Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111]

Fernández, Francisco M.

doi:10.1016/j.physleta.2008.05.071

Cited by 16 publications

(14 citation statements)

References 31 publications

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“…The convergence of the HPM has already been discussed in our earlier paper [21]; therefore, here we simply compare our results with those of Parand and Shahini [20]. Our estimate of u ′ (0) −1.588071022611375313 is by far more accurate than the one obtained by Parand and Shahini [20] Table 1 shows our earlier results [21], those of Parand and Shahini [20] and the numerical calculation of Kobayashi et al [9]. Those HPM results obtained from a relatively small [5/8] Padé approximant are more accurate than the numerical ones for x ≤ 1.…”

Section: The Hankel-padé Methodsmentioning

confidence: 71%

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Rational approximation to the Thomas–Fermi equations

Fernández

2011

Applied Mathematics and Computation

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Section: The Hankel-padé Methodsmentioning

confidence: 71%

“…In order to make this letter self contained we review the main features of the HPM [21]. It is convenient to define the variables t = x 2 and f (t) = u(t 2 ) 1/2 , so that the TF equation becomes…”

Section: The Thomas-fermi Equationmentioning

confidence: 99%

Rational approximation to the Thomas–Fermi equations

Fernández

2011

Applied Mathematics and Computation

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“…For example, the first of them are

\begin{matrix} falsefalsearray \\ arraycenter f_{0} = s, f_{1} = 1, f_{3} = \frac{1}{6} MathClass-open(M + β MathClass-close), f_{4} = \frac{1}{12} f_{2} MathClass-open(- 1 + 2 β + M MathClass-close), \\ arraycenter f_{5} = - \frac{1}{60} M - \frac{1}{60} β - \frac{1}{30} {f_{2}}^{2} + \frac{1}{120} M^{2} + \frac{1}{40} Mβ + \frac{1}{60} β^{2} + \frac{1}{15} β {f_{2}}^{2}, \dots . \end{matrix}

Now, we can construct rational function in order to have the correct limit at infinity according to the boundary conditions ; one would expect that N ≥ S . However, in order to obtain an accurate value of f 2 , it is more convenient to choose S = N + d , d = 0,1,2, … (for more details, see , so from , each Hankel determinant is a polynomial function of f 2 , and we expect that there is a sequence of roots

f_{2}^{[D MathClass-punc, d]}

, D = 2,3, … that converges toward the actual value of f ″ (0). In our case, because f 4 is the first nonzero coefficient that depends on f 2 , we use the Hankel sequences with d ≥ 3 to compute f ″ (0).…”

Section: Magnetohydrodynamic Flow Over a Nonlinear Stretching Sheetmentioning

confidence: 99%

“…Recently, Abbasbandy et al obtained the unknown skin friction coefficient of the boundary‐layer Falkner–Skan equation for wedge by using the Hankel–Padé method, and they showed the efficiency of this method in magnetohydrodynamic (MHD) flow problems. The Hankel–Padé method is successfully applied for the treatment of two‐point nonlinear equations that arise from some fields of physics .…”

Section: Introductionmentioning

confidence: 99%

Solutions of the magnetohydrodynamic flow over a nonlinear stretching sheet and nano boundary layers over stretching surfaces

Abbasbandy

Ghehsareh

2011

Numerical Methods in Fluids

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SUMMARY In this paper, a simple and efficient analytical method, Hankel–Padé method, is employed in solving boundary‐layer problems. Two important types of boundary‐layer problems, the problem of the boundary‐layer flow of an incompressible viscous fluid over a nonlinear stretching sheet and the problem of the nano boundary‐layer flows with Navier boundary condition, are considered. The analytical solutions of the governing nonlinear boundary‐layer problems are developed as rational approximation solutions. The comparison of the obtained results with other available results shows that, in general, the Hankel–Padé is much simpler and more accurate than other approaches that were proposed by other authors recently. Copyright © 2011 John Wiley & Sons, Ltd.

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“…Very recently, Abbasbandy and Hayat (2009), obtained the unknown skin friction coefficient of the boundary layer Falkner‐Skan equation for wedge by using the Hankel‐Padé method and they showed the efficiency of this method in MHD flow problems. The Hankel‐Padé method, like other recent methods based on Taylor series expansion (Bervillier et al , 2008a, b), has already been efficient for the treatment of several two‐point boundary value problems of nonlinear equations of some different fields of physics (Abbasbandy and Hayat, 2009; Amore and Fernández, 2007; Bervillier, 2009; Fernández, 2007).…”

Section: Introductionmentioning

confidence: 99%

Solutions for MHD viscous flow due to a shrinking sheet by Hankel‐Padé method

Abbasbandy

Ghehsareh²

2013

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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PurposeIn this paper, an analysis is performed to find the solution of a nonlinear ordinary differential equation that appears in a model for MHD viscous flow caused by a shrinking sheet.Design/methodology/approachThe cases of two dimensional and axisymmetric shrinking have been discussed. When the sheet is shrinking in the x‐direction, the analytical solutions are obtained by the Hankel‐Padé method. Comparison to exact solutions reveals reliability and high accuracy of the procedure, even in the case of multiple solutions. The case of sheet shrinking in the y‐direction is also considered, with success.FindingsWhen the sheet shrinks in the x‐direction, the analytical solutions are obtained by Hankel‐Padé method. Also, when the sheet shrinks in the y‐direction, the obtained results with Hankel‐Padé method are presented.Practical implicationsComparison to exact solutions reveals reliability and high accuracy of the procedure and convincingly could be used to obtain multiple solutions for certain parameter domains of this case of the governing nonlinear problem.Originality/valueThe numerical solutions are given for both two‐dimensional and axisymmetric shrinking sheets by using Hankel‐Padé method. It is clear that the Hankel‐Padé method is, by far, more simple, straightforward and gives reasonable results for large Hartman numbers and suction parameters.

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Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111]

Cited by 16 publications

References 31 publications

Rational approximation to the Thomas–Fermi equations

Rational approximation to the Thomas–Fermi equations

Solutions of the magnetohydrodynamic flow over a nonlinear stretching sheet and nano boundary layers over stretching surfaces

Solutions for MHD viscous flow due to a shrinking sheet by Hankel‐Padé method

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