A new perturbative approach to nonlinear problems

Bender, Carl M.; Milton, Kimball A.; Pinsky, Stephen S.; Simmons, L. M.

doi:10.1063/1.528326

Cited by 290 publications

(198 citation statements)

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“…In particular, the physical radius of a polytrope is given by the first zero of the associated polytrope function as shown in equation (11). It is of physical interest to determine the first zero as a function of the polytropic index n. A lot of studies on analytical approximants of the LEE have been performed using various techniques, including series expansion methods (see, e.g., Seidov & Kuzakhmedov 1977;Hunter 2001;Pascual 1977;Iacono & De Felice 2015), perturbation methods (see, e.g., Bender et al 1989;Seidov 2004), and empirical interpolation schemes (see, e.g., Liu 1996). Here, we review in depth the DEM of the LEE (Bender et al 1989;Seidov 2004), because it is one of the foundations of the SDEM, which we are going to propose in Section 4.…”

Section: Delta Expansion Methodsmentioning

confidence: 99%

“…In this regard, the normalised polytrope functionθ (x) and its associated radiusξ are not particularly superior to other solutions to the LEE. In the following discussion, we shall make use of such a symmetry to remedy the singularity problem encountered in the DEM considered by Bender et al (1989) and Seidov (2004).…”

Section: The Lane-emden Equationmentioning

confidence: 99%

“…In Section 2, we derive the LEE from the assumptions of hydrostatic equilibrium and the polytropic equation of state, with emphasis on the relationship between the freedom of the choice of the density scale and the scale invariance symmetry of the LEE. Next, in Section 3, we review the applications of the DEM, as proposed by Bender et al (1989) and Seidov (2004), to the LEE. In Section 4, we introduce the SDEM.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we adopt the DEM proposed by Bender et al (1989) and Seidov (2004) to handle the nonlinear nature of the LEE. In order to remedy the singular behaviour of the polytrope function outside the unperturbed polytrope, we propose to scale the radius of polytropes in such a way that both the perturbed and unperturbed polytropes share a common radius of π in the new length scale.…”

Section: Introductionmentioning

confidence: 99%

“…The LEE is a nonlinear differential equation except for n = 0 and 1. To generate analytical approximate solutions to the LEE, Bender et al (1989) applied the delta expansion method (DEM), which was proposed to solve a wide range of non-linear problems through expanding the non-linear term in a power series of its exponent, and succeeded in expanding the polytrope function into a power series of n − 1 about the exact solution at n = 1. As LEE admits a closed form solution at n = 0 as well, Seidov (2004) considered the DEM about the incompressible limit where n = 0, which leads to the expansion of the polytrope function into powers of n. However, both of these two attempts encountered a common problem, namely the expansion of the perturbed polytrope function becomes singular at the surface of the unperturbed polytrope where the unperturbed polytrope function vanishes.…”

Section: Introductionmentioning

confidence: 99%

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Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Yip

Chan

Leung

2016

Mon. Not. R. Astron. Soc.

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Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index n. Here we use the known exact solutions of the LEE at n = 0 and 1 to solve the equation perturbatively. We first introduce a scaled LEE (SLEE) where polytropes with different polytropic indices all share a common scaled radius. The SLEE is then solved perturbatively as an eigenvalue problem. Analytical approximants of the polytrope function, the radius and the mass of polytropes as a function of n are derived. The approximant of the polytrope function is well-defined and uniformly accurate from the origin down to the surface of a polytrope. The percentage errors of the radius and the mass are bounded by 8.1 × 10 −7 per cent and 8.5 × 10 −5 per cent, respectively, for n ∈ [0, 1]. Even for n ∈ [1, 5), both percentage errors are still less than 2 per cent.

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Section: Delta Expansion Methodsmentioning

confidence: 99%

Section: The Lane-emden Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Yip

Chan

Leung

2016

Mon. Not. R. Astron. Soc.

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Kinetic energy functional derivative for the Thomas-Fermi atom inD dimensions

March

Kais

1997

Int. J. Quant. Chem.

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The self-consistent Thomas᎐Fermi atom satisfying Poisson's equation in D dimensions has a functional derivative of the kinetic energy T with respect to the Ž . 2rD 1y2rD ground-state density n r proportional to n . But the Poisson equation relates n to ''reduced'' density derivatives n y1Ž d 2 n r d r 2 .. Thus ␦ Tr␦ n can be written also, quite compactly, solely in terms of these derivatives. An analytic solution to the Thomas᎐Fermi equation in D dimensions can be presented as an expansion about the known analytic solution at D s 2.

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Analytical and numerical investigations of spontaneous imbibition in porous media

Nooruddin

Blunt

2016

Water Resources Research

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We present semianalytical solutions for cocurrent displacements with some degree of countercurrent flow. The solution assumes a one-dimensional horizontal displacement of two immiscible incompressible fluids with arbitrary viscosities and saturation-dependent relative permeability and capillary pressures. We address the impact of the system length on the degree of countercurrent flow when there is no pressure drop in the nonwetting phase across the system, assuming negligible capillary back pressure at the inlet boundary of the system. It is shown that in such displacements, the fractional flow can be used to determine a critical water saturation, from which regions of both cocurrent and countercurrent flow are identified. This critical saturation changes with time as the saturation front moves into the porous medium. Furthermore, the saturation profile in the approach presented here is not necessarily a function of distance divided by the square root of time. We also present approximate solutions using a perturbative approach, which is valid for a wide range of flow conditions. This approach requires less computational power and is much easier to implement than the implicit integral solutions used in previous work. Finally, a comprehensive comparison between analytical and numerical solutions is presented. Numerical computations are performed using traditional finite-difference formulations and convergence analysis shows a generally slow convergence rate for water imbibition rates and saturation profiles. This suggests that most coarsely gridded simulations give a poor estimate of imbibition rates, while demonstrating the value of these analytical solutions as benchmarks for numerical studies, complementing Buckley-Leverett analysis.

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A new perturbative approach to nonlinear problems

Cited by 290 publications

References 10 publications

Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Perturbative solution to the Lane–Emden equation: an eigenvalue approach

Kinetic energy functional derivative for the Thomas-Fermi atom inD dimensions

Analytical and numerical investigations of spontaneous imbibition in porous media

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