ANA Advances in Numerical Analysis 1687-9570 1687-9562 Hindawi Publishing Corporation 263467 10.1155/2013/263467 263467 Research Article A New Extended Padé Approximation and Its Application Kalateh Bojdi Z. 1 Ahmadi-Asl S. 1 Aminataei A. 2 Han Weimin 1 Department of Mathematics Birjand University Birjand Iran birjand.ac.ir 2 Faculty of Mathematics K. N. Toosi University of Technology P.O. Box 16315-1618, Tehran Iran kntu.ac.ir 2013 10 12 2013 2013 19 06 2013 16 09 2013 07 10 2013 2013 Copyright © 2013 Z. Kalateh Bojdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We extend ordinary Padé approximation, which is based on a set of standard polynomials as {1,x,,xn}, to a new extended Padé approximation (Müntz Padé approximation), based on the general basic function set {1,xλ,x2λ,,xnλ}  (0<λ1) (the particular case of Müntz polynomials) using general Taylor series (based on fractional calculus) with error convergency. The importance of this extension is that the ordinary Padé approximation is a particular case of our extended Padé approximation. Also the parameterization (λ is the corresponding parameter) of new extended Padé approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research.

1. Introduction

Suppose that we are given a power series i=0cizi, represent a function f(z), so that (1)f(z)=i=0cizi. This expansion is the fundamental starting point of any analysis using Padé approximation. A Padé approximation is a rational fraction as (2)[LM]=a0+a1z++aLzLb0+b1z++bMzM, which has a Maclaurin expansion which agrees with (1) as far as possible. Notice that, in (2), there are L+1 numerator coefficients and M+1 denominator coefficients. There is a more or less irrelevance common factor between them, and for definiteness we take b0=1. So there are L+1 independent numerator coefficients and M independent denominator coefficients, making M+L+1 unknown coefficients in all. This number suggests that normally the [L/M] out to fit the power series (1) through the orders 1,z,,zL+M. For the notation of formal power series, we have (3)i=0cizi=a0+a1z++aLzLb0+b1z++bMzM+O(zL+M+1), and returning to (3) and cross-multiplying, we find that (4)(b0+b1z++bMzM)(c0+c1z+)=a0+a1z++aLzL+O(zL+M+1). Equating the coefficients of zL+1,zL+2,,zL+M from (4), we find (5)bMcL-M+1+bM-1cL-M+2++b0cL+1=0,bMcL-M+2+bM-1cL-M+3++b0cL+1=0,bMcL+bM-1cL-M+1++b0cL+M=0. If j<0, we define cj=0 for consistency. Since b0=1, (5) becomes a set of M linear equations for the M unknown denominator coefficients and we obtain (6)[cL-M+1cL-McLcL-M+2cL-M+1cL+1cLcL+1cL+M-1][bMbM-1b1]=-[cL+1cL+2cL+M], from which the bi may be found. The numerator coefficients a0,,aL follow immediately from (4) by equating the coefficients of 1,z,,zL as (7)a0=c0,a1=c1+c0,a2=c2+b1c1+b2c0,al=cl+i=0min{L,M}bicL-i. Thus (4) and (7) normally determine the Padé numerator and denominator which are called Padé equations and we have constructed an [L/M] Padé approximation which agrees with i=0cizi through order zL+M. If the given power series converge to the same function for |z|<R with (0<R<+), then a sequence of Padé approximation may converge for zD, where D is a domain longer than |z|<R .

Remark 1.

In general, Padé approximation does not exist (for an arbitrary function with respect to particular L and M) .

Theorem 2.

If the Padé approximation [L/M] exists, then it will be unique .

1.2. Müntz Polynomials

An interesting generalization of Weierstrass theorem (1885) (density of polynomials set in C[a,b], equipped with a supremum norm) goes back to the German mathematician Herman Müntz. He performed this generalization by a suitable tool which is named Müntz polynomials. For presenting his theorem, first we introduce a definition and a remark.

Definition 3.

Suppose λ0<λ1<; then, the set {xλ0,xλ1,} is named as a set of Müntz or Müntz polynomials .

Remark 4.

