Research Article A New Extended Padé Approximation and...

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2013 Article ID 263467 8 pageshttpdxdoiorg1011552013263467

Research ArticleA New Extended Padeacute Approximation and Its Application

Z Kalateh Bojdi1 S Ahmadi-Asl1 and A Aminataei2

1 Department of Mathematics Birjand University Birjand Iran2 Faculty of Mathematics K N Toosi University of Technology PO Box 16315-1618 Tehran Iran

Correspondence should be addressed to A Aminataei ataeikntuacir

Received 19 June 2013 Revised 16 September 2013 Accepted 7 October 2013

Academic Editor Weimin Han

Copyright copy 2013 Z Kalateh Bojdi et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We extend ordinary Pade approximation which is based on a set of standard polynomials as 1 119909 119909119899 to a new extendedPade approximation (Muntz Pade approximation) based on the general basic function set 1 119909120582 1199092120582 119909119899120582 (0 lt 120582 le 1) (theparticular case of Muntz polynomials) using general Taylor series (based on fractional calculus) with error convergency Theimportance of this extension is that the ordinary Pade approximation is a particular case of our extended Pade approximationAlso the parameterization (120582 is the corresponding parameter) of new extended Pade approximation is an important subject whichobtaining the optimal value of this parameter can be a good question for a new research

1 Introduction

Rational approximations of an arbitrary function are animportant topic in numerical analysis due to their highapplications in physical sciences chemistry engineering andother applied sciences [1 2]The Pade approximation is a par-ticular and classical type of rational fraction approximationThe idea of this approximation is to expand a function as aratio of two power series and determining both the numer-ator and denominator coefficients using the coefficients ofTaylor series expansion of a function 119891(119909) [1] The Padeapproximation is the best approximation of a function bya rational function of a given order [1] The technique wasdeveloped around 1890 by Henri Pade but it goes back toGeorge Freobenius who introduced the idea and investigatedthe features of rational approximations of power series ThePade approximation is usually superior when functions con-tain poles because the use of rational function allows them tobe well represented [1] The Pade approximation often givesbetter approximation of the function than truncating its Tay-lor series and it may still work where the Taylor series doesnot converge [1] For these reasons Pade approximation isused extensively in computer calculations The Pade approx-imation has also been used as an auxiliary function in Dio-phantine approximation and transcendental number theorythough for sharp results ad hoc methods in some sense

inspired by the Pade theory typically replace them Since itprovides an approximation to the function throughout thewhole complex plane the study of Pade approximation issimultaneously a topic in mathematical approximation the-ory and analytic function theory The generalized Padeapproximation is given in [2] For the connection of Padeapproximation with continued fractions and orthogonalpolynomials see [2] Also multivariate Pade approximationwas done by [3] Two versions of this approximation whichconcerned with interpolation problems are Hermite-Padeand Newton-Pade approximations and they are in fact ratio-nal interpolation problems [4ndash6] In this paper using thegeneralized Taylor series (based on fractional calculus) weextend ordinary Pade approximation to the general Muntzpade approximation The paper is organized as follows InSections 11 to 13 we introduce an ordinary Pade approx-imation (and their properties) present the Muntz polyno-mials (rational Muntz functions and suitable theorems) andalso generalize Taylor series (based on fractional calculus)respectively Also Muntz Pade approximation is presented inSection 2 The uniqueness and convergence analysis resultsare given in Section 3 Some numerical tests are presented inSection 4 Two applications of this extended Pade approxima-tion (Muntz Pade approximation) on the numerical solutionof sound and vibration problem and microwave heatingmodel in a slab using differential equations are given in

2 Advances in Numerical Analysis

Section 5 Finally we have monitored a brief conclusion inSection 6

11 Ordinary Pade Approximation Suppose that we are givena power series suminfin

119894=0119888119894119911119894 represent a function 119891(119911) so that

119891 (119911) =

infin

sum119894=0

119888119894119911119894

(1)

This expansion is the fundamental starting point of any anal-ysis using Pade approximation A Pade approximation is arational fraction as

[119871

119872] =

1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872 (2)

which has a Maclaurin expansion which agrees with (1) asfar as possible Notice that in (2) there are 119871 + 1 numeratorcoefficients and 119872 + 1 denominator coefficients There is amore or less irrelevance common factor between them andfor definiteness we take 119887

0= 1 So there are 119871+1 independent

numerator coefficients and 119872 independent denominatorcoefficientsmaking119872+119871+1unknown coefficients in allThisnumber suggests that normally the [119871119872] out to fit the powerseries (1) through the orders 1 119911 119911119871+119872 For the notationof formal power series we have

infin

sum119894=0

119888119894119911119894

=1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872

+ 119874 (119911119871+119872+1

) (3)

and returning to (3) and cross-multiplying we find that

(1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872

) (1198880+ 1198881119911 + sdot sdot sdot )

= 1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

+ 119874 (119911119871+119872+1

)

(4)

Equating the coefficients of 119911119871+1 119911119871+2 119911119871+119872 from (4) wefind

119887119872119888119871minus119872+1

+ 119887119872minus1119888119871minus119872+2

+ sdot sdot sdot + 1198870119888119871+1= 0

119887119872119888119871minus119872+2

+ 119887119872minus1119888119871minus119872+3

+ sdot sdot sdot + 1198870119888119871+1= 0

119887119872119888119871+ 119887119872minus1119888119871minus119872+1

+ sdot sdot sdot + 1198870119888119871+119872

= 0

(5)

If 119895 lt 0 we define 119888119895= 0 for consistency Since 119887

0= 1

(5) becomes a set of119872 linear equations for the119872 unknowndenominator coefficients and we obtain

[[[[

[

119888119871minus119872+1

119888119871minus119872

119888119871

119888119871minus119872+2

119888119871minus119872+1

119888119871+1

d

119888119871

119888119871+1

119888119871+119872minus1

]]]]

]

[[[[

[

119887119872

119887119872minus1

1198871

]]]]

]

