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Advances in Difference Equations

, 2010:623508 | Cite as

Transformations of Difference Equations II

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Research Article
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Abstract

This is an extension of the work done by Currie and Love (2010) where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with non-eigenparameter-dependent boundary conditions at the end points. In particular, we now consider boundary conditions which depend affinely on the eigenparameter together with various combinations of Dirichlet and non-Dirichlet boundary conditions. The spectra of the resulting transformed boundary value problems are then compared to the spectra of the original boundary value problems.

Keywords

Boundary Condition Difference Equation Initial Point Dirichlet Boundary Condition Characteristic Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

This paper continues the work done in [1], where we considered a weighted second-order difference equation of the following form:

with Open image in new window representing a weight function and Open image in new window a potential function.

This paper is structured as follows.

The relevant results from [1], which will be used throughout the remainder of this paper, are briefly recapped in Section 2.

In Section 3, we show how non-Dirichlet boundary conditions transform to affine Open image in new window -dependent boundary conditions. In addition, we provide conditions which ensure that the linear function (in Open image in new window ) in the affine Open image in new window -dependent boundary conditions is a Nevanlinna or Herglotz function.

Section 4 gives a comparison of the spectra of all possible combinations of Dirichlet and non-Dirichlet boundary value problems with their transformed counterparts. It is shown that transforming the boundary value problem given by (2.2) with any one of the four combinations of Dirichlet and non-Dirichlet boundary conditions at the end points using (3.1) results in a boundary value problem with one extra eigenvalue in each case. This is done by considering the degree of the characteristic polynomial for each boundary value problem.

It is shown, in Section 5, that we can transform affine Open image in new window -dependent boundary conditions back to non-Dirichlet type boundary conditions. In particular, we can transform back to the original boundary value problem.

To conclude, we outline briefly how the process given in the sections above can be reversed.

2. Preliminaries

Consider the second-order difference equation (1.1) for Open image in new window with boundary conditions

where Open image in new window and Open image in new window are constants, see [2]. Without loss of generality, by a shift of the spectrum, we may assume that the least eigenvalue, Open image in new window , of (1.1), (2.1) is Open image in new window .

We recall the following important results from [1]. The mapping Open image in new window defined for Open image in new window by Open image in new window , where Open image in new window is the eigenfunction of (1.1), (2.1) corresponding to the eigenvalue Open image in new window , produces the following transformed equation:
Moreover, Open image in new window obeying the boundary conditions (2.1) transforms to Open image in new window obeying the Dirichlet boundary conditions as follows:
Applying the mapping Open image in new window given by Open image in new window for Open image in new window , where Open image in new window is a solution of (2.2) with Open image in new window , where Open image in new window is less than the least eigenvalue of (2.2), (2.4), such that Open image in new window for all Open image in new window , results in the following transformed equation:

Here, we take Open image in new window , thus Open image in new window is defined for Open image in new window .

In addition, Open image in new window obeying the Dirichlet boundary conditions (2.4) transforms to Open image in new window obeying the non-Dirichlet boundary conditions as follows:

3. Non-Dirichlet to Affine

In this section, we show how Open image in new window obeying the non-Dirichlet boundary conditions (3.2), (3.13) transforms under the following mapping:

to give Open image in new window obeying boundary conditions which depend affinely on the eigenparamter Open image in new window . We provide constraints which ensure that the form of these affine Open image in new window -dependent boundary conditions is a Nevanlinna/Herglotz function.

Theorem 3.1.

Under the transformation (3.1), Open image in new window obeying the boundary conditions
for Open image in new window , transforms to Open image in new window obeying the boundary conditions

where Open image in new window , Open image in new window and Open image in new window . Here, Open image in new window and Open image in new window is a solution of (2.2) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.2), (3.2), and (3.13) such that Open image in new window for Open image in new window .

Proof.

The values of Open image in new window for which Open image in new window exists are Open image in new window . So to impose a boundary condition at Open image in new window , we need to extend the domain of Open image in new window to include Open image in new window . We do this by forcing the boundary condition (3.3) and must now show that the equation is satisfied on the extended domain.

