Journal of mathematics and computer Science 14 (2015) 124 - 142
Rational Laguerre Functions and Their Applications
A. Aminataei a , S. Ahmadi-Asl b , Z. KalatehBojdi b
a
Department of Applied Mathematics, Faculty of Mathematics, K. N. Toosi University of Technology,
P.O. Box 16315-1618, Tehran, Iran
b
Department of Mathematics, Birjand University, Birjand, Iran
ataei@kntu.ac.ir
Article history:
Received July 2014
Accepted August 2014
Available online November 2014
Abstract
In this work, we introduce a new class of rational basis functions defined on [a, b) and based on mapping
the Laguerre polynomials on the bounded domain [a, b) . By using these rational functions as basic
functions, we implement spectral methods for numerical solutions of operator equations. Also the
quadrature formulae and operational matrices (derivative, integral and product) with respect to these basis
functions are obtained. We show that using quadrature formulae based on rational Laguerre functions give
us very good results for numerical integration of rational functions and also implementing spectral methods
based on these basis functions for solving stiff systems of ordinary differential equationsgive us suitable
results. The details of the convergence rates of these basis functions for the solutions of operator equations
are carried out, both theoretically and computationally and the error analysis is presented in L2 пЂЁ[a, b) пЂ©
space norm.
Keywords: Rational Laguerre functions, Spectral methods, Quadrature formulae, Stiff system, Hilbert
space.
1. Introduction
Orthogonal polynomials play a prominent role in pure, applied and computational mathematics, as well
as in the applied sciences and also in the many fields of numerical analysis such as quadratures,
approximation theory and so on [1-4]. In particular case, these polynomials have an important role in the
spectral methods. These methods (spectral methods) have been successfully applied in the approximation
of partial, differential and integral equations. Three most widely used spectral versions are the Galerkin,
collocation and Tau methods. Their utility is based on the fact that if the solution sought is smooth, usually
only a few terms in an expansion of global basis functions are needed to represent it to high accuracy [511].
124
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
We must note to this point that numerical methods for ordinary, partial and integral differential equations
can be classified into the local and global categories. The finite-difference and finite-element methods are
based on local arguments, whereas the spectral methods are in a global class [12, 13]. Spectral methods,
in the context of numerical schemes for differential equations, belong to the family of weighted residual
methods, which are traditionally regarded as the foundation of many numerical methods such as finite
element, spectral, finite volume and boundary element methods. There are essentially three important
questions that we have to make to derive a spectral method: i: what expansion functions should be used?
ii: the form in which the approximation will be written? And finally iii: the procedure by which the solution
unknowns are determined? The first question or selecting the global basis functions for expansion of
functions has an important role in implementing spectral methods, and this choice depend on the
behavior of an exact solution of the problem. For instance, if an exact solution has periodic behavior, then
trigonometric functions are the suitable choices for basic functions [14]. Also in nonperiodic cases the
polynomial solutions of eigenvalue problems in ordinary differential equations, known as Sturm-Liouville
problems on the interval [пЂ1,1] are suitable choices. The Jacobi polynomials are the well-known class of
polynomial solutions of nonperiodic Sturm-Liouville problem exhibiting spectral convergence, of which
particular examples are Chebyshev polynomials of the firrst and second kinds, and Legendre polynomials
in which have high applications [8-15]. In the numerical solutions of ordinary, partial and integral
differential equations on the unbounded domains, also some basic functions are named as rational Jacobi
functions are used [16-18]. In this work, we introduce a new set of rational basis functions defined on
[a, b) and based on mapping the Laguerre polynomials on the bounded domain [a, b). using these
rational functions as basic functions, we implement spectral methods for numerical solution of the
operator equations. The importance and superiority of the new basis functions rather than other basis
functions such as polynomials are shown for one experiment with stiff solution. The details of the
convergence rates of these basis functions for solution of the operator equations are carried out, both
theoretically and computationally. This work is organized as follows. In section 2, we introduce the
Laguerre polynomials and rational Laguerre functions (generalized) and their properties such as their
quadrature formulae and etc. In section 3, we present the derivative, integral and product operational
matrices with respect to rational Laguerre functions. Section 4, is devoted to the approximation with
rational Laguerre functions and convergence rates of these basis functions for solutions of the operator
equations. One experiment is presented in section 5, for showing the accuracy of our development. Finally
in section 6, we have monitored a brief conclusion.
2. The Laguerre polynomials (generalized)
In this part, we define the Laguerre polynomials (generalized) and their properties such as their SturmLiouville ordinary differential equation, three terms recursion formula and etc. Let пЃЊ пЂЅ (0, пЂ«п‚Ґ), then
Laguerre polynomials are denoted by LпЃЎn ( x) (пЃЎ пЂѕ пЂ1), and they are the Eigen functions of the Sturm
Liouville problem
пѓ¦
пѓ¶п‚ў
x пЂпЃЎ e x пѓ§ xпЃЎ пЂ«1eпЂ x пЂЁ LпЃЎn ( x) пЂ©п‚ў пѓ· пЂ« пЃ¬n LпЃЎn ( x) пЂЅ 0, x пѓЋ пЃЊ,
пѓЁ
пѓё
(1)
with the eigenvalues пЃ¬n пЂЅ n [12, 13].
Laguerre polynomials are orthogonal in L2wпЃЎ (пЃЊ) space with the weight function wпЃЎ ( x) пЂЅ xпЃЎ eпЂ x , satisfy in
the following relation
пѓІ
пЂ«п‚Ґ
0
LпЃЎn ( x)LпЃЎm ( x) wпЃЎ ( x)dx пЂЅ пЃ§ nпЃЎ пЃ¤ m,n , пЃ§ nпЃЎ пЂЅ
125
пЃ‡(n пЂ« пЃЎ пЂ« 1)
,
пЃ‡(n пЂ« 1)
(2)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
where пЃ¤ m ,n is kronecker delta function. The explicit form of these polynomials is in the form
n
Ln ( x)   Ei xi ,
(3)
i пЂЅ0
where
пѓ¦ n пЂ«пЃЎ пѓ¶
i
пѓ§
пѓ· пЂЁ пЂ1пЂ©
nпЂi пѓё
EiпЃЎ пЂЅ пѓЁ
.
i!
(4)
These polynomials are satisfied in the following three terms recurrence formula
(n пЂ« 1) LпЃЎnпЂ«1 ( x) пЂЅ (2n пЂ« пЃЎ пЂ« 1 пЂ x) LпЃЎn ( x) пЂ (n пЂ« пЃЎ ) LпЃЎnпЂ1 ( x),
LпЃЎ0 ( x) пЂЅ 1, LпЃЎ1 ( x) пЂЅ пЃЎ пЂ« 1 пЂ x.
(5)
The case пЃЎ пЂЅ 0, leads to the classical Laguerre polynomials, which are used most frequently in practice
and will simply be denoted by Ln пЂЁ x пЂ© . An important property of the Laguerre polynomials is the following
derivative relation [12, 13]:
n пЂ1
 L ( x)    L ( x).
пЃЎ
n
п‚ў
i пЂЅ0
пЃЎ
i
(6)
Further, ( LпЃЎi ( x))( k ) are orthogonal with respect to the weight function wпЃЎ пЂ« k . i.e.
пѓІ
пЂ«п‚Ґ
0
( LпЃЎi )( k ) ( x)( LпЃЎj )( k ) ( x) wпЃЎ пЂ«k ( x)dx пЂЅ пЃ§ nпЃЎпЂпЂ«kkпЃ¤ i , j ,
(7)
where пЃ§ nпЃЎпЂпЂ«kk is defined in (2).