= minus

[[[[

[

119888119871+1

119888119871+2

119888119871+119872

]]]]

]

(6)

from which the 119887119894may be found The numerator coefficients

1198860 119886

119871follow immediately from (4) by equating the coef-

ficients of 1 119911 119911119871 as

1198860= 1198880

1198861= 1198881+ 1198880

1198862= 1198882+ 11988711198881+ 11988721198880

119886119897= 119888119897+

min119871119872sum119894=0

119887119894119888119871minus119894

(7)

Thus (4) and (7) normally determine the Pade numeratorand denominator which are called Pade equations and wehave constructed an [119871119872]Pade approximationwhich agreeswith suminfin

119894=0119888119894119911119894 through order 119911119871+119872 If the given power series

converge to the same function for |119911| lt 119877with (0 lt 119877 lt +infin)then a sequence of Pade approximation may converge for119911 isin 119863 where119863 is a domain longer than |119911| lt 119877 [1]

Remark 1 In general Pade approximation does not exist (foran arbitrary function with respect to particular 119871 and119872) [1]

Theorem 2 If the Pade approximation [119871119872] exists then itwill be unique [2]

12 Muntz Polynomials An interesting generalization ofWeierstrass theorem (1885) (density of polynomials set in119862[119886 119887] equipped with a supremum norm) goes back to theGerman mathematician Herman Muntz He performed thisgeneralization by a suitable tool which is named Muntzpolynomials For presenting his theorem first we introducea definition and a remark

Definition 3 Suppose 1205820lt 1205821lt sdot sdot sdot isin R then the set

1199091205820 1199091205821 is named as a set of Muntz or Muntz polyno-

mials [6]

Remark 4 Let (120582119894)infin

119894=0be any sequences of distinct real num-

bers and 119886 gt 0 then

sum119899

119894=0119886119894119909120582119894

sum119899

119894=0119887119894119909120582119894 119886119894 119887119894isin R 119899 isin N (8)

is dense in 119862[119886 119887] [6]

Theorem 5 (generalized Muntz theorem) If 0 le 1205820lt 1205821lt

sdot sdot sdot rarr +infin then the Muntz polynomials with respect to thesesequences are dense in 1198712[0 1] if and only if suminfin

119894=0120582119894

minus1

= infin

[6]

Remark 6 In this paper by considering 120582119896= 119896120582 119896 =

1 2 (0 lt 120582 le 1) which is a particular case of Muntzpolynomials we introduce a new and extended Muntz Padeapproximation Also it is clear that by considering 120582

119896= 119896 we

can obtain the classical Weierstrass theorem [6]

Advances in Numerical Analysis 3

13 Generalized Taylor Series In calculus Taylorrsquos theoremgives an approximation of a 119896-times differentiable function119891(119909) around a given point 119909

0by a 119896th order Taylor polyno-

mial as

119891 (119909) ≃

119899

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(9)

with the error

119890 (119909) =119891(119899+1)(120589)

(119899 + 1)(119909 minus 119909

0)119899+1

120589 isin (1199090 119909) (10)

and for analytical polynomials we have

119891 (119909) =

infin

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(11)

Taylorrsquos theorem also generalizes to multivariate and vectorvalued functions But the generalized version of this theoremfrom fractional calculus view or the theory of derivatives ofarbitrary order was done by [7] because of its high appli-cations in solving fractional ordinary differential equationsintegral ordinary differential equations and obtaining anoperational matrix and so forth For this purpose we needsome definitions in fractional calculus

Definition 7 TheCaputo fractional derivative of119891(119909)of order120582 gt 0 with 119886 ge 0 is defined as

(119863120582

119886119891) (119909) =

1

Γ (120573 minus 120582)int119909

119886

119891(120573)

(119905)

(119909 minus 119905)120582minus120573+1

119889119905 (12)

for 120573 minus 1 lt 120582 le 120573 120573 isin N 119909 ge 119886From Caputo fractional derivative we have

119863120582

119886119862 = 0

119863120582

119886119909120575

=

0 if 120575 isin 1 120573 minus 1 Γ (120575 + 1)

Γ (120575 + 1 minus 120582)(119909)120575minus120582

if 120575 isin N 120575 ge 120573

or 120575 gt 120573 minus 1(13)

where 120573 = lceil120582rceil and 120582 ge 0 However the Caputo fractionalderivative is a linear operator which means

119863120582

119886(120574119891 + 120579119892) = 120574119863

120582

119886(119891) + 120579119863

120582

119886(119892) (14)

For more literature review of fractional calculus see [8]

Theorem 8 Suppose that 119863119896120582119886119891(119909) isin 119862(119886 119887] for 119896 =

0 1 119899 + 1 where 0 lt 120582 le 1 then one has

119891 (119909) =

119899

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119909 minus 119886)

119896120582

+119863(119899+1)120582

119886119891 (120589)

Γ ((119899 + 1) 120582 + 1)(119909 minus 119886)

(119899+1)120582

(15)

where 119886 le 120589 le 119887 for all 119909 isin (119886 119887] and119863(119899)120582119886

= 119863120582

119886sdot 119863120582

119886sdot sdot sdot 119863120582

119886

(119899-times) [7]

If we consider 119886 = 0 then we obtain a fractional Maclau-rin series and in a similar manner as the arbitrary function119891(119911) that has an infinite Caputo differentiable at point 119886named as analytical functions on119863 in a fractional sense if

119891 (119911) =

infin

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119911 minus 119886)

119896120582

forall119911 isin 119863 (16)

2 Muumlntz Padeacute Approximations

Now by using the general Taylor series (based on fractionalcalculus) we generalize the classical and ordinary Padeapproximations to a new and extended Muntz Pade approx-imation For this goal we present the Muntz Pade approx-imation as a ratio of two Muntz polynomials constructedfrom the coefficients of generalized Taylor series expansionof a function Also in a similar approach from ordinaryPade approximation we prove the uniqueness of Muntz Padeapproximation Now in a similar manner from ordinary Padeapproximation suppose that 119891(119909) is an analytical function(from fractional calculus) in the neighborhood of 119886 = 0 andthen we can write 119891(119909)

