# A new kind of double Chebyshev polynomial approximation on unbounded domains

- Ayşe Betül Koç
^{1}Email author and - Aydın Kurnaz
^{1}

**2013**:10

**DOI: **10.1186/1687-2770-2013-10

© Koç and Kurnaz; licensee Springer. 2013

**Received: **2 October 2012

**Accepted: **4 January 2013

**Published: **22 January 2013

## 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 pseudospectral-collocation method matrix method unbounded domains## 1 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 [1–12]. One of the most important of those special functions is Chebyshev polynomials. The well-known first kind Chebyshev polynomials [1] are orthogonal with respect to the weight-function ${w}_{c}(x)=\frac{1}{\sqrt{1-{x}^{2}}}$ on the interval $[-1,1]$. These polynomials have many applications in different areas of interest, and a lot of studies are devoted to show the merits of them in various ways. One of the application fields of Chebyshev polynomials can appear in the solution of differential equations. For example, Chebyshev polynomial approximations have been used to solve ordinary differential equations with boundary conditions in [1], with collocation points in [13], the general class of linear differential equations in [14, 15], linear-integro differential equations with collocation points in [16], the system of high-order linear differential and integral equations with variable coefficients in [17, 18], and the Sturm-Liouville problems in [19].

Some of the fundamental ideas of Chebyshev polynomials in one-variable techniques have been extended and developed to multi-variable 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 Akyuz-Dascioglu [23] as well.

*et al.*[24] has proposed a modified type of Chebyshev polynomials as an alternative to the solutions of the problems given in a nonnegative real domain. In his study, the basis functions called rational Chebyshev polynomials are orthogonal in ${L}_{2}(0,\mathrm{\infty})$ and are defined by

Parand *et al.* and Sezer *et al.* successfully applied spectral methods to solve problems on semi-infinite 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 multi-variable case, especially, to two-variable problems.

## 2 Properties of double EC polynomials

Therefore, the exponential Chebyshev (EC) functions are recently defined in a similar fashion as follows [27].

*n*, the

*n*th derivative of a function $f\in {L}^{2}$ is also in ${L}^{2}$. Then an EC polynomial can be given by

where $y=\frac{{e}^{x}-1}{{e}^{x}+1}$ .

where ${c}_{m}=\{\begin{array}{l}2,m=0,\\ 1,m\ne 0\end{array}$ and ${\delta}_{mn}$ is the Kronecker function.

### Double EC functions

*et al.*[5] and Doha [11] have also mentioned a Chebyshev polynomial expression for an infinitely differentiable function $u(x,y)$ defined on the square $S(-1\le x,y\le 1)$ by

where ${T}_{r}(x)$ and ${T}_{s}(y)$ are Chebyshev polynomials of the first kind, and the double primes indicate that the first term is $\frac{1}{4}{a}_{0,0}$; ${a}_{m,0}$ and ${a}_{0,n}$ are to be taken as $\frac{1}{2}{a}_{m,0}$ and $\frac{1}{2}{a}_{0,n}$ for $m,n\ge 0$, respectively.

**Definition**

**Recurrence relation**The polynomial ${E}_{r,s}(x,y)$ satisfies the recurrence relations

**Multiplication**${E}_{i,j}(x,y)$ is said to be of higher order than ${E}_{m,n}(x,y)$ if $i+j>m+n$. Then the following result holds:

### Function approximation

*m*th and

*n*th terms, then it can be written in the matrix form

**A**is an unknown coefficient vector,

### Matrix relations of the derivatives of a function

**Proposition 1**

*Let*$u(x,y)$

*and*$(i,j)$

*th*-

*order derivative be given by*(2.12)

*and*(2.16),

*respectively*.

*Then there exists a relation between the double EC coefficient row vector*$\mathbf{E}(x,y)$

*and*$(i+j)$

*th*-

*order partial derivatives of the vector*${\mathbf{E}}^{(i,j)}(x,y)$

*of size*$1\times (m+1)(n+1)$

*as*

*where*${\mathbf{D}}_{x}$

*and*${\mathbf{D}}_{y}$

*are*$(m+1)(n+1)\times (m+1)(n+1)$

*operational matrices for partial derivatives given in the following forms*:

*and*

*Here*, **I** *and* **O** *are* $(m+1)(n+1)$ *identity and zero matrices*, *respectively*, *and* *T* *denotes the usual matrix transpose*.

*Proof*Taking the partial derivatives of ${E}_{0,s}$, ${E}_{1,s}$ and both sides of the recurrence relation (2.5) with respect to

*x*, we get

*y*, respectively, we write

We have noted here that ${E}_{r,s}^{(1,0)}(x,y)={E}_{r,s}^{(0,0)}(x,y)=0$ for $r>m$ and ${E}_{r,s}^{(0,1)}(x,y)={E}_{r,s}^{(0,0)}(x,y)=0$ for $s>n$.

where ${\mathbf{E}}^{(0,0)}(x,y)=\mathbf{E}(x,y)$ and ${({\mathbf{D}}_{x})}^{0}={({\mathbf{D}}_{y})}^{0}=\mathbf{I}$ and **I** denotes $(m+1)(n+1)$ identity matrix.