Evaluating (2.5) at Open image in new window for Open image in new window and using (3.3) gives the following:
Also from (3.1) for Open image in new window and Open image in new window , we obtain the following:
Substituting (3.2) into the above equation yields
Thus, (3.4) becomes
This may be slightly rewritten as follows
Also from (2.2), with Open image in new window , together with (3.2), we have the following:
Subtracting (3.9) from (3.8) and using the fact that Open image in new window results in
Equating coefficients of Open image in new window on both sides gives the following:
and equating coefficients of Open image in new window on both sides gives the following:

where Open image in new window , and recall Open image in new window .

Note that for Open image in new window , this corresponds to the results in [1] for Open image in new window .

Theorem 3.2.

Under the transformation (3.1), Open image in new window satisfying the boundary conditions
for Open image in new window , transforms to Open image in new window obeying the boundary conditions

where Open image in new window , Open image in new window , and Open image in new window . Here, Open image in new window is a solution to (2.2) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.2), (3.2), and (3.13) such that Open image in new window in the given interval, Open image in new window .

Proof.

Evaluating (3.1) at Open image in new window and Open image in new window gives the following:
By considering Open image in new window satisfying (2.2) at Open image in new window , we obtain that
Substituting (3.17) into (3.16) gives the following:
Now using (3.13) together with (3.15) yields
which in turn, by substituting into (3.13), gives the following:
Thus, by putting (3.19) and (3.20) into (3.18), we obtain
The equation above may be rewritten as follows:
Now, since Open image in new window is a solution to (2.2) for Open image in new window , we have that
Substituting (3.23) into (3.22) gives the following:

which is of the form (3.14), where Open image in new window , Open image in new window , and Open image in new window .

Note that if we require that Open image in new window in (3.3) be a Nevanlinna or Herglotz function, then we must have that Open image in new window . This condition provides constraints on the allowable values of Open image in new window .

Remark 3.3.

In Theorems 3.1 and 3.2, we have taken Open image in new window to be a solution of (2.2) for Open image in new window with Open image in new window less than the least eigenvalue of (2.2), (3.2), and (3.13) such that Open image in new window in Open image in new window . We assume that Open image in new window does not obey the boundary conditions (3.2) and (3.13) which is sufficient for the results which we wish to obtain in this paper. However, this case will be dealt with in detail in a subsequent paper.

Theorem 3.4.

If Open image in new window where Open image in new window is a solution to (2.2) for Open image in new window with Open image in new window less than the least eigenvalue of (2.2), (3.2), and (3.13) and Open image in new window in the given interval Open image in new window , then the values of Open image in new window which ensure that Open image in new window in (3.3), that is, which ensure that Open image in new window in (3.3) is a Nevanlinna function are

Proof.

From Theorem 3.1, we have that
Assume that Open image in new window , then to ensure that Open image in new window we require that either Open image in new window and Open image in new window or Open image in new window and Open image in new window . For the first case, since Open image in new window , we get Open image in new window and Open image in new window . For the second case, we obtain Open image in new window and Open image in new window , which is not possible. Thus, allowable values of Open image in new window for Open image in new window are

Since Open image in new window If Open image in new window , then we must have that either Open image in new window and Open image in new window or Open image in new window and Open image in new window . The first case of Open image in new window is not possible since Open image in new window and Open image in new window , Open image in new window , which implies that Open image in new window in particular for Open image in new window . For the second case, we get Open image in new window and Open image in new window which is not possible. Thus for Open image in new window , there are no allowable values of Open image in new window .

Also, if we require that Open image in new window from (3.14) be a Nevanlinna/Herglotz function, then we must have Open image in new window . This provides conditions on the allowable values of Open image in new window .

Corollary 3.5.

If Open image in new window where Open image in new window is a solution to (2.2) for Open image in new window with Open image in new window less than the least eigenvalue of (2.2), (3.2), and (3.13), and Open image in new window in the given interval Open image in new window , then

Proof.