Let (λi)i=0 be any sequences of distinct real numbers and a>0; then, (8){i=0naixλii=0nbixλi;ai,bi,n}, is dense in C[a,b] .

Theorem 5 (generalized Müntz theorem).

If 0λ0<λ1<+, then the Müntz polynomials with respect to these sequences are dense in L2[0,1], if and only if i=0λi-1= .

Remark 6.

In this paper, by considering λk=kλ, k=1,2,(0<λ1) which is a particular case of Müntz polynomials, we introduce a new and extended Müntz Padé approximation. Also, it is clear that by considering λk=k, we can obtain the classical Weierstrass theorem .

1.3. Generalized Taylor Series

In calculus, Taylor’s theorem gives an approximation of a k-times differentiable function f(x) around a given point x0 by a kth order Taylor polynomial as (9)f(x)k=0nf(k)(x0)k!(x-x0)k, with the error (10)e(x)=f(n+1)(ς)(n+1)!(x-x0)n+1,ς(x0,x), and for analytical polynomials, we have (11)f(x)=k=0f(k)(x0)k!(x-x0)k. Taylor’s theorem also generalizes to multivariate and vector valued functions. But the generalized version of this theorem from fractional calculus view or the theory of derivatives of arbitrary order was done by , because of its high applications in solving fractional ordinary differential equations integral ordinary differential equations and obtaining an operational matrix, and so forth. For this purpose, we need some definitions in fractional calculus.

Definition 7.

The Caputo fractional derivative of f(x) of order λ>0 with a0 is defined as (12)(Daλf)(x)=1Γ(β-λ)axf(β)(t)(x-t)λ-β+1dt, for β-1<λβ, β, xa.

From Caputo fractional derivative, we have (13)DaλC=0,        Daλxδ={0,ifδ{1,,β-1},Γ(δ+1)Γ(δ+1-λ)(x)δ-λ,ifδ,δβ,orδ>β-1, where β=λ and λ0. However, the Caputo fractional derivative is a linear operator which means (14)Daλ(γf+θg)=γDaλ(f)+θDaλ(g). For more literature review of fractional calculus, see .

Theorem 8.

Suppose that Dakλf(x)C(a,b], for k=0,1,,n+1, where 0<λ1; then, one has (15)f(x)=k=0nDakλf(a)Γ(λk+1)(x-a)kλ+Da(n+1)λf(ς)Γ((n+1)λ+1)(x-a)(n+1)λ, where aςb, for all x(a,b], and Da(n)λ=Daλ·DaλDaλ, (n-times) .

If we consider a=0, then we obtain a fractional Maclaurin series and in a similar manner as the arbitrary function f(z) that has an infinite Caputo differentiable at point a named as analytical functions on D in a fractional sense if (16)f(z)=k=0Dakλf(a)Γ(λk+1)(z-a)kλ,zD.

Now by using the general Taylor series (based on fractional calculus), we generalize the classical and ordinary Padé approximations to a new and extended Müntz Padé approximation. For this goal, we present the Müntz Padé approximation as a ratio of two Müntz polynomials constructed from the coefficients of generalized Taylor series expansion of a function. Also in a similar approach from ordinary Padé approximation, we prove the uniqueness of Müntz Padé approximation. Now in a similar manner from ordinary Padé approximation, suppose that f(x) is an analytical function (from fractional calculus) in the neighborhood of a=0, and then we can write f(x)(17)f(x)=k=0ckzλk,ck=DakλΓ(kλ+1), and also Müntz Padé approximation is defined as a ratio of two Müntz polynomials as (18)[LM]=a0+a1zλ++aLzLλb0+b1zλ++bMzMλ, and then we suppose that [L/M] (Müntz Padé approximation) has a Maclaurin series (17) or (19)i=0cizλi=a0+a1zλ++aLzLλb0+b1zλ++bMzMλ+O(zLλ+Mλ+1). By cross multiplying, we find that (20)(b0+b1zλ++bMzMλ)(c0+c1zλ+)=a0+a1zλ++aLzLλ+O(zLλ+Mλ+1). Equating the coefficients zλL+λ,zλL+2λ,,zλL+λM, from (20) we obtain a system of equations similar to (5) and we can obtain bi coefficients. Also immediately by equating the coefficients of 1,zλ,,zLλ, we can obtain ai coefficients in a recursion formula as (7). Thus by assumption of existence of [L/M] Müntz Padé approximation, for establishing the uniqueness of this approximation, we must prove that the matrix of system of (5) is nonsingular, but we perform the proof of the uniqueness of Müntz Padé approximation by a different approach in the next section.

