix into a block tridiagonal matrix, and each block is solved by the help of Thomas algorithm. We have successfully implemented this method to find the solution of the three dimensional Poisson’s equation in cylindrical coordinates system and it is found that the method can easily be applied and adapted to find a solution of some related applied problems. The method produced accurate results considering double precision. This method is direct and allows considerable savings in computer storage as well as execution speed.

Therefore, the method is suitable to apply to some three dimensional Poisson’s equations.

References