2.1 The rational Laguerre functions (generalized)
In this section, we present the rational Laguerre functions (generalized) as orthogonal basis functions. At
first we use a suitable change of variable formula, in which transform the un bounded domain (0, п‚Ґ)
(domain of Laguerre polynomials) to the bounded domain [a, b) . Using this change of variable formula
and substituting it in the Laguerre polynomials, we define a new set of rational orthogonal basis functions
in the bounded domain. For this purpose, let пЃ¦ ( x, h) be a strictly increasing function with respect to a
positive parameter h ; or
y пЂЅ пЃ¦ ( x, h), h пЂѕ 0, x пѓЋ[a, b),
y пѓЋ (0, пЂ«п‚Ґ).
(8)
(9)
However the condition пЃ¦ п‚ў( x, h) пЂѕ 0, ; is equivalent to the strictly increasing property of пЃ¦ ( x, h) .
Several types of transformation пЃ¦ ( x, h) are exist for mapping the unbounded domain to bounded domain
such as:
yпЂЅ
h( x пЂ a )
by пЂ« ha
, xпЂЅ
,
bпЂx
yпЂ«h
126
(10)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
in which transform between x пѓЋ[a, b) and y пѓЋ (0, п‚Ґ).
This transformation (10) is named as an algebraic transformation. Now using the change of variable
h( x пЂ a )
(algebraic transformation), we construct a new set of orthonormal basis functions in a particular
bпЂx
weighted
Sobolev
spaces
(on
the
interval
).
Let
[a, b)
пѓ¦ h( x пЂ a ) пѓ¶
(11)
RiпЃЎ ( x, h) пЂЅ LпЃЎi пѓ§
пѓ· , x пѓЋ[a, b),
пѓЁ bпЂx пѓё
then RiпЃЎ ( x, h) are the eigenfunctions of the Sturm-Liouville problem
пѓ¶п‚ў
( x пЂ« h)2 пЂпЃЎ x пѓ¦ ( x пЂ« h)2 пЃЎ пЂ«1 пЂ x пЃЎ
x e пѓ§
x e пЂЁ Ri ( x, h) пЂ©п‚ў пѓ· пЂ« пЃ¬n RiпЃЎ ( x, h) пЂЅ 0,
h(b пЂ a)
пѓЁ h(b пЂ a)
пѓё
(12)
with the corresponding eigenvalues пЃ¬n пЂЅ n. We call RiпЃЎ ( x, h) as the rational Laguerre functions and the
orthogonality relation for these functions is
пѓІ
1
пЂ1
RmпЃЎ ( x, h)RnпЃЎ ( x, h) wrпЃЎ ( x)dx пЂЅ пЃ§ nпЃЎ пЃ¤ m,n ,
(13)
h( x пЂ a) пЃЎ пЂ h (baпЂпЂ«xx )
wr ( x) пЂЅ (
) e
k ( x),
bпЂx
(14)
where
пЃЎ
k ( x) пЂЅ (
h( x пЂ a ) п‚ў пЃЎ
) , пЃ§ n is defined in (2) and пЃ¤ m,n is a kronecker delta function. Moreover, the recurrence
(b пЂ x)
relations (5) and (6), imply that
h( x пЂ a ) пѓ¶ пЃЎ
пѓ¦
пЃЎ
(n пЂ« 1) RnпЃЎпЂ«1 ( x, h) пЂЅ пѓ§ 2n пЂ« пЃЎ пЂ« 1 пЂ
пѓ· Rn ( x, h) пЂ (n пЂ« пЃЎ ) Rn пЂ1 ( x, h),
bпЂx пѓё
пѓЁ
h( x пЂ a )
R0пЃЎ ( x, h) пЂЅ 1, R1пЃЎ ( x, h) пЂЅ пЃЎ пЂ« 1 пЂ
,
bпЂx
n пЂ1
( x пЂ« h) 2
(
)  Rn ( x, h)    Ri ( x, h).
h(b пЂ a)
i пЂЅ0
Also let us consider the following notation
Rnh пЂЅ span{LпЃЎi (
h( x пЂ a )
), i пЂЅ 0,..., n},
bпЂx
(15)
(16)
in which is needed in this work later.
2.2 The quadrature formulae with respect to rational Laguerre functions (generalized)
In this part, we introduce the Gauss and Gauss-Radau quadrature formulae based on rational
Laguerre functions. For this purpose, let us first consider the following theorem.
Theorem 1.Let x (jпЃЎ ) , w(jпЃЎ ) be the nodes and weights associated with the Gauss or GaussRadauquadratures for Laguerre polynomials. Then
пѓІ
пЂ«п‚Ґ
0
n
f ( x) w ( x)dx   f ( x (j ) )w(j )  R n, ( x),
j пЂЅ0
127
(17)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
where
R n,пЃ¤ ( x) пЂЅ
f (2 nпЂ«пЃ¤ ) (пЃё )пЃ§ nпЃЎ
,
2n пЂ« пЃ¤
(18)
Further пЃ¤ пЂЅ 0 and 1, for Gauss and Gauss-Radauquadratures respectively, and for some пЃё пѓЋ[0, пЂ«п‚Ґ).
For the Gauss quadrature case, {x (jпЃЎ ) }nj пЂЅ1 are the zeros of LпЃЎn ( x) and
w(jпЃЎ ) пЂЅ пЂ
пЃ‡(n пЂ« пЃЎ пЂ« 1)
1
(n пЂ« 1)! L(пЃЎ ) ( x (пЃЎ ) ) пЂЁ L(пЃЎ ) пЂ©п‚ў ( x (пЃЎ ) )
n
j
n пЂ«1
j
x (jпЃЎ )
пЃ‡(n пЂ« пЃЎ пЂ« 1)
пЂЅ
, 0 п‚Ј j п‚Ј n,
(n пЂ« пЃЎ пЂ« 1)(n пЂ« 1)! пЂЁ L(пЃЎ ) ( x (пЃЎ ) ) пЂ©2
n
(19)
j
пЂЁ
пЂ©
п‚ў
and for the Gauss-Radau quadrature, {x (jпЃЎ ) }nj пЂЅ1 are the zeros of L(nпЃЎпЂ«)1 ( x) and
(пЃЎ пЂ« 1)пЃ‡ 2 (пЃЎ пЂ« 1)пЃ‡( N пЂ« 1)
w пЂЅ
,
пЃ‡( N пЂ« пЃЎ пЂ« 1)
пЃ‡( N пЂ« пЃЎ пЂ« 1)
1
пЃ‡( N пЂ« пЃЎ пЂ« 1)
1
пЂЅ
пЂЅ
,1 п‚Ј j п‚Ј n,
2
N !пЃ‡( N пЂ« пЃЎ пЂ« 1) пѓ© (пЃЎ ) п‚ў (пЃЎ ) пѓ№
N !пЃ‡( N пЂ« пЃЎ пЂ« 1) пѓ© L(пЃЎ ) ( x (пЃЎ ) ) пѓ№ 2
пѓ« N j пѓ»
пѓЄпѓ«пЂЁ LN пЂ© ( x j ) пѓєпѓ»
(пЃЎ )
0
w(jпЃЎ )
(20)
Proof: see [13].