Remark 9.

Let λ=1; then, we can obtain the ordinary Padé approximation.

3. Uniqueness and Convergence Analysis of Müntz Padé Approximation

In this section, we present the uniqueness and convergence analysis of Müntz Padé approximation in some theorems.

Theorem 10 (uniqueness).

When Müntz Padé approximation ([L/M]) exists, then it is unique for any formal power series.

Proof.

Assume that there are two such Müntz Padé approximations U(z)/V(z) and X(z)/Y(z), where the degrees of X and U are less than or equal to λL and that of Y and V are less than or equal to λM. Then by (19), we must have (21)X(z)Y(z)-U(z)V(z)=O(zLλ+Mλ+1), since both approximate the same series. If we multiply (21) by Y(z)V(z), we obtain (22)X(z)V(z)-Y(z)U(z)=O(zLλ+Mλ+1), but the left-hand side of (22) is a polynomial of degree at most λL+λM and thus is identically zero. Since neither Y nor V is identically zero, we conclude (23)X(z)Y(z)=U(z)V(z). Since, by definition, both X and Y and U and V are relatively prime and Y(0)=V(0)=1; we have shown that the two supposedly different Müntz Padé approximants are the same.

Theorem 11 (convergency).

Let f(z) be analytic in |z|R (in fractional sense); then, an infinite subsequence of [L/1] Müntz Padé approximants converges to f(z) uniformly in |z|R.

Proof.

By hypothesis, f(z) is analytic in |z|R (in fractional sense) and consequently within a large interval, |z|<ρ, with ρ>ρ>R. Let (24)f(z)=i=0cizλi,withci=O((ρ)-λi). The [L/1] Müntz Padé approximation is given by (25)[L1]=c0+c1zλ++cL-1zλ(L-1)+cLzλL1-cL+1zλ/cL. If a subsequence of coefficients {cLj,j=1,2,} are zero, then [(Lj-1)/1] are truncated fractional Maclaurin expansions which converge to f(z) uniformly in |z|R<ρ. So we assume that no infinite subsequence of {cL} vanishes and consider rL=cL+1/cL which is well defined for all sufficiently large L, because (26)cLzλL=O(Rρ)λL the sequence of [L/1] Müntz Padé approximation given by (25) converges uniformly in |z|<R unless there exists a sequence of values of L for which 1-cL+1z/cL=0 within |z|<ρ. Thus either a subsequence of the second row converges uniformly or else for some L0 and all L>L0, |cL/cL+1|>ρ. In the latter case (27)|cL0cL|=i=L0L-1|cici+1|>(ρ)λ(L-L0), contradicting with (24), so the proof is completed.

4. The Test Experiments

Now in this section, in the two subsections, we show the applications of the Müntz Padé approximation in the functional approximation (see Section 4.1) and fractional calculus fields. The advantage of using the Müntz Padé approximation is shown for numerical approximation of fractional differential equations in Section 4.2.

4.1. Müntz Padé Approximation and Functional Approximation

In this section, we obtain the Müntz Padé approximations of f(x)=ex, for different values of λ. Figure 1 shows the [3/5] Müntz Padé approximations of f(x)=ex and λ=0.4.

The [3/5] Müntz Padé approximation of f(x)=ex and exact function of λ=0.4.

Also the [3/5] Müntz Padé approximation of f(x)=ex, and λ=0.65 is shown in Figure 2.

Comparison between the [3/5] Müntz Padé approximation of f(x)=ex and exact function of λ=0.65.

