Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

Bervillier, C.; Boisseau, B.; Giacomini, Héctor

doi:10.1016/j.nuclphysb.2008.02.021

Cited by 33 publications

(57 citation statements)

References 43 publications

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“…The AO with different potential configurations have been explicitly considered to illustrate the efficiency and the limitations of a quasi-analytic method for solving nonlinear ODE named the mapping method [5]. As several other quasi-analytic methods encountered here and there in the literature of the AO it is based on a generic Taylor series in powers of the independent variable.…”

Section: Discussionmentioning

confidence: 99%

“…It is not by accident that the mapping method [5] discussed in the present paper has been elaborated in the context of the derivative expansion of the exact renormalization group equation (for reviews and recent pedagogical introductions see [6]). In that field, one is faced with coupled nonlinear differential equations and quasianalytical methods have soon been introduced [7,8] to solve them because they are much more user-friendly than purely numerical methods.…”

Section: Introductionmentioning

confidence: 99%

“…Methods of that kind had previously been developped in different contexts [10,12]. Among all these methods, the mapping method seems to be particularly efficient [5]. Considering first the logarithmic derivative of the wave function (which softens the singularity at infinity), the mapping method relies upon a conformal mapping onto the unit disc of an angular sector of the complex plane of the independent variable [5].…”

Section: Introductionmentioning

confidence: 99%

“…Among all these methods, the mapping method seems to be particularly efficient [5]. Considering first the logarithmic derivative of the wave function (which softens the singularity at infinity), the mapping method relies upon a conformal mapping onto the unit disc of an angular sector of the complex plane of the independent variable [5]. The angular sector is chosen so that the solution looked for is analytic in its interior or assumed to be so.…”

Section: Introductionmentioning

confidence: 99%

“…The mapping method introduced in [5] leans also on an analytic continuation after a logarithmic-derivative-like transformation but in addition, the infinite boundary is brought close to the origin using the following conformal transformation:…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier¹

2009

J. Phys. A: Math. Theor.

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Abstract. The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential V (x) = βx 2 + x 2m (m > 0). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable x. A conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for β ≥ 0 or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as β is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier¹

2009

J. Phys. A: Math. Theor.

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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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Functional renormalization group with a compactly supported smooth regulator function

Nándori

2013

J. High Energ. Phys.

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The functional renormalization group equation with a compactly supported smooth (CSS) regulator function is considered. It is demonstrated that in an appropriate limit the CSS regulator recovers the optimized one and it has derivatives of all orders. The more generalized form of the CSS regulator is shown to reduce to all major type of regulator functions (exponential, power-law) in appropriate limits. The CSS regulator function is tested by studying the critical behavior of the bosonized two-dimensional quantum electrodynamics in the local potential approximation and the sine-Gordon scalar theory for d < 2 dimensions beyond the local potential approximation. It is shown that a similar smoothing problem in nuclear physics has already been solved by introducing the so called Salamon-Vertse potential which can be related to the CSS regulator.

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Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

Cited by 33 publications

References 43 publications

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

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

Functional renormalization group with a compactly supported smooth regulator function

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