119891 (119909) =

infin

sum119896=0

119888119896119911120582119896

119888119896=

119863119896120582

119886

Γ (119896120582 + 1) (17)

and also Muntz Pade approximation is defined as a ratio oftwo Muntz polynomials as

[119871

119872] =

1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

(18)

and then we suppose that [119871119872] (Muntz Pade approxima-tion) has a Maclaurin series (17) or

infin

sum119894=0

119888119894119911120582119894

=1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

+ 119874 (119911119871120582+119872120582+1

) (19)

By cross multiplying we find that

(1198870+ 1198871119911120582

+ sdot sdot sdot + 119887119872119911119872120582

) (1198880+ 1198881119911120582

+ sdot sdot sdot )

= 1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

+ 119874 (119911119871120582+119872120582+1

)

(20)

Equating the coefficients 119911120582119871+120582 119911120582119871+2120582 119911120582119871+120582119872 from (20)we obtain a system of equations similar to (5) and wecan obtain 119887

119894coefficients Also immediately by equating the

coefficients of 1 119911120582 119911119871120582 we can obtain 119886119894coefficients in

a recursion formula as (7) Thus by assumption of existenceof [119871119872] Muntz Pade approximation for establishing theuniqueness of this approximation we must prove that thematrix of system of (5) is nonsingular but we perform theproof of the uniqueness of Muntz Pade approximation by adifferent approach in the next section

Remark 9 Let 120582 = 1 then we can obtain the ordinary Padeapproximation


3 Uniqueness and Convergence Analysis ofMuumlntz Padeacute Approximation

In this section we present the uniqueness and convergenceanalysis of Muntz Pade approximation in some theorems

Theorem 10 (uniqueness) WhenMuntz Pade approximation([119871119872]) exists then it is unique for any formal power series

Proof Assume that there are two such Muntz Pade approx-imations 119880(119911)119881(119911) and 119883(119911)119884(119911) where the degrees of 119883and119880 are less than or equal to 120582119871 and that of119884 and119881 are lessthan or equal to 120582119872 Then by (19) we must have

119883 (119911)

119884 (119911)minus119880 (119911)

119881 (119911)= 119874 (119911

119871120582+119872120582+1

) (21)

since both approximate the same series If wemultiply (21) by119884(119911)119881(119911) we obtain

119883 (119911)119881 (119911) minus 119884 (119911)119880 (119911) = 119874 (119911119871120582+119872120582+1

) (22)

but the left-hand side of (22) is a polynomial of degree atmost120582119871+120582119872 and thus is identically zero Since neither 119884 nor119881 isidentically zero we conclude

119883 (119911)

119884 (119911)=119880 (119911)

119881 (119911) (23)

Since by definition both119883 and 119884 and119880 and 119881 are relativelyprime and 119884(0) = 119881(0) = 1 we have shown that the twosupposedly different Muntz Pade approximants are the same

Theorem 11 (convergency) Let119891(119911) be analytic in |119911| le 119877 (infractional sense) then an infinite subsequence of [1198711]MuntzPade approximants converges to 119891(119911) uniformly in |119911| le 119877

Proof By hypothesis 119891(119911) is analytic in |119911| le 119877 (in fractionalsense) and consequently within a large interval |119911| lt 120588 with120588 gt 1205881015840

gt 119877 Let

119891 (119911) =

infin

sum119894=0

119888119894119911120582119894

with 119888119894= 119874((120588

1015840

)minus120582119894

) (24)

The [1198711]Muntz Pade approximation is given by

[119871

1] = 1198880+ 1198881119911120582

+ sdot sdot sdot + 119888119871minus1119911120582(119871minus1)

+119888119871119911120582119871

1 minus 119888119871+1119911120582119888119871

(25)

If a subsequence of coefficients 119888119871119895 119895 = 1 2 are zero then

[(119871119895minus 1)1] are truncated fractional Maclaurin expansions

which converge to 119891(119911) uniformly in |119911| le 119877 lt 1205881015840 So weassume that no infinite subsequence of 119888

119871 vanishes and con-

sider 119903119871= 119888119871+1119888119871which iswell defined for all sufficiently large

119871 because

119888119871119911120582119871

= 119874(119877

1205881015840)

120582119871

(26)

28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x

Figure 1 The [35] Muntz Pade approximation of 119891(119909) = 119890119909 andexact function of 120582 = 04

the sequence of [1198711] Muntz Pade approximation given by(25) converges uniformly in |119911| lt 119877 unless there exists asequence of values of 119871 for which 1minus119888

119871+1119911119888119871= 0within |119911| lt

1205881015840 Thus either a subsequence of the second row converges

uniformly or else for some 1198710and all 119871 gt 119871

0 |119888119871119888119871+1| gt 1205881015840 In

the latter case

10038161003816100381610038161003816100381610038161003816

1198881198710

119888119871

10038161003816100381610038161003816100381610038161003816=

119871minus1

prod119894=1198710

10038161003816100381610038161003816100381610038161003816

119888119894

119888119894+1

10038161003816100381610038161003816100381610038161003816gt (1205881015840

)120582(119871minus1198710)

(27)

contradicting with (24) so the proof is completed

4 The Test Experiments

Now in this section in the two subsections we show theapplications of the Muntz Pade approximation in the func-tional approximation (see Section 41) and fractional calculusfieldsThe advantage of using theMuntz Pade approximationis shown for numerical approximation of fractional differen-tial equations in Section 42

41 Muntz Pade Approximation and Functional Approxima-tion In this section we obtain the Muntz Pade approxima-tions of119891(119909) = 119890119909 for different values of 120582 Figure 1 shows the[35]Muntz Pade approximations of 119891(119909) = 119890119909 and 120582 = 04

Also the [35]Muntz Pade approximation of 119891(119909) = 119890119909and 120582 = 065 is shown in Figure 2