Now from (17), we can obtain the following result:
Let x (jпЃЎ ) , w(jпЃЎ ) be the nodes and weights associated with the Gauss or Gauss-Radau quadrature for
Laguerre polynomials. Then
пѓІ
b
a
n
f ( x) wr ( x)dx   f ( y (j ) )w(j )  R nh, ( x),
(21)
j пЂЅ0
where
R hn ,пЃ¤ ( x) пЂЅ
пѓ¦ пѓ¦ h( x пЂ a ) пѓ¶ пѓ¶
пѓ§ f пѓ§ b пЂ x пѓ·пѓ·
пѓёпѓё
пѓЁ пѓЁ
(2 n пЂ«пЃ¤ )
пЃ§ nпЃЎ
x пЂЅпЃ–
(2n пЂ« пЃ¤ )
,
(22)
and
y
(пЃЎ )
j
пЂЅ
h( x (jпЃЎ ) пЂ a)
b пЂ x (jпЃЎ )
.
(23)
Further, пЃ¤ пЂЅ 1, and 0; for Gauss and Gauss-Radau quadrature respectively, and for some пЃ– пѓЋ[0, пЂ«п‚Ґ).
Remark 1.As we see from (17), we can prove that the integration formula (21) is exact for all p пѓЋ R2hnпЂ«пЃ¤
where R2hn пЂ«пЃ¤ is defined in (16). However it is expected that for rational functions, using quadrature (21),
we obtain better results than other polynomial base methods, because of the similar behavior of a
function and its basis functions for approximation. Now for showing this fact and the accuracy of
quadrature formula (21), we present some experiments.
Experiment 1.Consider the following integral
128
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
пѓІ
0.5
0
x2 пЂ 2 x пЂ 3
dx,
пЂЁ x пЂ 1пЂ© пЂЁ x2 пЂ« 2 x пЂ« 2 пЂ©
(24)
then the obtained results from rational Laguerre gauss (RLG) and Legendre gauss (LG) methods are shown
in table 1.
Experiment 2.Consider the following integral
x2 пЂ« 1
пѓІпЂ0.5 x3 пЂ« x пЂ« 1dx,
0
(25)
then the obtained results from RLG and LG methods are shown in table 2.
3. The operational matrices with respect to rational Laguerrefunctions (generalized)
In this section, we obtain the operational matrices with respect to rational Laguerre functions.
To do this, first we introduce the concept of operational matrix.
3.1 The operational matrix
Definition 1.Suppose
пЃ¦ пЂЅ [пЃ¦0 , пЃ¦1 ,...,пЃ¦n ], (26)
where пЃ¦0 , пЃ¦1 ,..., пЃ¦n are the basis functions on the given interval [a, b] . The matrices Enп‚ґn and Fnп‚ґn are
named as the operational matrices of derivatives and integrals respectively if and only if
пѓІ
d
пЃ¦ (t )
dt
EпЃ¦ (t ),
x
FпЃ¦ (t ).
a
пЃ¦ (t )dt
(27)
Further assume g пЂЅ пЃ›g 0 , g1 ,..., g n пЃќ , named as the operational matrix of the product, if and only if
пЃ¦ ( x)пЃ¦ T ( x) GgпЃ¦ ( x).
129
(28)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
In other words, to obtain the operational matrix of a product, it is sufficient to find g i , j ,k , in the following
relation
iпЂ« j
g
пЃ¦i ( x)пЃ¦ j ( x)
k пЂЅ0
пЃ¦ ( x),
(29)
i , j ,k k
which is called the linearization formula [19]. Operational matrices are used in several areasof numerical
analysis and they hold particular importance in various subjects such as integral equations [20],
differential and partial differential equations [21] and etc. Also many textbooks and papers have
employed the operational matrices for spectral methods [9]. Now we present the following theorem.
Theorem 2.If we present f ( x) in the form
п‚Ґ
f ( x)   Dk( ) Rk ( x, h), x [a, b),
(30)
k пЂЅ0
where
Dk(пЃЎ ) пЂЅ (пЃ§ kпЃЎ )пЂ1
f , RkпЃЎ ( x, h)
wпЃЎr ( x )
.
(31)
Then
п‚Ґ
D  E 
i пЂЅk
( )
i
( ,i )
k
пЂЅ пЃk ,
(32)
where
пЃk пЂЅ
g ( k ) (0)
h( x пЂ a )
, g ( x) пЂЅ f (
),
k!
bпЂx
(33)
and Ek(пЃЎ , i ) is presented in (4).
h( x пЂ a )
in (3), then we have
bпЂx
k
n
(пЃЎ )
(пЃЎ , n ) пѓ¦ h( x пЂ a ) пѓ¶
Rn ( x, h)   Ek 
пѓ·.
пѓЁ bпЂx пѓё
k пЂЅ0
Proof: If we use the change of variable x п‚®
(34)
But (34), can be rewritten as
i
п‚Ґ
пѓ¦ k
пѓ¦ h ( x пЂ a ) пѓ¶ пѓ¶ пѓ¦ п‚Ґ ( пЃЎ ) (пЃЎ , i ) пѓ¶ пѓ¦ п‚Ґ ( пЃЎ ) ( пЃЎ , i ) пѓ¶ пѓ¦ h ( x пЂ a ) пѓ¶
f ( x)   Dk( )   Ei( ,k ) 
     Di E0     Di E1  
пѓ·
пѓ§ i пЂЅ0
b
пЂ
x
пѓЁ
пѓё пѓё пѓЁ i пЂЅ0
k пЂЅ0
пѓё пѓЁ i пЂЅ1
пѓёпѓЁ b пЂ x пѓё
пѓЁ
пѓ¦
пѓ¶ пѓ¦ h( x пЂ a ) пѓ¶
пѓ¦
( пЃЎ ) (пЃЎ , i ) пѓ¶ пѓ¦ h ( x пЂ a ) пѓ¶
   Di( ) E2( ,i )  
  ...     Di Ek  
пѓ· .
k пЂЅ0 пѓЁ i пЂЅk
пѓЁ i пЂЅ2
пѓёпѓЁ b пЂ x пѓё
пѓёпѓЁ b пЂ x пѓё
п‚Ґ
2
п‚Ґ
Now if we define
g ( x) пЂЅ f (
k
п‚Ґ
bx пЂ« ha
h( x пЂ a )
) пѓћ f ( x) пЂЅ g (
),
hпЂ« x
bпЂx
(35)
(36)
then from the Maclaurin series of g ( x), we have:
п‚Ґ
g ( k ) (0) k
x ,
k!
k пЂЅ0
g ( x)  
or
130
(37)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
(k )
пѓ¦ h( x пЂ a) пѓ¶ п‚Ґ g (0) пѓ¦ h( x пЂ a) пѓ¶
f ( x) пЂЅ g пѓ§
пЂЅ
 
пѓ§
пѓ· .
пѓЁ b пЂ x пѓё k пЂЅ0 k ! пѓЁ b пЂ x пѓё
k
(38)
Comparing (35) and (38), we get
п‚Ґ
 Di( ) Ek( , ,i ) 
i пЂЅk
g ( k ) (0)
,
k!
(39)
so the proof is completed.