Without loss of generality, we may shift the spectrum of (2.2) with boundary conditions (3.2), (3.13), such that the least eigenvalue of (2.2) with boundary conditions (3.2), (3.13) is strictly greater than Open image in new window , and thus we may assume that Open image in new window .

Since Open image in new window , we consider the two cases, Open image in new window and Open image in new window .

Now if Open image in new window , then the numerator of Open image in new window is strictly negative. Thus, in order that Open image in new window , we require that the denominator is strictly negative, that is, Open image in new window . So either Open image in new window and Open image in new window or Open image in new window and Open image in new window . As Open image in new window , we obtain that either Open image in new window and Open image in new window or Open image in new window and Open image in new window . These are the same conditions as we had on Open image in new window for Open image in new window . Thus, the sign of Open image in new window does not play a role in finding the allowable values of Open image in new window which ensure that Open image in new window , and hence we have the required result.

4. Comparison of the Spectra

In this section, we see how the transformation, (3.1), affects the spectrum of the difference equation with various boundary conditions imposed at the initial and terminal points.

By combining the results of [1, conclusion] with Theorems 3.1 and 3.2, we have proved the following result.

Theorem 4.1.

Assume that Open image in new window satisfies (2.2). Consider the following four sets of boundary conditions:

The transformation (3.1), where Open image in new window is a solution to (2.2) for Open image in new window , where Open image in new window is less than the least eigenvalue of (2.2) with one of the four sets of boundary conditions above, such that Open image in new window in the given interval Open image in new window , takes Open image in new window obeying (2.2) to Open image in new window obeying (2.5).

In addition,
  1. (i)
     
  2. (ii)

    Open image in new window obeying (4.2) transforms to Open image in new window obeying (4.5) and (3.14).

     
  3. (iii)

    Open image in new window obeying(4.3) transforms to Open image in new window obeying (3.3) and (4.6).

     
  4. (iv)

    Open image in new window obeying (4.4) transforms to Open image in new window obeying (3.3) and (3.14).

     

The next theorem, shows that the boundary value problem given by Open image in new window obeying (2.2) together with any one of the four types of boundary conditions in the above theorem has Open image in new window eigenvalues as a result of the eigencondition being the solution of an Open image in new window th order polynomial in Open image in new window . It should be noted that if the boundary value problem considered is self-adjoint, then the eigenvalues are real, otherwise the complex eigenvalues will occur as conjugate pairs.

Theorem 4.2.

The boundary value problem given by Open image in new window obeying (2.2) together with any one of the four types of boundary conditions given by (4.1) to (4.4) has Open image in new window eigenvalues.

Proof.

So setting Open image in new window , in (4.7), gives the following:
For the boundary conditions (4.1) and (4.2), we have that Open image in new window giving

where Open image in new window and Open image in new window are real constants, that is, a first order polynomial in Open image in new window .

Also Open image in new window in (4.7) gives that
Substituting in for Open image in new window , from above, we obtain

where again Open image in new window are real constants, that is, a quadratic polynomial in Open image in new window .

Thus, by an easy induction, we have that

where Open image in new window , Open image in new window and Open image in new window , Open image in new window are real constants, that is, an Open image in new window th and an Open image in new window th order polynomial in Open image in new window , respectively.

Now, (4.1) gives Open image in new window , that is,
So our eigencondition is given by

which is an Open image in new window th order polynomial in Open image in new window and, therefore, has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.2) with (4.1) has Open image in new window eigenvalues.

from which we obtain the following eigencondition:

This is again an Open image in new window th order polynomial in Open image in new window and therefore has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.2) with (4.2) has Open image in new window eigenvalues.

Now for the boundary conditions (4.3) and (4.4), we have that Open image in new window , thus (4.8) becomes

where Open image in new window and Open image in new window are real constants, that is, a first order polynomial in Open image in new window .

Using Open image in new window and Open image in new window from above, we can show that Open image in new window can be written as the following:

where again Open image in new window , Open image in new window are real constants, that is, a quadratic polynomial in Open image in new window .