42 Muntz Pade Approximation and Fractional CalculusThis section is devoted to presentation of some numericalsimulations obtained by applying the collocationmethod andbased on a new extended Pade approximation (Muntz Padeapproximation) The algorithm for numerical approximationof solutions to the initial value problems for the fractional dif-ferential equationswas implemented byMATLAB In the caseof nonlinear equations the MATLAB function 119891119904119900119897V119890 was


28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x

Figure 2 Comparison between the [35] Muntz Pade approxima-tion of 119891(119909) = 119890119909 and exact function of 120582 = 065

used for solving the nonlinear system In the case that theexact solution 119910 to a problem is known the dependence ofapproximation errors on the discretization parameter 119899 isestimated in 2-norm as

119890119899= radic

119899

sum119896=0

(119910119899(119909119896) minus 119910 (119909

119896))2

(28)

where 119910119899is the approximated solution corresponding to the

discretization parameter 119899

Experiment 1 We start with a simple nonlinear problem [8]

119863120572

(119910 (119909)) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(29)

where we have a nonlinear and nonsmooth right-hand sideThe solution 119910 has a smooth derivative of order 0 lt 120572 lt 1 [8]The analytical solution subject to the initial condition 119910(0) =0 is given by

119910 (119905) = 1199098

minus 31199094+1205722

+9

4119909120572

(30)

Now we approximate the exact solution of (29) or 119910120582(119909)with the [1198991]new and extended Pade approximation (MuntzPade approximation) as

119910120582

(119909) ≃ 119910120582

119899(119909) =

119899

sum119894=0

119888119894119909119894120582

(31)

and substituting it into (29) we obtain

Res(120572120582)119899

(119909) = 119863120572

(119910120582

119899(119909)) minus 119875 (120572 119910 (119909) 119909) ≃ 0 (32)

times10minus3

09

08

07

06

05

04

03

02

01

015 2 25 3 35 4 45 5

1

1

Figure 3 [121] ordinary Pade approximation coefficients of 120572 =05 120573 = 025 and 120574 = 075 of Experiment 1

where

119875 (120572 119910 (119909) 119909) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(33)

For the collocation points we use the first roots of the Jacobipolynomial 119875(120573120574)

119899(119909) [9 10] and then after enforcing the ini-

tial condition 119910(0) = 0 we obtain a system of nonlinear alge-braic equations and we use the MATLAB function 119891119904119900119897V119890 forsolving the nonlinear system Thus substituting the colloca-tion points into (32) yields

Res(120572120582)119899

(119909(120573120574)

119894) = 119863

120572

(119910120582

119899(119909(120573120574)

119894))

minus 119875 (120572 119910 (119909(120573120574)

119894) 119909(120573120574)

119894) ≃ 0

119894 = 0 119899

(34)

Now from (34) and its initial condition we have 119899+2 algebraicequations of 119899 + 1 unknown coefficients Thus for obtainingthe unknown coefficients we must eliminate one arbitraryequation from these 119899+2 equations But because of the neces-sity of holding the boundary conditions we eliminate the lastequation from (34) Finally replacing the last equation of (34)by the equation of initial condition we obtain a system of 119899+1equations of 119899+1 unknowns 119888

119894 By implementing the method

as presented for 119899 = 5 and also for different parametersof 120573 and 120574 we obtain the approximate solutions The [121]ordinary and Muntz Pade coefficients approximation of thisproblem are shown in Figures 3 and 4 respectively We haveobserved that this method (the new extended Pade approx-imation (Muntz Pade approximation)) is very efficient fornumerical approximation of the fractional ordinary differen-tial equations Also closer look at the results of the MuntzPade approximation scheme reveals that in this method ofsolution the coefficients decrease faster than the classical case


times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10

Figure 4 [121] new Pade approximation coefficients of 120582 = 05120572 = 05 120573 = 025 and 120574 = 075 of Experiment 1

This is the advantage on the application of the Muntz Padeapproximation to the fractional ordinary differential equa-tions

Experiment 2 Consider the nonlinear fractional integro-differential equation [11]

11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)

119892 (119909) =2 arcsin ℎ (radic119909)radic120587radic1 + 119909

minus 211990932

(36)

The exact solution of (35) is 119906(119909) = ln(1 + 119909) We implementthe collocationmethod based on the [101]newPade approx-imation of 120582 = 016 120573 = 1 and 120574 = 1 The obtained results ofour method are presented in Table 1

5 Application of MuumlntzPadeacute Approximation in Vibration andElectromagnetic Radiation Problems

In this section we present two applications of Muntz Padeapproximation for numerical approximation of some appli-cable ordinary differential equations

Experiment 3 Consider the following model problem [12ndash15]

119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)

Table 1 The comparison between the collocation method based on[101] Muntz Pade approximation and the exact solution of 120582 =016 120573 = 1 120574 = 1 of Experiment 2

119909 Exact Approximate Error00 00000000000 00000000000 000000000001 00953101798 00953103008 000000012102 018232155679 01823218407 000000028403 026236426446 02623649824 000000071804 033647223662 03364726886 000000045205 040546510810 040546528510 000000017706 047000362924 04700043522 000000072307 053062825106 05306290280 000000077708 058778666490 058778755290 000000088809 064185388617 064185410017 000000021410 069314718055 069314789955 0000000719

Table 2 Comparison between the [23] and [44] Muntz Padeapproximations of 120572 = 05 120576 = 04 and 120582 = 05 of Experiment 3

119905 Error of [23] Error of [44]00 00012 0000901 00010 minus0000402 00013 0000603 00022 minus0000204 00014 0000705 00012 00003

where 0 lt 120576 le 120572 le 1 [12] This model has high application inthe theory of sound and vibration [12] The exact solution tothis initial value problem has the form

119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)

Now for 120572 = 05 120576 = 04 and 120582 = 05 and using [23] and[44] Muntz Pade approximations we obtain the followingresults shown in Table 2

Also the obtained results for different kinds of MuntzPade approximations are shown in Figure 5

Experiment 4 Consider the following model problem [16]