□

**Remark** ${({\mathbf{D}}_{x})}^{i}{({\mathbf{D}}_{y})}^{j}={({\mathbf{D}}_{y})}^{j}{({\mathbf{D}}_{x})}^{i}$.

**Corollary**

*From 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 ${E}_{r,s}(x,y)$ of an unknown function and those of its derivatives ${E}_{r,s}^{(i,j)}(x,y)$ for different nonnegative integer values of *i* and *j*.

Here, ${u}^{(0,0)}(x,y)=u(x,y)$, ${u}^{(i,j)}(x,y)=\frac{{\partial}^{i+j}}{\partial {x}^{i}\partial {y}^{j}}u(x,y)$ and ${q}_{i,j}(x,y)$, $f(x,y)$, ${c}_{{t}_{i,j}}(x)$, $g(x)$, ${d}_{{t}_{i,j}}(y)$, $h(y)$ are known functions on the square $S(-\mathrm{\infty}<x,y<\mathrm{\infty})$. We now describe an approximate solution of this problem by means of double EC series as defined in (2.10). Our aim is to find the EC coefficients in the vector **A**. For this reason, we can represent the given problem and its conditions by a system of linear algebraic equations by using collocation points.

- (i)
${x}_{m}\to \mathrm{\infty}$ and ${y}_{n}\to \mathrm{\infty}$,

- (ii)
${x}_{0}\to -\mathrm{\infty}$ and ${y}_{n}\to -\mathrm{\infty}$.

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.

where ${\mathbf{Q}}_{i,j}$ denotes the diagonal matrix with the elements ${q}_{i,j}({x}_{k},{y}_{l})$ ($k=0,1,\dots ,m$; $l=0,1,\dots ,n$) and **F** denotes the column matrix with the elements $f({x}_{k},{y}_{l})$ ($k=0,1,\dots ,m$; $l=0,1,\dots ,n$).

**E**is the block matrix given by

which corresponds to a system of $(m+1)(n+1)$ linear algebraic equations with unknown double EC coefficients ${a}_{r,s}$.

It is also noted that the structures of matrices ${\mathbf{Q}}_{i,j}$ and **F** vary according to the number of collocation points and the structure of the problem. However, **E**, ${\mathbf{D}}_{x}$ and ${\mathbf{D}}_{y}$ do not change their nature for fixed values of *m* and *n* which are truncation limits of the EC series. In other words, the changes in **E**, ${\mathbf{D}}_{x}$ and ${\mathbf{D}}_{y}$ are just dependent on the number of collocation points.

**W**and write

**V**is an $h\times (m+1)(n+1)$ matrix and

**R**is an $h\times 1$ matrix, so that

*h*is the rank of all row matrices belonging to the given condition. The augmented matrices of the conditions become

Finally, the vector **A** (thereby the coefficients ${a}_{r,s}$) is determined by applying some numerical methods (*e.g.*, Gauss elimination) designed especially to solve the system of linear equations. Therefore, the approximate solution can be obtained. In other words, it gives the double EC series solution of the problem (3.1) with given conditions.

## 4 Illustration

Now, we give an example to show the ability and efficiency of the double EC polynomial approximation method.

**Example**

It is known that the exact solution of the problem is $u(x,y)=\frac{{e}^{x+y}-{e}^{x}-{e}^{y}+1}{({e}^{x}+1)({e}^{y}+1)}$.

**Absolute errors of Example at different points**

x | y | $\mathit{m}\mathbf{=}\mathit{n}\mathbf{=}\mathbf{15}$ |
---|---|---|

4.5056 | 4.5056 | 3.31 E-08 |

4.5056 | 2.248 | 2.09 E-08 |

4.5056 | 1.618 | 1.48 E-08 |

4.5056 | −2.248 | 3.18 E-08 |

4.5056 | −4.5056 | 3.38 E-08 |

3.0970 | 4.5056 | 1.58 E-08 |

3.0970 | 1.618 | 6.00 E-10 |

3.0970 | −0.209 | 1.90 E-10 |

2.248 | 3.0970 | 4.40 E-09 |

2.248 | −3.0970 | 5.30 E-09 |

1.6183 | −0.2098 | 3.50 E-10 |

0.2098 | −0.2098 | 1.80 E-10 |

−0.2098 | −0.2098 | 1.00 E-10 |

−2.248 | −1.098 | 1.90 E-09 |

−3.0970 | 2.248 | 2.20 E-09 |

−3.0970 | −2.248 | 1.30 E-09 |

## 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 multi-variable problems. It is also noted that the double EC-collocation method is very effective and has a direct ability to solve multi-variable (especially two-variable) 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 non-zero entries lain along two subdiagonals.

## Declarations

### 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, 20-24 June, (2012).

## Authors’ Affiliations

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