Theorem 3.Suppose RпЃЎ ( x, h) пЂЅ [ R0пЃЎ ( x, h), R1пЃЎ ( x, h),..., RnпЃЎ ( x, h)], where the elements RiпЃЎ ( x, h) are the
rational Laguerre functions, then we have
пѓ© R0пЃЎ ( x, h) пѓ№
пѓЄ пЃЎ
пѓє
R1 ( x, h) пѓє
пѓЄ
пЃЎ
D( n пЂ«1)п‚ґ( n пЂ«1) пѓЄ
пѓє,
пѓЄ
пѓє
пѓЄпѓ« RnпЃЎ ( x, h) пѓєпѓ»
d
пѓ© R0пЃЎ ( x, h), R1пЃЎ ( x, h), п‚ј, RnпЃЎ ( x, h) пѓ№пѓ»
dx пѓ«
(40)
where DiпЃЎ (the ith-column of DnпЃЎп‚ґn ) is obtained from this linear system (with upper triangular matrix)
пѓ© E0(пЃЎ ,0)
пѓЄ
пѓЄ 0
пѓЄ
пѓЄ
пѓЄ 0
пѓЄ 0
пѓЄ
пѓЄпѓ« 0
п‚ј
E1(пЃЎ ,1)
E0(пЃЎ ,i пЂ1)
п‚ј
E0(пЃЎ ,i )
E1(пЃЎ ,i )
п‚ј
0
0
Ei(пЂпЃЎ1,i пЂ1)
0
0
Ei(пЂпЃЎ1,i )
Ei(пЃЎ ,i )
п‚ј
E0(пЃЎ ,i пЂ«1) пѓ№ пѓ© di ,0 пѓ№ пѓ© пЃ0 пѓ№
пѓєпѓЄ
пѓє пѓЄ
пѓє
E1(пЃЎ ,i пЂ«1) пѓє пѓЄ di ,1 пѓє пѓЄ пЃ1 пѓє
пѓєпѓЄ
пѓє пѓЄ
пѓє
пѓєпЂЅпѓЄ
пѓє.
(пЃЎ , i пЂ«1) пѓє пѓЄ
Ei пЂ1 пѓє пѓЄ
пѓє пѓЄ
пѓє
Ei(пЃЎ ,i пЂ«1) пѓє пѓЄ di ,i пѓє пѓЄ пЃi пѓє
пѓєпѓЄ
пѓє пѓЄ
пѓє
Ei(пЂ«пЃЎ1,i пЂ«1) пѓєпѓ» пѓЄпѓ« di ,i пЂ«1 пѓєпѓ» пѓЄпѓ« пЃiпЂ«1 пѓєпѓ»
(41)
Proof:We know that there exist d i , j such that:
(h пЂ« x)2 пЂЁ LпЃЎi ( x) пЂ©п‚ў
i пЂ«1
  di , j Lj ( x),
hb пЂ ha
j пЂЅ0
h( x пЂ a )
but if in (42), we apply the change of variable x п‚®
, we have
bпЂx
hb пЂ ha пѓ¦ пЃЎ пѓ¦ h( x пЂ a) пѓ¶ пѓ¶п‚ў i пЂ«1
пѓ¦ пЃЎ h( x пЂ a) пѓ¶п‚ў
L
)пѓ· ,
    di , j  Li (
2 пѓ§ i пѓ§
(b пЂ x) пѓЁ пѓЁ b пЂ x пѓё пѓё j пЂЅ0
bпЂx пѓё
пѓЁ
(42)
(43)
or
i пЂ«1
( Ri ( x, h))   di , j Rj ( x, h).
(44)
j пЂЅ0
Combining (44) and (3), yields:
пЂЁR
пЃЎ
i
i пЂ«1
( x, h)    di , j E
п‚ў
j
j пЂЅ0 k пЂЅ0
(пЃЎ , j )
k
пѓ¦ h( x пЂ a ) пѓ¶
пѓ§
пѓ·.
пѓЁ bпЂx пѓё
Now noting to theorem 2, in this case пЂЁ f пЂЁ x пЂ© пЂЅ R ' пЂЁ x, h пЂ© пЂ© , we get:
131
k
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A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
i пЂ«1
d
j пЂЅk
g ( k ) (0)
, k пЂЅ 0,1,..., i пЂ« 1,
k!
(пЃЎ , j )
пЂЅ
i , j Ek
(46)
thus from (46), we can obtain the following linear system:
пѓ© E0(пЃЎ ,0)
пѓЄ
пѓЄ 0
пѓЄ
пѓЄ
пѓЄ 0
пѓЄ 0
пѓЄ
пѓЄпѓ« 0
п‚ј
E1(пЃЎ ,1)
E0(пЃЎ ,i пЂ1)
п‚ј
E0(пЃЎ ,i )
E1(пЃЎ ,i )
п‚ј
0
0
Ei(пЂпЃЎ1,i пЂ1)
0
0
Ei(пЂпЃЎ1,i )
Ei(пЃЎ ,i )
п‚ј
Solving the linear system (47), concludes:
пЃi пЂ«1
,
E
d
di ,i пЂ«1 пЂЅ
di , j пЂЅ
пЃi пЂ
Ei(пЂ«пЃЎ1,i пЂ«1)
E0(пЃЎ ,i пЂ«1) пѓ№ пѓ© di ,0 пѓ№ пѓ© пЃ0 пѓ№
пѓєпѓЄ
пѓє пѓЄ
пѓє
E1(пЃЎ ,i пЂ«1) пѓє пѓЄ di ,1 пѓє пѓЄ пЃ1 пѓє
пѓєпѓЄ
пѓє пѓЄ
пѓє
пѓєпЂЅпѓЄ
пѓє.
(пЃЎ ,i пЂ«1) пѓє пѓЄ
Ei пЂ1 пѓє пѓЄ
пѓє пѓЄ
пѓє
Ei(пЃЎ ,i пЂ«1) пѓє пѓЄ di ,i пѓє пѓЄ пЃi пѓє
пѓєпѓЄ
пѓє пѓЄ
пѓє
Ei(пЂ«пЃЎ1,i пЂ«1) пѓєпѓ» пѓЄпѓ« di ,i пЂ«1 пѓєпѓ» пѓЄпѓ« пЃiпЂ«1 пѓєпѓ»
j пЂ«1
( ,k )
j
i ,k
k пЂЅi пЂ«1
(пЃЎ , j )
j
E
(47)
(48)
; j пЂЅ 0,1,..., i.
Also with similar strategy, we can obtain the integral operational matrix of rational Laguarrefunctions.
This means if we consider
i пЂ«3
(h пЂ« t )2 пЃЎ
L
(
t
)
dt
пЂЅ
ci , j RпЃЎj (u, h); , пЂ1 п‚Ј u п‚Ј 1.

пѓІ0 hb пЂ ha i
j пЂЅ0
h(t пЂ a)
, on right hand side of (49), gives:
Employing the change of variable t п‚®
bпЂt
h (t пЂa )
i пЂ«3
пЃЎ h(t пЂ a )
b пЂt
L
(
)
dt
пЂЅ
ci , j LпЃЎj (u ).

i
пѓІa
bпЂt
j пЂЅ0
bt пЂ ha
Now if we define x пЂЅ
then (50) transforms to
tпЂ«h
i пЂ«3
x
x пЂ1
пЃЎ t пЂ1
L
(
)
dt
пЂЅ
ci , j LпЃЎj (
),

i
пѓІa t пЂ« 1
x пЂ«1
j пЂЅ0
u
(49)
(50)
(51)
or
пѓІ
i пЂ«3
Ri (t , h) dt   ci , j R j ( x, h).
x
a
(52)
j пЂЅ0
Combining (52) and (3), yields
пѓІ
x
0
i пЂ«3
j
Ri (t , h) dt   ci , j E
пЃЎ
(пЃЎ , j )
k
j пЂЅ0 k пЂЅ0
пѓ¦ h( x пЂ a ) пѓ¶
пѓ§
пѓ·.