1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)

This equation has high application in the theory of electro-magnetic radiation and describes the steady state reaction-diffusion equations with source term that arise in modelingmicrowave heating in an infinite slab with isothermal walls[16] Also 120582 and 119896 represent the thermal absorptivity and


t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB

Figure 5 A the obtained results of [44] Muntz Pade approxima-tion of experiment 3 B the obtained results of [23] Muntz Padeapproximation of Experiment 3

Table 3 Comparison between the [23] and [44] Muntz Padeapproximations of 119896 = 1 and 120582 = 05 of Experiment 4

119910 Error of [23] Error of [44]00 00002 0000101 minus00033 minus0001102 00014 0000903 00003 0000104 00026 minus0001205 00023 00021

electric field decay rate parameters respectively The exactsolution to this boundary value problem has the form

119879 (119910) = (1198691(2 (radic120582119896)) 119884

0(minus2 (radic120582119896radic119890119896119910)))

times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)

where 1198690 1198691are Bessel functions of the first kind and119884

01198841are

Bessel functions of the second kind Now for 119896 = 1 and 120582 =05 and using [23] and [44] Muntz Pade approximationswe obtain the following results shown in Table 3

6 Conclusion

In this paper using the general Taylor series (based on frac-tional calculus) we extend the ordinary Pade approximation

to the general Muntz Pade approximation The importanceof this extension is that the ordinary Pade approximation isa particular case of our Muntz Pade approximation (120582 = 1)We have applied the method in the application of functionalapproximation fractional exponent and vibration and elec-tromagnetic radiationmodel problems and have obtained theresults with a good order of accuracy Also the uniquenessresults and error analysis have been presented completelyIn addition the test experiments have been presented forshowing the applicability and validity of the newMuntz Padeapproximation

Acknowledgments

The authors are grateful to the reviewers and the editor Pro-fessor Dr Weimin Han for their helpful comments and sug-gestions which indeed improved the quality of this paper

References

[1] G A Baker and P Graves-Morris Pade ApproximantsAddison-Wesley 1981

[2] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

[3] A Cuyt ldquoMultivariate Pade approximants revisitedrdquo BIT vol26 no 1 pp 71ndash79 1986

[4] LWuytack ldquoOn the osculatory rational interpolation problemrdquoMathematics and Computers in Simulation vol 29 pp 837ndash8431975

[5] A A M Cuyt and B M Verdonk ldquoGeneral order Newton-Pade approximants for multivariate functionsrdquo NumerischeMathematik vol 43 no 2 pp 293ndash307 1984

[6] P Borwein andT ErdelyiPolynomials and Polynomials Inequal-ities vol 161 ofGraduate Texts in Mathematics Springer BerlinGermany 1995

[7] Z M Odibat and N T Shawagfeh ldquoGeneralized Taylorrsquos for-mulardquoAppliedMathematics andComputation vol 186 no 1 pp286ndash293 2007

[8] K Diethelm The Analysis of Fractional Differential EquationsSpringer Berlin Germany 2010

[9] D FunaroPolynomial Approximations of Differential EquationsSpringer 1992

[10] J S Hesthaven S Gottlieb andD Gottlieb SpectralMethods forTime-Dependent Problems Cambridge University 2009

[11] P Mokhtary and F Ghoreishi ldquoThe 1198712-convergence of the Leg-endre spectral tau matrix formulation for nonlinear fractionalintegro differential equationsrdquo Numerical Algorithms vol 58no 4 pp 475ndash496 2011

[12] I V Andrianov and J Awrejcewicz ldquoAnalysis of jump phenom-ena using Pade approximationsrdquo Journal of Sound and Vibra-tion vol 260 no 3 pp 577ndash588 2003

[13] I Andrianov and J Awrejcewicz ldquoSolutions in the Fourier seriesform Gibbs phenomena and Pade approximantsrdquo Journal ofSound and Vibration vol 245 no 4 pp 753ndash756 2001

[14] I V Andrianov and J Awrejcewicz ldquoIterative determination ofhomoclinic orbit parameters and Pade approximantsrdquo Journalof Sound and Vibration vol 240 no 2 pp 394ndash397 2001


[15] G Kudra and J Awrejcewicz ldquoTangents hyperbolic approxima-tions of the spatial model of friction coupled with rolling resis-tancerdquo International Journal of Bifurcation andChaos vol 21 pp2905ndash2917 2011

[16] O D Makinde ldquoSolving microwave heating model in a slabusing Hermite-Pade approximation techniquerdquo Applied Ther-mal Engineering vol 27 pp 599ndash603 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Section 5 Finally we have monitored a brief conclusion inSection 6

11 Ordinary Pade Approximation Suppose that we are givena power series suminfin

119894=0119888119894119911119894 represent a function 119891(119911) so that

119891 (119911) =

infin

sum119894=0

119888119894119911119894

(1)

This expansion is the fundamental starting point of any anal-ysis using Pade approximation A Pade approximation is arational fraction as

[119871

119872] =

1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872 (2)

which has a Maclaurin expansion which agrees with (1) asfar as possible Notice that in (2) there are 119871 + 1 numeratorcoefficients and 119872 + 1 denominator coefficients There is amore or less irrelevance common factor between them andfor definiteness we take 119887

0= 1 So there are 119871+1 independent

numerator coefficients and 119872 independent denominatorcoefficientsmaking119872+119871+1unknown coefficients in allThisnumber suggests that normally the [119871119872] out to fit the powerseries (1) through the orders 1 119911 119911119871+119872 For the notationof formal power series we have

infin

sum119894=0

119888119894119911119894

=1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872

+ 119874 (119911119871+119872+1

) (3)

and returning to (3) and cross-multiplying we find that

(1198870+ 1198871119911 + sdot sdot sdot + 119887

119872119911119872

) (1198880+ 1198881119911 + sdot sdot sdot )

= 1198860+ 1198861119911 + sdot sdot sdot + 119886

119871119911119871

+ 119874 (119911119871+119872+1

)

(4)