4.2. Müntz Padé Approximation and Fractional Calculus

This section is devoted to presentation of some numerical simulations obtained by applying the collocation method and based on a new extended Padé approximation (Müntz Padé approximation). The algorithm for numerical approximation of solutions to the initial value problems for the fractional differential equations was implemented by MATLAB. In the case of nonlinear equations, the MATLAB function fsolve was used for solving the nonlinear system. In the case that the exact solution y to a problem is known, the dependence of approximation errors on the discretization parameter n is estimated in 2-norm as (28)en=k=0n(yn(xk)-y(xk))2, where yn is the approximated solution corresponding to the discretization parameter n.

Experiment 1.

We start with a simple nonlinear problem  (29)Dα(y(x))=40320Γ(9-α)x8-α-3Γ(5+α/2)Γ(5-α/2)x4-α/2+94Γ(α+1)+(32xα/2-x4)3-(y(x))3/2, where we have a nonlinear and nonsmooth right-hand side. The solution y has a smooth derivative of order 0<α<1 . The analytical solution subject to the initial condition y(0)=0 is given by (30)y(t)=x8-3x4+α/2+94xα. Now we approximate the exact solution of (29) or yλ(x), with the [n/1] new and extended Padé approximation (Müntz Padé approximation) as (31)yλ(x)ynλ(x)=i=0ncixiλ, and substituting it into (29), we obtain (32)Resn(α,λ)(x)=Dα(ynλ(x))-P(α,y(x),x)0, where (33)P(α,y(x),x)=40320Γ(9-α)x8-α-3Γ(5+α/2)Γ(5-α/2)x4-α/2  +94Γ(α+1)+(32xα/2-x4)3-(y(x))3/2. For the collocation points, we use the first roots of the Jacobi polynomial Pn(β,γ)(x) [9, 10], and then after enforcing the initial condition y(0)=0, we obtain a system of nonlinear algebraic equations and we use the MATLAB function fsolve for solving the nonlinear system. Thus, substituting the collocation points into (32) yields (34)Resn(α,λ)(xi(β,γ))=Dα(ynλ(xi(β,γ)))-P(α,y(xi(β,γ)),xi(β,γ))0,22222222222222222i=0,,n. Now from (34) and its initial condition, we have n+2 algebraic equations of n+1 unknown coefficients. Thus for obtaining the unknown coefficients, we must eliminate one arbitrary equation from these n+2 equations. But because of the necessity of holding the boundary conditions, we eliminate the last equation from (34). Finally, replacing the last equation of (34) by the equation of initial condition, we obtain a system of n+1 equations of n+1 unknowns ci. By implementing the method as presented, for n=5 and also for different parameters of β and γ, we obtain the approximate solutions. The [12/1] ordinary and Müntz Padé coefficients approximation of this problem are shown in Figures 3 and 4, respectively. We have observed that this method (the new extended Padé approximation (Müntz Padé approximation)) is very efficient for numerical approximation of the fractional ordinary differential equations. Also, closer look at the results of the Müntz Padé approximation scheme reveals that in this method of solution the coefficients decrease faster than the classical case. This is the advantage on the application of the Müntz Padé approximation to the fractional ordinary differential equations.

[ 12 / 1 ] ordinary Padé approximation coefficients of α=0.5, β=0.25, and γ=0.75 of Experiment 1.

[ 12 / 1 ] new Padé approximation coefficients of λ=0.5, α=0.5, β=0.25, and γ=0.75 of Experiment 1.

Experiment 2.

Consider the nonlinear fractional integro-differential equation  (35)D1/2u(x)=f(x)u(x)+g(x)+x0xu2(t)dt,2222222222222222222222122y(0)=0, where (36)f(x)=2x+2x3/2-(x+x3/2)ln(1+x),g(x)=2arcsinh(x)π1+x-2x3/2. The exact solution of (35) is u(x)=ln(1+x). We implement the collocation method based on the [10/1] new Padé approximation of λ=0.16, β=1, and γ=1. The obtained results of our method are presented in Table 1.

