scholarly journals A New Approach for a Class of the Blasius Problem via a Transformation and Adomian’s Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdelhalim Ebaid ◽  
Nwaf Al-Armani
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Padé Approximation ◽  
Current Approach ◽  
Pade Approximation ◽  
Flow Problems ◽  
Blasius Problem ◽  
Classical Boundary ◽  
Layer Flow ◽  
Conditions At Infinity

The main feature of the boundary layer flow problems is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such problems. To the best of the authors’ knowledge, two procedures were used extensively in the past two decades to deal with the boundary conditions at infinity, either the Padé approximation or the direct numerical codes. However, an intensive work is needed to perform the calculations using the Padé technique. Regarding this point, a new idea is proposed in this paper. The idea is based on transforming the unbounded domain into a bounded one by the help of a transformation. Accordingly, the original differential equation is transformed into a singular differential equation with classical boundary conditions. The current approach is applied to solve a class of the Blasius problem and a special class of the Falkner-Skan problem via an improved version of Adomian’s method (Ebaid, 2011). In addition, the numerical results obtained by using the proposed technique are compared with the other published solutions, where good agreement has been achieved. The main characteristic of the present approach is the avoidance of the Padé approximation to deal with the infinity boundary conditions.

scholarly journals Melting anomaly and occurrence of a liquid–liquid phase transition

2014 ◽  
Vol 70 (a1) ◽  
pp. C1336-C1336
Author(s):  
Kazuhiro Fuchizaki ◽  
Nozomu Hamaya
Keyword(s):  
Differential Equation ◽  
Phase Transition ◽  
Liquid Phase ◽  
Padé Approximation ◽  
Melting Curve ◽  
Qualitative Change ◽  
Melting Line ◽  
Pade Approximation ◽  
Liquid Phase Transition ◽  
Liquid Phases

A liquid–liquid phase transition is discussed in view of singularity in a melting line. Possible existence of a liquid–liquid phase transition has been discussed thus far in relation to existence of a maximum in a melting curve [1]. We claim that it is the existence of a breakpoint, rather than a maximum, that can potentially influence the occurrence of a liquid-to-liquid transition. Our in situ synchrotron x-ray diffraction measurements revealed that the molecular crystal tin tetraiodide has an unusual melting curve. It rapidly increases with a pressure up to about 1.5 GPa, at which it abruptly breaks. The melting curve becomes almost flat on the high-pressure side of the breakpoint (with a slight maximum at about 3 GPa). Kechin proposed the differential equation [2], which can be handled with an aid of Padé approximation, to capture the overall aspect of such a melting curve. We could show that the melting curve obtained as the solution to the differential equation became closer to the actual melting curve with the improvement of the degree of the Padé approximation. Kevin's proposal thus seems to be appropriate to handle the differential equation provided the slope of the melting line is everywhere continuous. We believe that this is not the case for the melting curve in question at the breakpoint, as inferred from the nature of breakdown of the Kraut–Kennedy and the Magalinskiii–Zubov relationships [3]. We propose that the breakdown of these relationships is rather a manifestation of such a qualitative change in the intermolecular interaction as electronic (bonding) transformation in the liquid state. The breakpoint may then be a triple point among the crystalline phase and two liquid phases, whose existence has been confirmed.

Padé approximation in time-domain boundary conditions of porous surfaces

2007 ◽  
Vol 122 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Vladimir E. Ostashev ◽  
Sandra L. Collier ◽  
D. Keith Wilson ◽  
David F. Aldridge ◽  
Neill P. Symons ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Time Domain ◽  
Domain Boundary ◽  
Padé Approximation ◽  
Pade Approximation ◽  
Porous Surfaces

scholarly journals The Numerical Method for Solving Differential Equations of Lane-Emden Type by Padé Approximation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammed Yiğider ◽  
Khatereh Tabatabaei ◽  
Ercan Çelik
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Padé Approximation ◽  
Pade Approximation ◽  
One Dimensional ◽  
Series Form ◽  
Differential Transformation

Numerical solution differential equation of Lane-Emden type is considered by Padé approximation. We apply these method to two examples. First differential equation of Lane-Emden type has been converted to power series by one-dimensional differential transformation, then the numerical solution of equation was put into Padé series form. Thus, we have obtained numerical solution differential equation of Lane-Emden type.

Associated Laguerre and Hermite polynomials

1984 ◽  
Vol 96 (1-2) ◽  
pp. 15-37 ◽  
Author(s):  
Richard Askey ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Padé Approximation ◽  
Hermite Polynomials ◽  
Pade Approximation ◽  
Hypergeometric Differential Equation ◽  
Orthogonality Relations

SynopsisExplicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Padé approximation to Ψ(a + 1, b; x)/Ψ(a, b; x), where Ψ(a, b; x) is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity.

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