Thus, by induction,

where Open image in new window , Open image in new window and Open image in new window , Open image in new window are real constants, thereby giving an Open image in new window th and an Open image in new window th order polynomial in Open image in new window , respectively.

Now, (4.3) gives Open image in new window , that is,
So our eigencondition is given by

which is an Open image in new window th order polynomial in Open image in new window and, therefore, has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.2) with (4.3) has Open image in new window eigenvalues.

Lastly, (4.4) gives Open image in new window , that is,
from which we obtain the following eigencondition:

This is again an Open image in new window th order polynomial in Open image in new window and therefore has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.2) with (4.4) has Open image in new window eigenvalues.

In a similar manner, we now prove that the transformed boundary value problems given in Theorem 4.1 have Open image in new window eigenvalues, that is, the spectrum increases by one in each case.

Theorem 4.3.

The boundary value problem given by Open image in new window obeying (2.5), Open image in new window , together with any one of the four types of transformed boundary conditions given in (i) to (iv) in Theorem 4.1 has Open image in new window eigenvalues. The additional eigenvalue is precisely Open image in new window with corresponding eigenfunction Open image in new window , as given in Theorem 4.1.

Proof.

The proof is along the same lines as that of Theorem 4.2. By Theorem 3.1, we have extended Open image in new window , such that Open image in new window exists for Open image in new window .

Since Open image in new window obeys (2.5), we have that, for Open image in new window ,
For the transformed boundary conditions in (i) and (ii) of Theorem 4.1, we have that (4.5) is obeyed, and as in Theorem 4.2, we can inductively show that
and also by [1], we can extend the domain of Open image in new window to include Open image in new window if necessary by forcing (4.6) and then

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window , Open image in new window are real constants, that is, an Open image in new window th, Open image in new window th, and Open image in new window th order polynomial in Open image in new window , respectively.

Now for (i), the boundary condition (4.6) gives the following:
Therefore, the eigencondition is

which is an Open image in new window th order polynomial in Open image in new window and thus has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.5) with transformed boundary conditions (i), that is, (4.5) and (4.6), has Open image in new window eigenvalues.

Also, for (ii), from the boundary condition (3.14), we get
Therefore, the eigencondition is

which is an Open image in new window th order polynomial in Open image in new window and thus has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.5) with transformed boundary conditions (ii), that is, (4.5) and (3.14), has Open image in new window eigenvalues.

Putting Open image in new window in (4.24), we get
For the boundary conditions in (iii) and (iv), we have that (3.3) is obeyed, thus,

where Open image in new window , Open image in new window , and Open image in new window are real constants.

Putting Open image in new window in (4.24), we get
which, by using (3.3) and Open image in new window , can be rewritten as follows:

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window are real constants.

Thus, inductively we obtain
Also, by [1], we can again extend the domain of Open image in new window to include Open image in new window , if needed, by forcing (4.6), thus,

where all the coefficients of Open image in new window are real constants.

The transformed boundary conditions (iii) mean that (4.6) is obeyed, thus, our eigencondition is

which is an Open image in new window th order polynomial in Open image in new window and thus has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.5) with transformed boundary conditions (iii), that is, (3.3) and (4.6), has Open image in new window eigenvalues.

Also, the transformed boundary conditions in (iv) give (3.14) which produces the following eigencondition:

which is an Open image in new window th order polynomial in Open image in new window and thus has Open image in new window roots. Hence, the boundary value problem given by Open image in new window obeying (2.5) with transformed boundary conditions (iv), that is, (3.3) and (3.14), has Open image in new window eigenvalues.

Lastly, we have that (3.1) transforms eigenfunctions of any of the boundary value problems in Theorem 4.2 to eigenfunctions of the corresponding transformed boundary value problem, see Theorem 4.2. In particular, if Open image in new window are the eigenvalues of the original boundary value problem with corresponding eigenfunctions Open image in new window , then Open image in new window are eigenfunctions of the corresponding transformed boundary value problem with eigenvalues Open image in new window . Since we know that the transformed boundary value problem has Open image in new window eigenvalues, it follows that Open image in new window constitute all the eigenvalues of the transformed boundary value problem, see [1].

