Renormalization Group Method and Reductive Perturbation Method

It is shown that the renormalization group method does not necessarily eliminate all secular terms in perturbation series to partial differential equations and a functional subspace of renormalizable secular solutions corresponds to a choice of scales of independent variables in the reductive perturbation method.Comment: 5 pages, late

Anharmonic oscillators energies via artificial perturbation method

A new pseudoperturbative (artificial in nature) methodical proposal [15] is used to solve for Schrodinger equation with a class of phenomenologically useful and methodically challenging anharmonice oscillator potentials V(q)=\alpha_o q^2 + \alpha q^4. The effect of the [4,5] Pade' approximant on the leading eigenenergy term is studied. Comparison with results from numerical (exact) and several eligible (approximation) methods is made.Comment: 22 pages, Latex file, to appear in the Eur. Phys. J.

Testing the heating method with perturbation theory

The renormalization constants present in the lattice evaluation of the topological susceptibility can be non-perturbatively calculated by using the so-called heating method. We test this method for the $O(3)$ non-linear $\sigma$-model in two dimensions. We work in a regime where perturbative calculations are exact and useful to check the values obtained from the heating method. The result of the test is positive and it clarifies some features concerning the method. Our procedure also allows a rather accurate determination of the first perturbative coefficients.Comment: 15 pages, LaTeX file, needs RevTeX style. Tarred, gzipped, uuencode

Modeling of micro flows using perturbation method

This paper was presented at the 3rd Micro and Nano Flows Conference (MNF2011), which was held at the Makedonia Palace Hotel, Thessaloniki in Greece. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, Aristotle University of Thessaloniki, University of Thessaly, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute.A new method for modeling micro flows is presented in this research. The basis of this method is the development of governing continuum equations on fluid dynamics using perturbation expansion of the velocity, pressure, density and temperature fields in dependence of Knudsen number. In the present work, we use three-term perturbation expansions and reach three order of equations O(1), O(Kn), O(Kn2). Required boundary conditions (BC) for solving each order of these equations are obtained by substitution of the perturbation expansions into the general boundary conditions for the velocity slip and temperature jump. This set of equations is discretized in two-dimensional state on a staggered grid using the finite volume method. A three-part computer program has been produced for solving the set of discretized equations. Each part of this code, solve one order of the equations with the SIMPLE algorithm. Incompressible slip micro Poiseuille and micro Couette flows are solved either analytically or numerically using the perturbation method (PM). Good agreement is found between analytical and numerical results in the low Knudsen numbers, whereas numerical results deviate from analytical results by increasing the Knudsen number. The results of perturbation method are also compared with the results obtained from different slip models

A hybrid-perturbation-Galerkin technique which combines multiple expansions

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is parameter in the problem formulation and that a perturbation method can be sued to construct one or more expansions in this perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes which replace and improve upon the gauge functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied separately, while combining some of their better features. The proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem and a nonlinear free oscillation problem. The results obtained from the hybrid methods are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed