ON THE EXPONENTIAL CHEBYSHEV APPROXIMATION IN UNBOUNDED DOMAINS: A COMPARISON STUDY FOR SOLVING HIGH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Abstract: In this paper we investigate an improved scheme for exponential Chebyshev (EC) collocation method. The improved scheme of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain. This technique transforms the given differential equation and mixed conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.


Introduction
Chebyshev polynomials are one of the most important special functions, which are widely used in numerical analysis.The well-known Chebyshev first kind polynomials T n (x) [5] are orthogonal with respect to the weight-function w(x) = 1/ √ 1 − x 2 on the interval [-1, 1] and the recurrence relation is One of the applications of Chebyshev polynomials is the solution of ordinary differential equations with boundary conditions.Many studies are considered on the interval [−1, 1] in which Chebyshev polynomials are defined.Therefore, this limitation causes a failure of the Chebyshev approach in the problems that are naturally defined on larger domains, especially including infinity.Under a transformation that maps the interval [−1, 1] into a semi-infinite domain [0, ∞), Boyd [2], [3], Parand et al. [6], [7], and Sezer [9], [10] successfully applied spectral methods to solve problems on semi-infinite intervals.In their studies the basis functions called rational Chebyshev functions, and defined by Recently, Koc, and Kurnaz [4] have proposed modified type of Chebyshev polynomials as an alternative to the solutions of the problems given in all real domain.In their studies, the basis functions called exponential Chebyshev functions E n (x) that are orthogonal in (−∞,∞).This kind of extension tackles the problems over the whole real domain.Therefore, we introduce a new improved type of exponential Chebyshev functions.

Definition and Properties of Exponential Chebyshev (EC) Functions
The exponential Chebyshev EC functions are defined by where the corresponding recurrence relation is

Orthogonality of EC Functions
In the next Proposition we define the form of the weight function needed to guarantee the orthogonality of EC function.
Proposition 1.The weight function w(x) corresponding to EC functions, such that they are orthogonal in the interval (−∞,∞) is given by √ e x / (e x + 1) , with the orthogonal condition where and δ nm is the Kronecker function [4].Also the product relation of EC functions used in the derivative relations is given by

Function Expansion in Term of EC Functions
A function f (x) well defined over the interval (−∞,∞), can be expanded as where 5) is a truncated to N < ∞ in terms of the EC functions as where E(x) is 1 × (N + 1) vector with elements E n (x) and A is an unknown coefficient vector as, The (k )th-order derivative of f (x) can be written as where

The Derivatives of EC Functions
Proposition 2. The relation between the row vector E(x) and its (k)thorder derivative is given as where, D is the (N + 1) × (N + 1) operational matrix for the derivative, and the general form of the matrix D is a tridiagonal matrix which is obtained from Proof.Derivatives of the EC functions can be found by differentiating relation (2), and by the help of (4) we get and ) (1) − (E n−1 (x)) (1) ], that can be written as By using the relations (10)-( 12) and by the help of product relation (4) for {n = 0, 1, . . ., N }, then we get the previous equalities form (N +1)×(N +2) matrix then we make a truncation to the last column to get square operational matrix D given in (10), then to obtain the matrix E (k) (x) we can use the relations ( 14) as then we can write Proposition 2 and its proof are derived a regular scheme for the relation between the vector E(x) and its (k )th-order derivative, a similar proof found in [4] with less details.Now we turn to introduce a new improved scheme of the vector E(x) and its (k )th-order derivative that leads us to get equality sign in (15), as explained in next proposition Proposition 3. The derivatives of the vector can be expressed with equality sign by where D is (N + 1) × (N + 1) operational matrix for the derivative given in (10), and B(x) is 1 × (N + 1) row vector which is an actual term to get the equality sign of (16), that was truncated in (9).This added term will improve the obtained approximate solutions as will be shown in the numerical examples in section 6.And B(x) is deduced as shown below: Consequently, to obtain the matrix E (k) (x), we can use the relation ( 16) as where . For example at N = 4 the two matrices D and B(x ) takes the form

