
Journal of Applied Sciences and Environmental Management
World Bank assisted National Agricultural Research Project (NARP)  University of Port Harcourt
ISSN: 11198362
Vol. 11, Num. 4, 2007, pp. 147149

Journal of Applied Sciences and Environmental Management, Vol. 11, No.
4, 2007, pp. 147149
Exact Solutions for Chebyshev Equations by using the Asymptotic
Iteration
Method
SOUS A. J. *, M. ALHAWARI^{1}
Department of Mathematics, AlQuds Open University, Nablus Email: melhem_fan@hotmail.com
^{1}Department of Mathematics, Aljouf University, Askaka, Saudi Arabia
* Corresponding author: Sous A. J
Code Number: ja07112.
ABSTRACT:
The asymptotic iteration method is used in order to solve the Chebyshev
differential equations, and to reproduce the Chebyshev polynomials of the first
and second kinds respectively. It is shown that the asymptotic iteration method
is valid for any degree .
The Chebyshev polynomials are important in many areas of mathematics, and
physics. Particularly in the approximation theory since the roots of the Chebyshev
polynomials of the first kind are used in the polynomial interpolation [1].
In the study of differential equations, Chebyshev polynomials arise as the
solution to the Chebyshev differential equations
(1x^{2}) yn''(x) x yn'(x) +n^{2} y_{n}(x) =0, (1)
and
[[(1x^{2}) yn''(x)3x yn'(x)+n(n+2) yn(x)=0,]] (2)
where n = 0 , 1, 2 , 3 , ... for the polynomials T_{n} (x), U_{n} (x)
of the first and second kinds respectively [2,8]. These
equations are special case of the SturmLiouville equation [3].
Chebyshev polynomials are used virtually in the field of numerical analysis,
and it holds particular importance in different subjects including orthogonal
polynomials, and polynomial approximation. Ellgendi [4] has extensively shown
how Chebyshev polynomials can be used to solve linear integral equations, integrodifferential
equations, and ordinary differential equations. Various methods for solving
linear and nonlinear ordinary differential equations [5, 6, 7] were devised
at about
the same time and were based on the discrete orthogonality relationships of
the Chebyshev polynomials.
In the literature [2, 8], the Chebyshev differential equations has been solved
very heavily using the power series solution method. The reader may face several
problems in following the power series solution technique, in which guessing
the solution in many cases is very difficult task. Therefore, we applied a
new method, the asymptotic iteration method (AIM) [9] to solve this kind of
differential
equations, where we don’t need to use the recurrence relation to find the
general solution. This method is very easy to implement in the case of Chebyshev
differential equations. The results of this method are very accurate. Moreover,
the reader can obtain the solutions without a strong background in mathematics.
The paper is organized as follows: in section 2 we will describe the AIM to solve
the Chebyshev differential equations. In section 3 our analytical results for
the Chebyshev polynomials, and then we conclude and remark therein.
2. Formalism of the asymptotic iteration
method for the Chebyshev differential
equations
The starting point to apply the AIM is to rewrite equations (1) and (2) in
the following form:
yn '' (x)= k_{0} (x) y_{n} (x) + z_{0} (x)
y_{n} (x) (3)
Where k_{0} (x) and z_{0} (x) are defined for equations (1),
and (2) as:
Note that for equation equation
(2)a = b = 1.
In order to find a general solution to equation (3) we rely on the symmetric
structure of the right hand side of equation (3). Thus if we differentiate
equation (3) with respect to , we obtain [912]
y"n ( x)= k_{1} (x) y_{n} (x) + z_{1} (x)
y_{n} (x) (5)
where
Likewise, the calculations of the second derivative of equation (3) yield
where
Thus for (j+1) and (j+2)^{th}
derivatives, j = 1, 2, 3, ..we have
and
respectively, where
The ratio of the ( j + 1 ) and ( j + 2 ) th derivatives
can be expressed as
For sufficiently large j , we can introduce the”asymptotic” aspect of the method, that is
Thus equation (12) can be reduced to
which yields
Where C_{n1} is the integration constant, and the
righthand of equation (15) follows from equation
(11), and the definition of φ (x) . Substituting
equation (15) into equation (9) we obtain a
firstorder differential equation
This, in turn, yields the general solution to the
equation (3)
Maple software producing a constant of the form
The results of our calculations with different values
of n are given, so that the reader may, if so
inclined reproduce our results.
Case (1): The first few Chebyshev polynomials of the first kind are
Case (2): The first few Chebyshev polynomials of the second kind are
In all cases, we have only considered the sixth order of polynomials of the
first and second kinds. This was so to make a clear comparison between the
results of this method and the results of [2, 8]. The obtained polynomials
are all in excellent agreement with the exact ones.
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