5. Affine to Non-Dirichlet

In this section, we now show that the process in Section 3 may be reversed. In particular, by applying the following mapping:

we can transform Open image in new window obeying affine Open image in new window -dependent boundary conditions to Open image in new window obeying non-Dirichlet boundary conditions.

Theorem 5.1.

Consider the boundary value problem given by Open image in new window satisfying (2.5) with the following boundary conditions:
The transformation (5.1), for Open image in new window , where Open image in new window is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue Open image in new window , yields the following equation:
In addition, Open image in new window obeying (5.2) and (5.3) transforms to Open image in new window obeying the non-Dirichlet boundary conditions

where Open image in new window and Open image in new window .

Proof.

The fact that Open image in new window , obeying (2.5), transforms to Open image in new window , obeying (5.4), was covered in [1, conclusion]. Now, Open image in new window is defined for Open image in new window and is extended to Open image in new window by (5.2). Thus, Open image in new window is defined for Open image in new window giving that (5.4) is valid for Open image in new window . For Open image in new window and Open image in new window , (5.1) gives the following:
Setting Open image in new window in (2.5) gives the following:
which by using (5.2) becomes
Since Open image in new window is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue Open image in new window , we have that
and hence
Substituting (5.11) and (5.13) into (5.8) and using (5.2), we obtain
Since Open image in new window , everything can be written over the common denominator Open image in new window . Taking out Open image in new window and simplifying, we get
Substituting (5.2) into (5.9) gives the following:
Hence, by putting (5.16) into (5.17), we get

So to impose the boundary condition (5.7), it is necessary to extend the domain of Open image in new window by forcing the boundary condition (5.7). We must then check that Open image in new window satisfies the equation on the extended domain.

Evaluating (5.4) at Open image in new window and using (5.7) give the following:
Using (5.1) with Open image in new window and Open image in new window together with (5.3), we obtain
Substituting the above two equations into (5.19) yields
Since Open image in new window is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue Open image in new window we have that Open image in new window . Thus, the above equation can be simplified to
Also (2.5) evaluated at Open image in new window for Open image in new window together with (5.3) gives
Adding (5.22) to (5.23) and using the fact that Open image in new window yields
By substituting in for Open image in new window and Open image in new window , it is easy to see that all the Open image in new window terms cancel out. Next, we examine the coefficients of Open image in new window , and using Open image in new window , we obtain that the coefficient of Open image in new window is
which equals Open image in new window by (2.5) evaluated at Open image in new window . Thus, only the terms in Open image in new window remain. First, we note that by substituting in for Open image in new window , Open image in new window and Open image in new window we get
Thus, equating coefficients of Open image in new window gives the following:
Since Open image in new window , we can divide and solve for Open image in new window to obtain

Note that the case of Open image in new window , that is, a non-Dirichlet boundary condition, gives Open image in new window , that is, Open image in new window which corresponds to the results obtained in [1].

If we set Open image in new window , with Open image in new window a solution of (2.2) for Open image in new window where Open image in new window less than the least eigenvalue of (2.2), (3.2), and (3.13) and Open image in new window in the given interval Open image in new window , then Open image in new window is an eigenfunction of (2.5), (5.2), and (5.3) corresponding to the eigenvalue Open image in new window . To see that Open image in new window satisfies (2.5), see [1, Lemma Open image in new window ] with, as previously, Open image in new window , and now Open image in new window . Then, by construction, Open image in new window obeys (5.2). We now show that Open image in new window obeys (5.3). Let Open image in new window ,
Now Open image in new window is a solution of (2.2) for Open image in new window , thus,

Remark 5.2.

For Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window as above, the transformation (5.1), in Theorem 5.1, results in the original given boundary value problem. In particular, we obtain that in Theorem 5.1   Open image in new window and Open image in new window , see [1, Theorem Open image in new window .2]. In addition,

That is, the boundary value problem given by Open image in new window satisfying (2.5) with boundary conditions (5.2), (5.3) transforms under (5.1) to Open image in new window obeying (2.2) with boundary conditions (3.2), (3.13) which is the original boundary value problem.