Application of the Introduced two Schemes for Studying Higher-order Ordinary Differential Equations
The form of high-order linear non-homogeneous differential equations with variable coefficients in unbounded domains is with the mixed conditions where the P k (x) and f (x) are continuous functions on the interval (−∞, ∞), d k ij , b j and α i are appropriate constants or b j may tends to±∞.Now, we consider that the approximate solution y N (x) to the exact solution y(x) of Eq. ( 19) defined by expression (6) and its (k )th-order derivative defined by expression (7) as and substituting the relation (15) into expression (22), we have the regular scheme of the (k )th-order derivative of the solution function y N (x) of the higher-order differential equations as and by substituting the relation (18) into expression (22), we get the improved scheme of the (k )th-order derivative of the solution y(x) as

Fundamental Matrix Relations
Let us define the collocation points [4], so that −∞ < x i < ∞, as and at the boundaries (i = 0, i = N ) x 0 → ∞, x N → −∞, since the EC functions are convergent at both boundaries ±∞, namely their values are ±1, the appearance of infinity in the collocation points does not cause a loss in the method.Then, we substitute the collocation points (25) into Eq.( 19) to obtain The system (26) can be written in the matrix form where By putting the collocation points x i in the regular scheme (23), we have the system y . . . .
And the fundamental matrix will be in the form Next the corresponding matrix form for the condition can be written as follows Now by using the improved scheme (24) the fundamental matrix is Next the corresponding matrix form for the condition can be written as follows where

Method of Solution
The fundamental matrices ( 27) and (29) for Eq. ( 19) corresponding to a system of (N +1) algebraic equations for the (N +1) unknown coefficientsa 0 , a 1 , ... a N .We can write Eq. ( 19) as so that, for regular scheme (23) and, for the improved scheme (24) W is we can obtain the matrix form for the mixed conditions by means of ( 28) and (30) as where, for (28) and, for (30) To obtain the solution of Eq. ( 19) under the given condition by replacing the rows matrices (32) by the last m rows of the matrix (31) we have required augmented matrix or the corresponding matrix equation Thus, the coefficient a i , i = 0, 1, ..., N are uniquely determined by Eq. ( 33), (34)

Test Examples
Example 1.Consider the following equation and the boundary conditions y(x) → 0 when |x| → ∞.
For N =4, the collocation points are x 0,4 → ±∞, Here P 2 is the identity matrix, and matrices P 0 , P 1 , E, B are in the form , and the augmented matrix for the conditions with N =4 for x → ∞ is then, the fundamental matrix takes the form for the regular scheme, and for the improved takes the form After the augmented matrices of the two systems and conditions are computed, we obtain the solutions for the regular scheme, , where, then the improved scheme gives for N =4 where the operator L = d 2 dx 2 − 1 , and the boundary conditions y(x) → 0 when |x| → ∞.The exact solution found in [1] by Fourier transform as where F and F −1 are Fourier and inverse Fourier transform, we apply our present method to Eq. (36), Table .1 shows the approximate and exact solutions at different N, x ∈ [−3, 3] if we take f (x) = −2 sec h 3 (x).In Table .2 L 2 and L ∞ norms are presented; we can see that for grater N good accuracy is achieved, where the improved scheme has accuracy better than regular scheme.The norms: x where the operator L = d 3 dx 3 − 1 , and the condition isy(x) → 0 when |x| → ∞.The exact solution taken to be same as the previous example i.e. y(x) = sec h(x), then the function f (x) is sec h(x) −1 + 5 sec h 2 (x) tanh(x) − tanh 3 (x) , by applying our present method to Eq. (37), Table .3 shows comparing the L 2 at different N where h=0.1 and x ∈ [−3 , 3].Now, we observe from example three, that the increasing order of differentiation offset by an increase in the truncation in the regular scheme, while the

Conclusion
In this paper a new improved exponential Chebyshev (EC) collocation method is investigated.The improved method introduced to solve high-order linear ordinary differential equations in unbounded domain.The proposed differential equations and the given conditions were transformed to matrix equation with unknown EC coefficients.This technique is considered to be a modification of the similar presented in [4], [9], [10] and [8].On the other hand, the EC functions approach deals directly with infinite boundaries without singularities.This variant for our method gave us freedom to solve differential equations with boundary conditions tends to infinity.Illustrative examples are used to demonstrate the applicability and the effectiveness of the proposed technique.

Table 1 :
Comparing the approximate and exact solution

Table 2 :
Error norms for Example 2 Example 3. Now we consider the following differential equation

Table 3 :
Error norms for Example 3 improved scheme does not have a truncation, so the values of L 2 worth almost such as example two.