Equating the coefficients of 119911119871+1 119911119871+2 119911119871+119872 from (4) wefind

119887119872119888119871minus119872+1

+ 119887119872minus1119888119871minus119872+2

+ sdot sdot sdot + 1198870119888119871+1= 0

119887119872119888119871minus119872+2

+ 119887119872minus1119888119871minus119872+3

+ sdot sdot sdot + 1198870119888119871+1= 0

119887119872119888119871+ 119887119872minus1119888119871minus119872+1

+ sdot sdot sdot + 1198870119888119871+119872

= 0

(5)

If 119895 lt 0 we define 119888119895= 0 for consistency Since 119887

0= 1

(5) becomes a set of119872 linear equations for the119872 unknowndenominator coefficients and we obtain

[[[[

[

119888119871minus119872+1

119888119871minus119872

119888119871

119888119871minus119872+2

119888119871minus119872+1

119888119871+1

d

119888119871

119888119871+1

119888119871+119872minus1

]]]]

]

[[[[

[

119887119872

119887119872minus1

1198871

]]]]

]

= minus

[[[[

[

119888119871+1

119888119871+2

119888119871+119872

]]]]

]

(6)

from which the 119887119894may be found The numerator coefficients

1198860 119886

119871follow immediately from (4) by equating the coef-

ficients of 1 119911 119911119871 as

1198860= 1198880

1198861= 1198881+ 1198880

1198862= 1198882+ 11988711198881+ 11988721198880

119886119897= 119888119897+

min119871119872sum119894=0

119887119894119888119871minus119894

(7)

Thus (4) and (7) normally determine the Pade numeratorand denominator which are called Pade equations and wehave constructed an [119871119872]Pade approximationwhich agreeswith suminfin

119894=0119888119894119911119894 through order 119911119871+119872 If the given power series

converge to the same function for |119911| lt 119877with (0 lt 119877 lt +infin)then a sequence of Pade approximation may converge for119911 isin 119863 where119863 is a domain longer than |119911| lt 119877 [1]

Remark 1 In general Pade approximation does not exist (foran arbitrary function with respect to particular 119871 and119872) [1]

Theorem 2 If the Pade approximation [119871119872] exists then itwill be unique [2]

12 Muntz Polynomials An interesting generalization ofWeierstrass theorem (1885) (density of polynomials set in119862[119886 119887] equipped with a supremum norm) goes back to theGerman mathematician Herman Muntz He performed thisgeneralization by a suitable tool which is named Muntzpolynomials For presenting his theorem first we introducea definition and a remark

Definition 3 Suppose 1205820lt 1205821lt sdot sdot sdot isin R then the set

1199091205820 1199091205821 is named as a set of Muntz or Muntz polyno-

mials [6]

Remark 4 Let (120582119894)infin

119894=0be any sequences of distinct real num-

bers and 119886 gt 0 then

sum119899

119894=0119886119894119909120582119894

sum119899

119894=0119887119894119909120582119894 119886119894 119887119894isin R 119899 isin N (8)

is dense in 119862[119886 119887] [6]

Theorem 5 (generalized Muntz theorem) If 0 le 1205820lt 1205821lt

sdot sdot sdot rarr +infin then the Muntz polynomials with respect to thesesequences are dense in 1198712[0 1] if and only if suminfin

119894=0120582119894

minus1

= infin

[6]

Remark 6 In this paper by considering 120582119896= 119896120582 119896 =

1 2 (0 lt 120582 le 1) which is a particular case of Muntzpolynomials we introduce a new and extended Muntz Padeapproximation Also it is clear that by considering 120582

119896= 119896 we

can obtain the classical Weierstrass theorem [6]




mial as

119891 (119909) ≃

119899

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(9)

with the error

119890 (119909) =119891(119899+1)(120589)

(119899 + 1)(119909 minus 119909

0)119899+1

120589 isin (1199090 119909) (10)


119891 (119909) =

infin

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(11)



(119863120582

119886119891) (119909) =

1

Γ (120573 minus 120582)int119909

119886

119891(120573)

(119905)

(119909 minus 119905)120582minus120573+1

119889119905 (12)


119863120582

119886119862 = 0

119863120582

119886119909120575

=

0 if 120575 isin 1 120573 minus 1 Γ (120575 + 1)

Γ (120575 + 1 minus 120582)(119909)120575minus120582

if 120575 isin N 120575 ge 120573

or 120575 gt 120573 minus 1(13)


119863120582

119886(120574119891 + 120579119892) = 120574119863

120582

119886(119891) + 120579119863

120582

119886(119892) (14)




119891 (119909) =

119899

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119909 minus 119886)

119896120582

+119863(119899+1)120582

119886119891 (120589)

Γ ((119899 + 1) 120582 + 1)(119909 minus 119886)

(119899+1)120582

(15)


= 119863120582

119886sdot 119863120582

119886sdot sdot sdot 119863120582

119886

(119899-times) [7]


119891 (119911) =

infin

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119911 minus 119886)

119896120582

forall119911 isin 119863 (16)



119891 (119909) =

infin

sum119896=0

119888119896119911120582119896

119888119896=

119863119896120582

119886

Γ (119896120582 + 1) (17)


[119871

119872] =

1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

(18)


infin

sum119894=0

119888119894119911120582119894

=1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

+ 119874 (119911119871120582+119872120582+1

) (19)


(1198870+ 1198871119911120582

+ sdot sdot sdot + 119887119872119911119872120582

) (1198880+ 1198881119911120582

+ sdot sdot sdot )

= 1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

+ 119874 (119911119871120582+119872120582+1

)

(20)











119883 (119911)

119884 (119911)minus119880 (119911)

119881 (119911)= 119874 (119911

119871120582+119872120582+1

) (21)


119883 (119911)119881 (119911) minus 119884 (119911)119880 (119911) = 119874 (119911119871120582+119872120582+1

) (22)


119883 (119911)

119884 (119911)=119880 (119911)