пѓЁ bпЂx пѓё
k
(53)
Now noting to theorem 2, we get:
i пЂ«3
c
j пЂЅk
i, j
пѓ¦ h( x пЂ a ) пѓ¶
пѓ§
пѓ· пЂЅ пЃk ; k пЂЅ 0,1,..., i пЂ« 3.
пѓЁ bпЂx пѓё
k
(пЃЎ , пЃў , j )
k
E
From (54), we obtain the following linear system:
132
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A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
пѓ© E0(пЃЎ ,0)
пѓЄ
пѓЄ 0
пѓЄ
пѓЄ
пѓЄ 0
пѓЄ 0
пѓЄ
пѓЄпѓ« 0
п‚ј
E1(пЃЎ ,1)
E0(пЃЎ ,i пЂ«1)
п‚ј
E0(пЃЎ ,i пЂ« 2)
E1(пЃЎ ,i пЂ« 2)
п‚ј
0
0
Ei(пЂпЃЎ1,i пЂ«1)
0
0
Ei(пЂпЃЎ1,i пЂ« 2)
Ei(пЂ«пЃЎ2,i пЂ« 2)
п‚ј
Solving the linear system (55), concludes:
ci ,i пЂ«3 пЂЅ
ci , j пЂЅ
пЃi пЂ
пЃ3пЂ«1
E0(пЃЎ ,i пЂ«3) пѓ№ пѓ© ci ,0 пѓ№ пѓ© пЃ0 пѓ№
пѓєпѓЄ
пѓє пѓЄ
пѓє
E1(пЃЎ ,i пЂ«3) пѓє пѓЄ ci ,1 пѓє пѓЄ пЃ1 пѓє
пѓєпѓЄ
пѓє пѓЄ
пѓє
пѓє
пѓЄ
пѓєпЂЅпѓЄ
пѓє.
(пЃЎ ,i пЂ« 3)
Ei пЂ1 пѓє пѓЄ
пѓє пѓЄ
пѓє
(пЃЎ ,i пЂ« 3) пѓє пѓЄ
пѓє
пѓЄ
ci ,i пЂ« 2
пЃi пЂ« 2 пѓє
Ei пЂ« 2
пѓє
пѓЄ
пѓє
пѓЄ
пѓє
Ei(пЂ«пЃЎ3,i пЂ«3) пѓєпѓ» пѓЄпѓ« ci ,i пЂ«3 пѓ»пѓє пѓЄпѓ« пЃi пЂ«3 пѓєпѓ»
,
Ei(пЂ«пЃЎ3,i пЂ«3)
j пЂ«1
E
(55)
( ,k )
j
i ,k
(56)
c
k пЂЅi пЂ«1
(пЃЎ , j )
j
E
; j пЂЅ 0,1,..., i пЂ« 2.
Therefore
пѓІ пѓ©пѓ« R
пЃЎ
1
( x, h), R2пЃЎ ( x, h), п‚ј, RnпЃЎ ( x, h) пѓ№пѓ»dx
пѓ© R1пЃЎ ( x, h) пѓ№
пѓЄ пЃЎ
пѓє
R1 ( x, h) пѓє
пѓЄ
пЃЎ
I ( n пЂ«1)п‚ґ( n пЂ«1) пѓЄ
пѓє,
пѓЄ
пѓє
пѓЄпѓ« R1пЃЎ ( x, h) пѓєпѓ»
(57)
in which (the ith-column of I nпЃЎп‚ґn is obtained from the linear system (55)). Finally for obtaining the
operational matrix of product, we use these two important formulas
mпЂ« n
Ln ( x) Lm ( x)   ci (m, n,  ,  ) Lm  ( x),
(58)
i пЂЅ0
and
n
пѓ¦ пЃЎ пЂ пЃў пЂ« n пЂ i пЂ« 1пѓ¶ пЃў
Ln ( x)   
пѓ·Li ( x),
nпЂi
i пЂЅ0 пѓЁ
пѓё
where
ci (m, n, пЃЎ , пЃў ) пЂЅ пЂЁ пЂ1пЂ©
m пЂ« n пЂ«пЃЎ
i
(59)
пѓ¦ i пѓ¶пѓ¦ m пЂ« пЃЎ пѓ¶пѓ¦ n пЂ« пЃЎ пѓ¶
 пѓ§ k пѓ·пѓ§ n пЂ k пЂ« i пѓ·пѓ§ m пЂ k пѓ· .
k пЂЅ0
пѓЁ пѓёпѓЁ
пѓёпѓЁ
пѓё
(60)
By combining the relations (58) and (59), we obtain
mпЂ« n
Ln ( x) Lm ( x)   M i (m, n,  ) Lk ( x),
(61)
i пЂЅ0
where
and
i
пѓ¦ пЃЎ пЂ« i пЂ k пЂ« 1пѓ¶
M i (m, n,  )   di (m, n,  ) 
пѓ·,
iпЂk пѓё
k пЂЅ0
пѓЁ
(62)
i
пѓ¦ пЃЎ пЂ« i пЂ k пЂ« 1пѓ¶
di (m, n,  )  ci (m, n,  ,  ) 
пѓ·.
iпЂk пѓё
k пЂЅ0 пѓЁ
(63)
133
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
Using the change of variable x п‚®
h( x пЂ a )
we obtain
bпЂx
mпЂ« n
i
пѓ¦ пЃЎ пЂ« i пЂ k пЂ« 1пѓ¶ пЃЎ
Rn ( x, h) Rm ( x, h)   di (m, n,  ) 
пѓ· Rk ( x, h).
iпЂk пѓё
i пЂЅ0
k пЂЅ0 пѓЁ
(64)
Therefore
пѓ© R0пЃЎ ( x, h) пѓ№
пѓЄ пЃЎ
пѓє
R1 ( x, h) пѓє
пѓЄ
пЃЎ
пЃЎ
пЃЎ
пѓ©пѓ« R0 ( x, h), R1 ( x, h), п‚ј, Rn ( x, h) пѓ№пѓ»
пѓЄ
пѓє
пѓЄ
пѓє
пѓЄпѓ« RnпЃЎ ( x, h) пѓєпѓ»
пЃЎ
where Pi ,1 пЂЅ
пѓ© R0пЃЎ ( x, h) пѓ№
пѓЄ пЃЎ
пѓє
R1 ( x, h) пѓє
пѓЄ
пЃЎ
P( n пЂ«1)п‚ґ(1) пѓЄ
пѓє,
пѓЄ
пѓє
пѓЄпѓ« RnпЃЎ ( x, h) пѓєпѓ»
(65)
n пЂi
 M ( j, j, ).
j пЂЅ0
4. Approximation with rational Laguerre functions (generalized)
Now we present some approximation properties of rational Laguerre functions. Let us considerthe finite
dimensional approximation space
Rnh пЂЅ span{RiпЃЎ ( x, h), i пЂЅ 0,..., n},
(66)
in which RiпЃЎ ( x, h) are orthogonal with respect to wrпЃЎ (defined on [a, b) ) and consider the orthogonal
projection Pn( R,h,пЃЎ ) (u ) : L2wпЃЎ [ a ,b ) п‚® пЃЈ n such that
r
( Pn( R,h,пЃЎ ) (u) пЂ u, vn )w пЂЅ 0, пЂўvn пѓЋ Rnh .
