Abstract
In this study, a new solution scheme for the partial differential equations with variable coefficients defined on a large domain, especially including infinities, has been investigated. For this purpose, a spectral basis, called exponential Chebyshev (EC) polynomials, has been extended to a new kind of double Chebyshev polynomials. Many outstanding properties of those polynomials have been shown. The applicability and efficiency have been verified on an illustrative example.
MSC: 35A25.
Keywords:
partial differential equations; pseudospectralcollocation method; matrix method; unbounded domains1 Introduction
The importance of special functions and orthogonal polynomials occupies a central
position in the numerical analysis. Most common solution techniques of differential
equations with these polynomials can be seen in [112]. One of the most important of those special functions is Chebyshev polynomials. The
wellknown first kind Chebyshev polynomials [1] are orthogonal with respect to the weightfunction
Some of the fundamental ideas of Chebyshev polynomials in onevariable techniques have been extended and developed to multivariable cases by the studies of Fox et al.[1], Basu [20], Doha [21] and Mason et al.[5]. In recent years, the Chebyshev matrix method for the solution of partial differential equations (PDEs) has been proposed by Kesan [22] and AkyuzDascioglu [23] as well.
On the other hand, all of the above studies are considered on the interval
Parand et al. and Sezer et al. successfully applied spectral methods to solve problems on semiinfinite intervals [25,26]. These approaches can be identified as the methods of rational Chebyshev Tau and rational Chebyshev collocation, respectively. However, this kind of extension also fails to solve all of the problems over the whole real domain. More recently, we have introduced a new modified type of Chebyshev polynomials that is developed to handle the problems in the whole real range called exponential Chebyshev (EC) polynomials [27].
In this study, we have shown the extension of the EC polynomial method to multivariable case, especially, to twovariable problems.
2 Properties of double EC polynomials
The wellknown first kind Chebyshev polynomials are orthogonal in the interval
Therefore, the exponential Chebyshev (EC) functions are recently defined in a similar fashion as follows [27].
Let
be a function space with the weight function
where
This definition leads to the threeterm recurrence equation for EC polynomials
This definition also satisfies the orthogonality condition [27]
where
Double EC functions
Basu [20] has given the product
where
Definition
Based on Basu’s study, now we introduce double EC polynomials in the following form:
where
Recurrence relation The polynomial
If the function
and we have
Multiplication
Function approximation
Let
where
If
with
and A is an unknown coefficient vector,
Matrix relations of the derivatives of a function
and its matrix form is
where
Proposition 1Let
where
and
Here, IandOare
Proof Taking the partial derivatives of
and
By using the relations (2.18)(2.20) for
Similarly, taking the partial derivatives of
and
Then with the help of the relations (2.22)(2.24), the elements
We have noted here that
From (2.21) and (2.25), the following equalities hold for
and
where
Then utilizing the equalities in (2.26) and (2.27), the explicit relation between the double EC polynomial row vector and those of its derivatives has been proved as follows:
or
□
Remark
CorollaryFrom Eqs. (2.16) and (2.17), it is clear that the derivatives of the function are expressed in terms of double EC coefficients as follows:
3 Collocation method with double EC polynomials
In the process of obtaining the numerical solutions of partial differential equations
with the double EC method, the main idea or major step is to evaluate the necessary
Chebyshev coefficients of the unknown function. So, in Section 2, we give the explicit
relations between the polynomials
In this section, we consider the higherorder linear PDE with variable coefficients of a general form
with the conditions mentioned in [23] as three possible cases:
and/or
and/or
Here,
Now, the collocation points can be determined in the inner domain as
and at the boundaries
(i)
(ii)
Since EC polynomials are convergent at both boundaries, namely their values are either 1 or −1, the appearance of infinity in the collocation points does not cause a loss in the method.
Therefore, when we substitute the collocation points into the problem (3.1), we get
The system (3.6) can be written in the matrix form as follows:
where
Putting the collocation points into derivatives of the unknown function as in Eq. (2.28) yields
where E is the block matrix given by
and for
Therefore, from Eq. (3.7), we get a system of the matrix equation for the PDE
which corresponds to a system of
It is also noted that the structures of matrices
Briefly, we can denote the expression in the parenthesis of (3.10) by W and write
Then the augmented matrix of Eq. (3.11) becomes
Applying the same procedure for the given conditions (3.2)(3.4), we have
Then these can be written in a compact form
where V is an
Consequently, (3.12) together with (3.17) can be written in a new augmented matrix form
This form can be achieved by replacing some rows of (3.12) by the rows of (3.17)
accordingly, or adding those rows to the matrix (3.12) provided that
Finally, the vector A (thereby the coefficients
4 Illustration
Now, we give an example to show the ability and efficiency of the double EC polynomial approximation method.
Example
Let us consider the linear partial differential equation
with the conditions
It is known that the exact solution of the problem is
Absolute errors of the proposed procedure at the grid points are tabulated for
Table 1. Absolute errors of Example at different points
Contour plots of the exact solutions and the approximate solutions are given for the
region
5 Conclusion
In this article, a new solution scheme for the partial differential equation with variable coefficients defined on unbounded domains has been investigated and EC polynomials have been extended to double EC polynomials to solve multivariable problems. It is also noted that the double ECcollocation method is very effective and has a direct ability to solve multivariable (especially twovariable) problems in the infinite domain. For computational purposes, this approach also avoids more computations by using sparse operational matrices and saves much memory. On the other hand, the double EC polynomial approach deals directly with infinite boundaries, and their operational matrices are of few nonzero entries lain along two subdiagonals.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ABK wrote the first draft and AK corrected and improved the final version. All authors read and approved the final draft.
Acknowledgements
This study was supported by the Research Projects Center (BAP) of Selcuk University. The authors would like to thank the editor and referees for their valuable comments and remarks which led to a great improvement of the article. Also, ABK and AK would like to thank the Selcuk University and TUBITAK for their support. We note here that this study was presented orally at the International Conference on Applied Analysis and Algebra (ICAAA 2012), Istanbul, 2024 June, (2012).
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