We now verify that Open image in new window . Let
Also, Open image in new window . Dividing through by Open image in new window and using Open image in new window together with Open image in new window gives the following:
Thus, using (5.35) and (5.36), the numerator of Open image in new window is simplified to
The denominator of Open image in new window can be simplified using Open image in new window to

hence Open image in new window .

Finally, substituting in for Open image in new window , we obtain

Thus, Open image in new window , that is, Open image in new window .

Note that
We now substitute in for Open image in new window and Open image in new window into the equation for Open image in new window and use (5.42) and (5.43) to obtain that

that is, Open image in new window .

To summarise, we have the following.

Consider Open image in new window obeying (2.5) with one of the following 4 types of boundary conditions:
  1. (a)

    non-Dirichlet and non-Dirichlet, that is, (4.5) and (4.6);

     
  2. (b)

    non-Dirichlet and affine, that is, (4.5) and (3.14);

     
  3. (c)

    affine and non-Dirichlet, that is, (3.3) and (4.6);

     
  4. (d)

    affine and affine, that is, (3.3) and (3.14).

     

By Theorem 4.3, each of the above boundary value problems have Open image in new window eigenvalues.

Now, the transformation (5.1), with Open image in new window an eigenfunction of (2.5) with boundary conditions (a) ((b), (c), (d), resp.) corresponding to the eigenvalue Open image in new window , transforms Open image in new window obeying (2.5) to Open image in new window obeying (2.2) and transforms the boundary conditions as follows:
  1. (1)

    boundary conditions (a) transform to Open image in new window and Open image in new window ;

     
  2. (2)

    boundary conditions (b) transform to Open image in new window and (3.13);

     
  3. (3)

    boundary conditions (c) transform to (3.2) and Open image in new window ;

     
  4. (4)

    boundary conditions (d) transform to (3.2) and (3.13).

     

By Theorem 4.2, we know that the above transformed boundary value problems in Open image in new window each have Open image in new window eigenvalues. In particular, if Open image in new window are the eigenvalues of (2.5), (a) ((b), (c), (d), resp.) with eigenfunctions Open image in new window , then Open image in new window and Open image in new window are eigenfunctions of (2.2), (1) ((2), (3), (4), resp.) with eigenvalues Open image in new window . Since we know that these boundary value problems have Open image in new window eigenvalues, it follows that Open image in new window constitute all the eigenvalues.

6. Conclusion

To conclude, we outline (the details are left to the reader to verify) how the entire process could also be carried out the other way around. That is, we start with a second order difference equation of the usual form, given in the previous sections, together with boundary conditions of one of the following forms:
  1. (i)

    non-Dirichlet at the initial point and affine at the terminal point;

     
  2. (ii)

    affine at the initial point and non-Dirichlet at the terminal point;

     
  3. (iii)

    affine at the initial point and at the terminal point.

     
We can then transform the above boundary value problem (by extending the domain where necessary, as done previously) to an equation of the same type with, respectively, transformed boundary conditions as follows:
  1. (A)

    Dirichlet at the initial point and non-Dirichlet at the terminal point;

     
  2. (B)

    non-Dirichlet at the initial point and Dirichlet at the terminal point;

     
  3. (C)

    non-Dirichlet at the initial point and at the terminal point.

     

It is then possible to return to the original boundary value problem by applying a suitable transformation to the transformed boundary value problem above.

Notes

Acknowledgments

The authors would like to thank Professor Bruce A. Watson for his useful input and suggestions. This work was supported by NRF Grant nos. TTK2007040500005 and FA2007041200006.

References

  1. 1.
    Currie S, Love A: Transformations of difference equations I. Advances in Difference Equations 2010, 2010:-22.Google Scholar
  2. 2.
    Atkinson FV: Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, vol. 8. Academic Press, New York, NY, USA; 1964:xiv+570.Google Scholar

Copyright information

© S. Currie and A. D. Love. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandWitsSouth Africa

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