(67)
From (67) and the orthogonality relation of rational Laguerre functions, we can write
N
Pn( R,h, ) (u )   un Rn ( x, h),
(68)
n пЂЅ0
where
un пЂЅ
b
1
пЃ§n
пѓІ u ( x) R
пЃЎ
n
( x, h) w( x)dx.
(69)
a
Now consider the general operator equation
with respect to boundary condition
L u пЂ« Nu пЂЅ g ( x), x пѓЋ[a, b),
(70)
Bu пЂЅ пЃ¬,
(71)
where L , Nband B are the linear, nonlinear and boundary operators respectively related to a suitable
Hilbert space, [a, b) is the domain (one dimensional) of approximation, пЃ¬ is an arbitrary constant and
g ( x) пѓЋ L2wпЃЎ ([a, b)) is an arbitrary function. Let Rnh be an approximation space with weight function wrпЃЎ ( x)
r
134
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
and a set of collocation points {x j }nj пЂЅ0 , shifted roots of Laguerre polynomial with degree n . he spectralcollocation method for the operator equation (70) defined in the following form: Find пЃ¦ ( x) пѓЋ Rnh or
n
 ( x)   ai Ri ( x, h) such that
i пЂЅ0
L пЃ¦ ( x j ) пЂ« NпЃ¦ ( x j ) пЂЅ g ( x j ), j пЂЅ 0,..., n,
(72)
B (пЃ¦ ( x)) пЂЅ пЃ¬.
If we suppose that the boundary conditions give us the m equations, then from equation (72), we obtain
m пЂ« n пЂ« 1 algebraic equations and so for obtaining the n пЂ« 1 unknowns, we must eliminate the m equation
and add boundary conditions, which must be solved for unknown {ai }inпЂЅ0 coefficients. Also the spectralGalerkin method for the operator equation (70) is defined in the following form:
Find пЃ¦ ( x) пѓЋ Rnh or пЃ¦ ( x) пЂЅ
n
 a R ( x, h) such that  ( x) are satisfied in the boundary conditions (71)
i пЂЅ0
i
i
automatically and
(L пЃ¦ ( x), p( x))wпЃЎ пЂ« ( NпЃ¦ ( x), p( x))wпЃЎ пЂЅ ( g ( x), p( x))wпЃЎ , пЂўp( x) пѓЋ Rnh .
r
r
(73)
r
Also from equation (73), we obtain n пЂ« 1 algebraic equations which must be solved for unknown {ai }inпЂЅ0
coefficients. In this work, by using MATLAB software (version 2013), we solve these algebraic systems.
4.1 The error analysis
Now in this section, we present the details of the convergence rates of rational Laguerre functions
(generalized) as basic functions. First we consider the following theorem.
Theorem 4.For any f пѓЋ BmпЃЎ (0, пЂ«п‚Ґ) and m п‚і 0,
d l (пЃЎ )
пЂЁ PN пЂЁ f пЂ© пЂ f пЂ©
dx
п‚Ј cN
wпЃЎ пЂ«l
пѓ¦ l пЂm пѓ¶
пѓ§
пѓ·
пѓЁ 2 пѓё
dm
пЂЁu пЂ©
dx m
, 0 п‚Ј l п‚Ј m,
(74)
wпЃЎ пЂ«m
where
пѓ¬
пѓј
dk
BmпЃЎ (0, п‚Ґ) пЂЅ пѓ f : k ( f ) пѓЋ L2wпЃЎ пЂ«k пЂЁ 0, п‚Ґ пЂ© , 0 п‚Ј k п‚Ј m пѓЅ ,
пѓ® dx
пѓѕ
(75)
Pn(пЃЎ ) is the L2wпЃЎ ( x ) -orthogonal projection operator associated with the Laguerre polynomials(generalized)
and c is a positive constant.
Proof: See [13].
135
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
Theorem 5. For any u пѓЋ GпЃЎm ([a, b)) with m п‚і 0
Pn( R,h,пЃЎ ) (u ) пЂ u
п‚Ј cN пЂ m/2
wпЃЎ ( x )
r
Dm
(u )
Dx m
wпЃЎr пЂ«m
,
(76)
and for m п‚і 1,
d
( PN( R ,h,пЃЎ ) (u ) пЂ u )
dx
пЃЎ пЂ«1
wr
п‚Ј cN (1пЂm)/2
( x)
Dm
(u )
Dx m
wпЃЎr пЂ«m
,
(77)
where
GпЃЎm ([a, b)) пЂЅ {u :
dk
(u ) пѓЋ L2wпЃЎ пЂ«k ([a, b)),0 п‚Ј k п‚Ј m},
k
r
dx
Dk
(u ) is defined as
c is a positive constant and the operator
Dx k
п‚ў
п‚ў
п‚ў
Dk
пѓ¦ h( x пЂ a) пѓ¶ d пѓ¦пѓ§ пѓ¦ h( x пЂ a) пѓ¶ d пѓ¦ пѓ¦ пѓ¦ h( x пЂ a) пѓ¶ d пѓ¶ пѓ¶ пѓ¶пѓ·
(u ) пЂЅ пѓ§
пѓ§ .... пѓ§ пѓ§
пѓ· .... пѓ· .
пѓ·
пѓ§
пѓ·
пѓ·
Dx k
пѓЁ b пЂ x пѓё dx пѓ§пѓЁ пѓЁ b пЂ x пѓё dx пѓ§пѓЁ пѓ§пѓЁ пѓЁ b пЂ x пѓё dx пѓ·пѓё пѓ·пѓё пѓ·пѓё
(78)
(79)
k times
Proof: Let U h ( x) пЂЅ
п‚Ґ
U
i пЂЅ0
п‚Ґ
пЃЎ
пЃЎ
h ,i Li ( x) and uh ( x)   uh ,i Ri ( x, h), then from definition, it is easyto show the
i пЂЅ0
relation between the coefficients of the rational Laguerre functions (generalized) and Laguerre
polynomials expansions (generalized) as
uh , i пЂЅ
u, RiпЃЎ ( x, h)
wпЃЎr ( x )
пЃ§i
пЃЎ
пЂЅ
u, LпЃЎi ( x)
пЃ§i
пЃЎ
wпЃЎ ( x )
пЂЅ U h ,i .
(80)
Also we need for expressing the error estimates, to introduce an operator
D
пѓ¦ h( x пЂ a) пѓ¶п‚ў du
(u ) пЂЅ пѓ§
.
пѓ·
Dx
пѓЁ b пЂ x пѓё dx
Repeating formula (79) leads to
пѓ¦
пѓ¶
d kU h ( x) пѓ¦ h( x пЂ a) пѓ¶п‚ў d пѓ§ пѓ¦ h( x пЂ a) пѓ¶п‚ў d пѓ¦пѓ§ пѓ¦пѓ§ пѓ¦ h( x пЂ a) пѓ¶п‚ў d пѓ¶пѓ· пѓ¶пѓ· пѓ· D k
пЂЅпѓ§
.... пѓ§
.... пЂЅ
(u ).