The comparison between the collocation method based on [10/1] Müntz Padé approximation and the exact solution of λ=0.16, β=1, γ=1, of Experiment 2.

x Exact Approximate Error
0.0 0.0000000000 0.0000000000 0.000000000
0.1 0.0953101798 0.0953103008 0.000000121
0.2 0.18232155679 0.1823218407 0.000000284
0.3 0.26236426446 0.2623649824 0.000000718
0.4 0.33647223662 0.3364726886 0.000000452
0.5 0.40546510810 0.40546528510 0.000000177
0.6 0.47000362924 0.4700043522 0.000000723
0.7 0.53062825106 0.5306290280 0.000000777
0.8 0.58778666490 0.58778755290 0.000000888
0.9 0.64185388617 0.64185410017 0.000000214
1.0 0.69314718055 0.69314789955 0.000000719
5. Application of Müntz Padé Approximation in Vibration and Electromagnetic Radiation Problems

In this section, we present two applications of Müntz Padé approximation for numerical approximation of some applicable ordinary differential equations.

Experiment 3.

Consider the following model problem : (37)dxdt=αx+εx2,x(0)=1, where 0<εα1 . This model has high application in the theory of sound and vibration . The exact solution to this initial value problem has the form (38)x(t)=αexp(αt)(α+ε-εexp(αt)). Now for α=0.5, ε=0.4, and λ=0.5 and using [2/3] and [4/4] Müntz Padé approximations, we obtain the following results shown in Table 2.

Also the obtained results for different kinds of Müntz Padé approximations are shown in Figure 5.

Comparison between the [2/3] and [4/4] Müntz Padé approximations of α=0.5, ε=0.4, and λ=0.5 of Experiment 3.

t Error of [2/3] Error of [4/4]
0.0 0.0012 0.0009
0.1 0.0010 −0.0004
0.2 0.0013 0.0006
0.3 0.0022 −0.0002
0.4 0.0014 0.0007
0.5 0.0012 0.0003

A: the obtained results of [4/4] Müntz Padé approximation of experiment 3. B: the obtained results of [2/3] Müntz Padé approximation of Experiment 3.

Experiment 4.

Consider the following model problem : (39)d2Tdy2+λe-kyT=0, with (40)dTdy(0)=0,T(1)=1. This equation has high application in the theory of electromagnetic radiation and describes the steady state reaction-diffusion equations with source term that arise in modeling microwave heating in an infinite slab with isothermal walls . Also λ and k represent the thermal absorptivity and electric field decay rate parameters, respectively. The exact solution to this boundary value problem has the form (41)T(y)=(J1(2(λ/k))Y0(-2(λ/keky)))×(J1(2(λ/k))Y0(-2(e-k/2λ/k))22222+Y1(-2(λ/k))J0(2(e-k/2λ/k)))-1+(Y1(2(λ/k))J0(-2(λ/keky)))×(J1(2(λ/k))Y0(-2(e-k/2λ/k))22222+Y1(-2(λ/k))J0(2(e-k/2λ/k)))-1, where J0, J1 are Bessel functions of the first kind and Y0, Y1 are Bessel functions of the second kind. Now for k=1, and λ=0.5 and using [2/3] and [4/4] Müntz Padé approximations, we obtain the following results shown in Table 3.

Comparison between the [2/3] and [4/4] Müntz Padé approximations of k=1 and λ=0.5 of Experiment 4.

y Error of [2/3] Error of [4/4]
0.0 0.0002 0.0001
0.1 −0.0033 −0.0011
0.2 0.0014 0.0009
0.3 0.0003 0.0001
0.4 0.0026 −0.0012
0.5 0.0023 0.0021
6. Conclusion

In this paper, using the general Taylor series (based on fractional calculus), we extend the ordinary Padé approximation to the general Müntz Padé approximation. The importance of this extension is that the ordinary Padé approximation is a particular case of our Müntz Padé approximation (λ=1). We have applied the method in the application of functional approximation, fractional exponent, and vibration and electromagnetic radiation model problems and have obtained the results with a good order of accuracy. Also the uniqueness results and error analysis have been presented completely. In addition, the test experiments have been presented for showing the applicability and validity of the new Müntz Padé approximation.

Acknowledgments

The authors are grateful to the reviewers and the editor Professor Dr. Weimin Han for their helpful comments and suggestions which indeed improved the quality of this paper.