119881 (119911) (23)




gt 119877 Let

119891 (119911) =

infin

sum119894=0

119888119894119911120582119894

with 119888119894= 119874((120588

1015840

)minus120582119894

) (24)


[119871

1] = 1198880+ 1198881119911120582


+119888119871119911120582119871

1 minus 119888119871+1119911120582119888119871

(25)






119871 because

119888119871119911120582119871

= 119874(119877

1205881015840)

120582119871

(26)

28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119871+1119911119888119871= 0within |119911| lt



0 |119888119871119888119871+1| gt 1205881015840 In

the latter case

10038161003816100381610038161003816100381610038161003816

1198881198710

119888119871

10038161003816100381610038161003816100381610038161003816=

119871minus1

prod119894=1198710

10038161003816100381610038161003816100381610038161003816

119888119894

119888119894+1

10038161003816100381610038161003816100381610038161003816gt (1205881015840

)120582(119871minus1198710)

(27)








28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119890119899= radic

119899

sum119896=0

(119910119899(119909119896) minus 119910 (119909

119896))2

(28)




119863120572

(119910 (119909)) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(29)


119910 (119905) = 1199098

minus 31199094+1205722

+9

4119909120572

(30)


119910120582

(119909) ≃ 119910120582

119899(119909) =

119899

sum119894=0

119888119894119909119894120582

(31)


Res(120572120582)119899

(119909) = 119863120572

(119910120582

119899(119909)) minus 119875 (120572 119910 (119909) 119909) ≃ 0 (32)

times10minus3

09

08

07

06

05

04

03

02

01

015 2 25 3 35 4 45 5

1

1


where

119875 (120572 119910 (119909) 119909) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(33)




Res(120572120582)119899

(119909(120573120574)

119894) = 119863

120572

(119910120582

119899(119909(120573120574)

119894))

minus 119875 (120572 119910 (119909(120573120574)

119894) 119909(120573120574)

119894) ≃ 0

119894 = 0 119899

(34)





times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10




11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)


minus 211990932

(36)





119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)






119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)




1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)



t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












mial as

119891 (119909) ≃

119899

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(9)

with the error

119890 (119909) =119891(119899+1)(120589)

(119899 + 1)(119909 minus 119909

0)119899+1

120589 isin (1199090 119909) (10)


119891 (119909) =

infin

sum119896=0

119891(119896)

(1199090)

119896(119909 minus 119909

0)119896

(11)



(119863120582

119886119891) (119909) =

1

Γ (120573 minus 120582)int119909

119886

119891(120573)

(119905)

(119909 minus 119905)120582minus120573+1

119889119905 (12)


119863120582

119886119862 = 0

119863120582

119886119909120575

=

0 if 120575 isin 1 120573 minus 1 Γ (120575 + 1)

Γ (120575 + 1 minus 120582)(119909)120575minus120582

if 120575 isin N 120575 ge 120573

or 120575 gt 120573 minus 1(13)


119863120582

119886(120574119891 + 120579119892) = 120574119863

120582

119886(119891) + 120579119863

120582

119886(119892) (14)




119891 (119909) =

119899

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119909 minus 119886)

119896120582

+119863(119899+1)120582

119886119891 (120589)

Γ ((119899 + 1) 120582 + 1)(119909 minus 119886)

(119899+1)120582

(15)


= 119863120582

119886sdot 119863120582

119886sdot sdot sdot 119863120582

119886

(119899-times) [7]


119891 (119911) =

infin

sum119896=0

119863119896120582

119886119891 (119886)

Γ (120582119896 + 1)(119911 minus 119886)

119896120582

forall119911 isin 119863 (16)



119891 (119909) =

infin

sum119896=0

119888119896119911120582119896

119888119896=

119863119896120582

119886

Γ (119896120582 + 1) (17)


[119871

119872] =

1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

(18)


infin

sum119894=0

119888119894119911120582119894

=1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

1198870+ 1198871119911120582 + sdot sdot sdot + 119887

119872119911119872120582

+ 119874 (119911119871120582+119872120582+1

) (19)


(1198870+ 1198871119911120582

+ sdot sdot sdot + 119887119872119911119872120582

) (1198880+ 1198881119911120582

+ sdot sdot sdot )

= 1198860+ 1198861119911120582

+ sdot sdot sdot + 119886119871119911119871120582

+ 119874 (119911119871120582+119872120582+1

)

(20)











119883 (119911)

119884 (119911)minus119880 (119911)

119881 (119911)= 119874 (119911

119871120582+119872120582+1

) (21)


119883 (119911)119881 (119911) minus 119884 (119911)119880 (119911) = 119874 (119911119871120582+119872120582+1

) (22)


119883 (119911)

119884 (119911)=119880 (119911)

119881 (119911) (23)




gt 119877 Let

119891 (119911) =

infin

sum119894=0

119888119894119911120582119894

with 119888119894= 119874((120588

1015840

)minus120582119894

) (24)


[119871

1] = 1198880+ 1198881119911120582


+119888119871119911120582119871

1 minus 119888119871+1119911120582119888119871

(25)






119871 because

119888119871119911120582119871

= 119874(119877

1205881015840)

120582119871

(26)

28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119871+1119911119888119871= 0within |119911| lt



0 |119888119871119888119871+1| gt 1205881015840 In

the latter case

10038161003816100381610038161003816100381610038161003816

1198881198710

119888119871

10038161003816100381610038161003816100381610038161003816=

119871minus1

prod119894=1198710

10038161003816100381610038161003816100381610038161003816

119888119894

119888119894+1

10038161003816100381610038161003816100381610038161003816gt (1205881015840

)120582(119871minus1198710)

(27)








28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119890119899= radic

119899

sum119896=0

(119910119899(119909119896) minus 119910 (119909

119896))2

(28)




119863120572

(119910 (119909)) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(29)


119910 (119905) = 1199098

minus 31199094+1205722

+9

4119909120572

(30)


119910120582

(119909) ≃ 119910120582

119899(119909) =

119899

sum119894=0

119888119894119909119894120582

(31)