пѓ·
пѓ§
пѓ·
пѓ·
k
dx k
пѓЁ b пЂ x пѓё dx пѓ§ пѓЁ b пЂ x пѓё dx пѓ§ пѓ§ пѓЁ b пЂ x пѓё dx пѓ· пѓ· пѓ· Dx
пѓё пѓёпѓё
пѓЁ пѓЁ
пѓЁ
k times
(82)
Let Pn(пЃЎ ) be the L2wпЃЎ ( x ) - orthogonal projection operator associated with the Laguerre polynomials
(generalized), then by Parseval identity [7], theorem (3) and formula (82), we have
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(81)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
PN( R ,h,пЃЎ ) (u ) пЂ u
(пЃЎ )
N
пЂЅ P
2
wпЃЎr ( x )
(U s ) пЂ U s
п‚Ґ
пЂЁ uh,i пЂ© пЃ§ iпЃЎ пЂЅ

пЂЅ
i пЂЅ N пЂ«1
2
wпЃЎ ( x )
п‚Ґ
2
п‚Ј cN
пЂm
Us
пЃ»
 U   
2
h ,i
i пЂЅ N пЂ«1
( m) 2
п‚Ј cN
wпЃЎ ( x )
i
пЂm
пЃ»
пЃЅ
пЃЅ
(83)
2
Dk
(u )
Dx k
.
wпЃЎr ( x )
Next, we deduce from (7) (orthogonality of ( LпЃЎi ( x))п‚ў that ( RiпЃЎ ( x, h))п‚ў is L2wпЃЎ orthogonal, and
пЃЎ
( Ri ( x, h))п‚ў
2
wпЃЎr
2
пЃЎ
пЂЅ ( Li ( x))п‚ў
wпЃЎ
r
пЂЅ пЃ¬i пЃ§ i ,
пЃЎ
пЃЎ
(84)
where пЃ¬iпЃЎ is the eigenvalue of the Laguerre Sturm-Liouville problem (1). Therefore by (80) and
theorem (3), we have
d
PN( R ,h,пЃЎ ) (u ) пЂ u пЂ©
пЂЁ
dx
2
пЂЅ
wпЃЎr
п‚Ґ

i пЂЅ N пЂ«1
пЃЎ
i
п‚Ґ
пЃ§ i пЂЁ u h ,i пЂ© пЂЅ

2
пЃЎ
i пЂЅ N пЂ«1
пЃЎ
i
пЃ§ i пЂЁU h ,i пЂ©
пЃЎ
cN (1пЂ m ) u
2
( m) 2
wпЃЎ пЂ«1
d
пЂЅ
PN(пЃЎ ) (u ) пЂ u пЂ©
пЂЁ
dx
п‚Ј cN (1пЂ m )
m
D
u
Dx m
2
п‚Ј
wпЃЎ
2
(85)
.
wпЃЎ пЂ«m
Thus the proof is completed.
Now we present a theorem which express the error analysis of projection operator of exact solution of
(70) based on Galerkin basis functions.
Theorem 6. Suppose L and N are the linear and nonlinear operators with respect to (70) wherein L
пЂ1
has inversion, L and N have these properties
N ( x) пЂ N ( y) п‚Ј N x пЂ y ,
LпЂ1 ( x) п‚Ј LпЂ1 x ,
(86)
(87)
and f , g пѓЋ GпЃЎm ([a, b)), (m п‚і 1) are satisfied in equation (70), then
f пЂ PN( R,h,пЃЎ ) ( f )
wr (пЃЎ )
п‚Ј c(1пЂ LпЂ1
N )T ,
(88)
where
T пЂЅ N пЂ m/2 LпЂ1
Dm
N ( PN( R,h,пЃЎ ) ( f )) пЂ©
m пЂЁ
Dx
wпЃЎr пЂ«m
.
(89)
Proof: From equation (70) and inevitability of L we have
f пЂ« LпЂ1 N f пЂЅ LпЂ1 ( g ) пЂ« k ( x), (90)
where k ( x) is a function dependent on an operator L.. Now by substituting PN( R,h,пЃЎ ) ( f ) in (90)
we have
пЂЁ
PN( R,h,пЃЎ ) ( f ) пЂ« LпЂ1 N PN( R,h,пЃЎ ) ( f )
пЂ©
LпЂ1 ( g ) пЂ« k ( x).
(91)
Also the nonlinear term N PN( R,h,пЃЎ ) ( f ) is approximated by
( R , h ,пЃЎ )
N
NP
(f)
PN( R,h,пЃЎ ) ( N PN( R,h,пЃЎ ) ( f )),
(92)
so from (92) and (91), we obtain
PN( R,h,пЃЎ ) ( f ) пЂ« LпЂ1PN( R,h,пЃЎ ) пЂЁ N PN( R,h,пЃЎ ) ( f ) пЂ© LпЂ1 ( g ) пЂ« k ( x).
Now by subtracting (90) and (93), we have
пЂЁ f пЂP
( R , h ,пЃЎ )
N
( f ) пЂ© (LпЂ1PN( R,h,пЃЎ ) пЂЁ NPN( R,h,пЃЎ ) ( f ) пЂ© пЂ LпЂ1 N ( f )).
137
(93)
(94)
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
From (94), we have
пЂЁ f пЂP
( R , h ,пЃЎ )
N
( f ) пЂ© LпЂ1 PN( R ,h,пЃЎ ) пЂЁ N PN( R, h,пЃЎ ) ( f ) пЂ© пЂ
(95)
LпЂ1 N PN( R ,h,пЃЎ ) ( f ) пЂ« LпЂ1 N PN( R,h,пЃЎ ) ( f ) пЂ LпЂ1 N ( f ),
so using properties (86) and (87), we obtain
f пЂ PN( R ,h,пЃЎ ) ( f )
wпЃЎr
пЂЁ
LпЂ1 PN( R , h,пЃЎ ) пЂЁ N PN( R , h,пЃЎ ) ( f ) пЂ© пЂ N PN( R, h,пЃЎ ) ( f )
пЂ« LпЂ1 пЂЁ N PN( R ,h,пЃЎ ) ( f ) пЂ N ( f ) пЂ© п‚Ј LпЂ1
LпЂ1 N
пЂЁP
( R , h ,пЃЎ )
N
пЂЁ NP
( R , h ,пЃЎ )
N
пЂ©
пЂ©
( f ) пЂ© пЂ N PN( R , h,пЃЎ ) ( f ) пЂ«
(96)
PN( R ,h,пЃЎ ) ( f ) пЂ ( f ) .
Finally using theorem (4), we obtain
f пЂ PN( R,h,пЃЎ ) ( f )
wr (пЃЎ )
п‚Ј c(1пЂ LпЂ1
N )T ,
(97)
where
T пЂЅ N пЂ m/2 LпЂ1
Dm
N ( PN( R,h,пЃЎ ) ( f ))
Dx m
wпЃЎr пЂ«m
.
(98)
Thus the proof is completed.
Remark 2.In the similar discussion, we can obtain a new rational basis functions in the interval (a, b)
using Hermit polynomials [5] and transform function
yпЂЅ
hx
,
( x пЂ a)(b пЂ x)
(99)
in which is named as an algebraic transformation, and transform the interval y пѓЋ (пЂп‚Ґ, пЂ«п‚Ґ) to
x пѓЋ[a, b).
5. The test experiment
Functions with stiffness structures arise in many applied sciences such as in engineering, physics and
kx
applied mathematics. For instance, when exact solution of problems contains terms of the form e ,
where k is a negative real part of a complex number problem, gives meaningless results [22-28]. In general,
the methods designed for non-stiff problems such as spectral methods (Galerkin, Tau and collocation and
based on classical polynomials) [7-11], homotopy analysis method [29], vibrational iteration method [3032] when applied to stiff systems of differential equations tend to be very slow and can give bad
(divergent) results in solution. Now in this section, a test experiment is presented where its solution has
stiffness structure. In this case, using spectral collocation method based on classical Jacobi polynomials
with free parameters (пЃЎ , пЃў ) we obtain very bad results in global interval [0, 1], because of the stiffness
structures of the exact solutions although if we use adaptive spectral collocation method based on
classical Jacobi polynomials in sub-intervals with small distances, we can obtain better results. But using
spectral method based on new rational basis functions, we show the accuracy and superiority of our new
basis functions rather than other basis functions such as polynomials in global interval.
Experiment 3.Consider the following system of differential equations [33], our new basis functions rather
than other basis functions such as polynomials in global interval.
138
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
пѓ© xп‚ў(t ) пѓ№ пѓ¦ пЂ21 19 пЂ20 пѓ¶ пѓ© x(t ) пѓ№
пѓЄ yп‚ў(t ) пѓє пЂЅ пѓ§ 19 пЂ21 20 пѓ· пѓЄ y(t ) пѓє ,
пѓ·пѓЄ
пѓЄ
пѓє пѓ§
пѓє
пѓЄпѓ« zп‚ў(t ) пѓєпѓ» пѓЁпѓ§ 40 пЂ40 40 пѓёпѓ· пѓ«пѓЄ z (t ) пѓ»пѓє
(100)
1 2t пЂ40t
пѓ©e пЂ« e пЂЁ cos(40t ) пЂ« sin(40t ) пЂ© пѓ№пѓ» ,
2пѓ«
1
y (t ) пЂЅ пѓ©пѓ«e пЂ2t пЂ e пЂ40t пЂЁ cos(40t ) пЂ sin(40t ) пЂ© пѓ№пѓ» ,
2
пЂ1
z (t ) пЂЅ пѓ©пѓ«eпЂ2t пЂ« eпЂ40t пЂЁ cos(40t ) пЂ sin(40t ) пЂ© пѓ№пѓ» .
2
(101)
пѓ©1.0 пѓ№
with initial condition x(0) пЂЅ пѓЄ0.0 пѓє .
пѓЄ пѓє
пѓЄпѓ« пЂ1 пѓєпѓ»
The exact solution is
x(t ) пЂЅ
Now we approximate the exact solution of (100) based on rational Laguarre functions (generalized), or
  R (t , h)   R
( x, h ) пЂ© пЃҐ ,
m
x(t )
i пЂЅ0
i
T
,m
i
  R (t , h)   R
n
y (t )
i
i пЂЅ0
,n
i
  R (t , h)   R
p
z (t )
i пЂЅ0
i
,p
i
( x, h ) пЂ© пЃў ,
T
(102)
( x, h ) пЂ© пЃ§ ,
T
where пЃҐ i , пЃўi , пЃ§ i , are the unknown coefficients,
пЃҐ пЂЅ пЃ›пЃҐ 0 ,..., пЃҐ m пЃќ , пЃў пЂЅ пЃ› пЃў0 ,..., пЃўn пЃќ , пЃ§ пЂЅ пѓ©пѓ«пЃ§ 0 ,..., пЃ§ p пѓ№пѓ» ,
T
T
T
(103)
and
RпЃЎ ,i ( x, h) пЂЅ [ R0пЃЎ ( x, h), R1пЃЎ ( x, h),..., RiпЃЎ ( x, h)]T .
(104)
In this work, we assume m пЂЅ n пЂЅ p, and we can consider the matrix form of equation (100) as follows
пЂЁR
пЂЁR
пЂЁR
пЃЎ ,n
пЃЎ ,n
пЃЎ ,n
пЂЁпЂЁ D пЂ« 21I пЂ© пЃҐ пЂ19пЃў пЂ« 20пЃ§ пЂ© 0,
( x, h) пЂ© пЂЁ пЂ19пЃҐ пЂ пЂЁ D пЂ« 21I пЂ© пЃў пЂ 20пЃ§ пЂ© 0,
( x, h) пЂ© пЂЁ пЂ40пЃҐ пЂ« 40пЃў пЂ« пЂЁ D пЂ 40 I пЂ© пЃ§ пЂ© 0,
( x, h ) пЂ©
T
(пЃЎ )
T
nп‚ґn
(пЃЎ )
T
nп‚ґn
(пЃЎ )
(105)
nп‚ґn
or
пѓ¦ пЂЁ D (пЃЎ ) пЂ« 21I nп‚ґn пЂ©
пЂ19 I nп‚ґn
пѓ§
H (пЃЎ ,n ) ( x, h) пѓ§
пЂ19 I nп‚ґn
пЂ пЂЁ D (пЃЎ ) пЂ« 21I nп‚ґn пЂ©
пѓ§
пѓ§
пЂ40 I nп‚ґn
40 I nп‚ґn
пѓЁ
139
пѓ¶
пѓ· пѓ©пЃҐ пѓ№
пѓ· пѓЄпЃў пѓє
пЂ20 I nп‚ґn
пѓ·пѓЄ пѓє
(пЃЎ )
пЂЁ D пЂ 40I nп‚ґn пЂ© пѓ·пѓё пѓЄпѓ«пЃ§ пѓєпѓ»
20 I nп‚ґn
пѓ©0пѓ№
пѓЄ0 пѓє , (106)
пѓЄ пѓє
пѓЄпѓ«0пѓєпѓ»
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
where
T
T
T
H (пЃЎ ,n ) ( x, h) пЂЅ пѓ©пѓЄпЂЁ RпЃЎ ,n ( x, h) пЂ© , пЂЁ RпЃЎ ,n ( x, h) пЂ© , пЂЁ RпЃЎ ,n ( x, h) пЂ© пѓ№пѓє ,
пѓ«
пѓ»
(107)
and D(пЃЎ ) , I nп‚ґn are operational matrices of derivative and identity of order n, respectively. Also the matrix
form of initial conditions is in the following form
пѓ©пЃҐ пѓ№ пѓ©1 пѓ№
(108)
H
(0, h) пѓЄпѓЄ пЃў пѓєпѓє пЂЅ пѓЄпѓЄ0 пѓєпѓє .
пѓЄпѓ«пЃ§ пѓєпѓ» пѓЄпѓ« пЂ1пѓєпѓ»
Now we choose n пЂ« 1 collocation points xi , i пЂЅ 0,..., n, ; and by vanishing (106) in these collocation points,
(пЃЎ , n )
we obtain 3n algebraic equations for 3n unknown coefficients. But for imposing the initial conditions, we
eliminate the 3 latter equations of linear system (106) and add the 3 initial conditions (108). The obtained
algebraic system is linear and solving it, is easy from implementation point of view.
140
A. Aminataei, S. Ahmadi-Asl, Z. KalatehBojdi /J. Math. Computer Sci. 14 (2015) 124 - 142
6. Conclusion
In this work, we have presented a new rational basis functions defined on [a, b) and based on mapping
Laguerre polynomials to bounded domain [a, b) . Using these rational functions as basic functions, we
have implemented spectral methods for numerical solution of operator equations and we have shown
that in some cases (operator equations with fractional solutions), these basis functions are obtained
better results than other basis functions (experiment 3). Also the quadrature formulae and operational
matrices (derivative, integral and product) with respect to these basis functions are obtained. The
importance and superiority of the new basis functions rather than other basis functions such as
polynomials is shown for quadrature formulae and for numerical solution of operator equations in some
experiments (experiments 1 and 2).
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