Res(120572120582)119899

(119909) = 119863120572

(119910120582

119899(119909)) minus 119875 (120572 119910 (119909) 119909) ≃ 0 (32)

times10minus3

09

08

07

06

05

04

03

02

01

015 2 25 3 35 4 45 5

1

1


where

119875 (120572 119910 (119909) 119909) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(33)




Res(120572120582)119899

(119909(120573120574)

119894) = 119863

120572

(119910120582

119899(119909(120573120574)

119894))

minus 119875 (120572 119910 (119909(120573120574)

119894) 119909(120573120574)

119894) ≃ 0

119894 = 0 119899

(34)





times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10




11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)


minus 211990932

(36)





119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)






119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)




1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)



t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119883 (119911)

119884 (119911)minus119880 (119911)

119881 (119911)= 119874 (119911

119871120582+119872120582+1

) (21)


119883 (119911)119881 (119911) minus 119884 (119911)119880 (119911) = 119874 (119911119871120582+119872120582+1

) (22)


119883 (119911)

119884 (119911)=119880 (119911)

119881 (119911) (23)




gt 119877 Let

119891 (119911) =

infin

sum119894=0

119888119894119911120582119894

with 119888119894= 119874((120588

1015840

)minus120582119894

) (24)


[119871

1] = 1198880+ 1198881119911120582


+119888119871119911120582119871

1 minus 119888119871+1119911120582119888119871

(25)






119871 because

119888119871119911120582119871

= 119874(119877

1205881015840)

120582119871

(26)

28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119871+1119911119888119871= 0within |119911| lt



0 |119888119871119888119871+1| gt 1205881015840 In

the latter case

10038161003816100381610038161003816100381610038161003816

1198881198710

119888119871

10038161003816100381610038161003816100381610038161003816=

119871minus1

prod119894=1198710

10038161003816100381610038161003816100381610038161003816

119888119894

119888119894+1

10038161003816100381610038161003816100381610038161003816gt (1205881015840

)120582(119871minus1198710)

(27)








28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119890119899= radic

119899

sum119896=0

(119910119899(119909119896) minus 119910 (119909

119896))2

(28)




119863120572

(119910 (119909)) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(29)


119910 (119905) = 1199098

minus 31199094+1205722

+9

4119909120572

(30)


119910120582

(119909) ≃ 119910120582

119899(119909) =

119899

sum119894=0

119888119894119909119894120582

(31)


Res(120572120582)119899

(119909) = 119863120572

(119910120582

119899(119909)) minus 119875 (120572 119910 (119909) 119909) ≃ 0 (32)

times10minus3

09

08

07

06

05

04

03

02

01

015 2 25 3 35 4 45 5

1

1


where

119875 (120572 119910 (119909) 119909) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(33)




Res(120572120582)119899

(119909(120573120574)

119894) = 119863

120572

(119910120582

119899(119909(120573120574)

119894))

minus 119875 (120572 119910 (119909(120573120574)

119894) 119909(120573120574)

119894) ≃ 0

119894 = 0 119899

(34)





times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10




11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)


minus 211990932

(36)





119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)






119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)




1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)



t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










28

26

24

22

2

18

16

14

12

1

0 01 02 03 04 05 06 07 08 09 1

ExactApproximate

x



119890119899= radic

119899

sum119896=0

(119910119899(119909119896) minus 119910 (119909

119896))2

(28)




119863120572

(119910 (119909)) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(29)


119910 (119905) = 1199098

minus 31199094+1205722

+9

4119909120572

(30)


119910120582

(119909) ≃ 119910120582

119899(119909) =

119899

sum119894=0

119888119894119909119894120582

(31)


Res(120572120582)119899

(119909) = 119863120572

(119910120582

119899(119909)) minus 119875 (120572 119910 (119909) 119909) ≃ 0 (32)

times10minus3

09

08

07

06

05

04

03

02

01

015 2 25 3 35 4 45 5

1

1


where

119875 (120572 119910 (119909) 119909) =40320

Γ (9 minus 120572)1199098minus120572

minus 3Γ (5 + 1205722)

Γ (5 minus 1205722)1199094minus1205722

+9

4Γ (120572 + 1) + (

3

21199091205722

minus 1199094

)3

minus (119910 (119909))32

(33)




Res(120572120582)119899

(119909(120573120574)

119894) = 119863

120572

(119910120582

119899(119909(120573120574)

119894))

minus 119875 (120572 119910 (119909(120573120574)

119894) 119909(120573120574)

119894) ≃ 0

119894 = 0 119899

(34)





times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10




11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)


minus 211990932

(36)





119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)






119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)




1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)



t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times10minus3

09

08

07

06

05

04

03

02

01

15 2 25 3 35 4 45 5

1

10




11986312

119906 (119909) = 119891 (119909) 119906 (119909) + 119892 (119909) + radic119909int119909

0

1199062

(119905) 119889119905

119910 (0) = 0

(35)

where

119891 (119909) = 2radic119909 + 211990932

minus (radic119909 + 11990932

) ln (1 + 119909)


minus 211990932

(36)





119889119909

119889119905= 120572119909 + 120576119909

2

119909 (0) = 1 (37)






119909 (119905) =120572 exp (120572119905)

(120572 + 120576 minus 120576 exp (120572119905)) (38)




1198892

119879

1198891199102+ 120582119890minus119896119910

119879 = 0 (39)

with

119889119879

119889119910(0) = 0 119879 (1) = 1 (40)



t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t

y

17

16

15

14

13

12

11

1

0 005 01 015 02 025 03 035 04 045 05

AB





119879 (119910) = (1198691(2 (radic120582119896)) 119884


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

+ (1198841(2 (radic120582119896)) 119869


times (1198691(2 (radic120582119896)) 119884

0(minus2 (119890

minus1198962radic120582119896))

+1198841(minus2 (radic120582119896)) 119869

0(2 (119890minus1198962radic120582119896)))

minus1

(41)


01198841are


6 Conclusion



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A New Extended Padé Approximation and...

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