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The solution of a singularly perturbed problem corresponds to a function with large gradients. Therefore the question of interpolation and numerical differentiation of such functions is relevant. The interpolation based on Lagrange polynomials on uniform mesh is widely applied. However, it is known that the use of such interpolation for the function with large gradients leads to estimates that are not uniform with respect to the perturbation parameter and therefore leads to errors of order O(1). To obtain the estimates that are uniform with respect to the perturbation parameter, we can use the polynomial interpolation on a fitted mesh like the piecewise-uniform Shishkin mesh or we can construct on uniform mesh the interpolation formula that is exact on the boundary layer components. In this paper the numerical differentiation formulas for functions with large gradients based on the interpolation formulas on the uniform mesh, which were proposed by A.I. Zadorin, are investigated. The formulas for the first and the second derivatives of the function with two or three interpolation nodes are considered. Error estimates that are uniform with respect to the perturbation parameter are obtained in the particular cases. The numerical results validating the theoretical estimates are discussed. [ABSTRACT FROM AUTHOR]

His paper investigated the influence of frequency intensity parameters on the controllability of structures with closely-spaced natural frequencies. The perturbation equations were established for a closed-loop system to study the relationship between frequencies, mode damping ratios and perturbation parameters. The calculation results showed that the damping ratios of two modes are decreasing as the decreasing of intensity parameter values of the structure with closely spaced natural frequencies, indicating that the controllability of corresponding modes was losing. The uncertainties in mass and stiffness matrices were also discussed in this paper. [ABSTRACT FROM AUTHOR]

This presentation is meant as a survey of recent papers which exactly evaluate a well-known functional formulation of Non-Abelian QCD, as well as an exact definition of its Renormalization that permits simple approximations to be used for the calculation of high-energy particle reactions. [ABSTRACT FROM AUTHOR]

In this paper, an approximate analytical solution of fractional 2D Navier-Stokes equation (NSE) is obtained using homotopy-perturbation method (HPM). The fractional 2D NSE can be represented in the form of fractional Riccati differential equation (FRDE). Then the FRDE is solved by HPM. The HPM solution is almost equal to the exact solution when a=1. A numerical example is given to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new iterative method for solving initial problem for linear and nonlinear partial differential equations is presented. Convergence analysis of iterative process is included. The proposed iterative scheme does not require any discretization, linearization or small perturbation and, compared to e.g. Adomian decomposition method or homotopy perturbation method, there is no need for calculation of generally n-dimensional integrals or derivatives, respectivelly. The approximate solution is calculated in the form of a sequence with easily computable terms. Depending on the form of the sequence it is possible to identify exact solution. Illustrative example is presented to demonstrate reliability and performance of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy perturbation method (HPM) combined with Padé approximant is applied to generate the approximate analytical solution of Korteweg de-Vries (KdV) equation. The results show that the combined method gives highly accurate approximate analytical solution. It is also found that this method can achieve more suitable results in comparison to HPM alone. The approximate solution is also compared with the exact solution and it is found to be in good agreement. [ABSTRACT FROM AUTHOR]

Numerical methods of successive approximations can find the solution of single nonlinear equations. These includes numerical-iterative and hybrids methods such as Newton's method, Secant method, Muller's method, x = g(x) method and Regula Falsi method. One of the recent studies is the higher order homotopy Taylor-perturbation methods via start-system (HHTPss) that have shown promising results. Theoretical considerations such as stability, consistency and convergence have yet being given much attention. Thus, this paper considers the theoretical aspects of the HHTPss numerical schemes used to solve the nonlinear equations. Theoretical and numerical results obtained relating to the stability, consistency and convergence of the schemes are verified by data from computational experiments, Maple14. [ABSTRACT FROM AUTHOR]

In previous studies, the effectiveness of the second-order quarter-sweep finite difference approximation equations has been shown in solving singularly perturbed boundary value problems. In this paper, however, we investigate the application of the fourth-order quarter-sweep finite difference approximation equation based on the fourth-order standard central difference scheme. To solve the problems numerically, discretization of the singularly perturbed problems via second-order and fourth-order finite difference schemes is proposed to form the corresponding system of linear algebraic equations. For comparison purpose, we also discuss on how to derive the basic formulation and implementation for the family of Successive Over-Relaxation (SOR) iterative methods such as FSSOR, HSSOR and QSSOR in solving the corresponding linear systems generated from the fourth-order discretization schemes based on full, half- and quarter-sweep cases. Some numerical tests were conducted to show that the accuracy of fourth-order finite difference schemes via the corresponding GS methods is more accurate than second-order schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some results concerning viability of nonlinear reaction-diffusion systems governed by multi-valued perturbations of m—dissipative operators. [ABSTRACT FROM AUTHOR]

In this paper we show how recent results concerning the stability of semi-coercive variational inequalities on reflexive Banach spaces, obtained by the authors in [3] can be applied to establish the existence of an elastic equilibrium to any small uniform perturbation of statical loads in frictional unilateral linear elasticity. The Fenchel duality is one of the key techniques that we use. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this paper optimal homotopy asymptotic method (OHAM) with the help of Caputo derivative is used for time fractional Sharma-Tasso-Olever equation. Obtained results are compared with HPM and ADM. Numerical results shows that optimal homotopy asymptotic method gives more accurate solution than that of HPM and ADM even at 1st order approximation. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new optimal homotopy approach, namely the Optimal Homotopy Perturbation Method (OHPM), to thin film flow of a fourth grade fluid down a vertical cylinder. This approach does not depend upon any small or large parameters contradistinguishing from other perturbation methods. The main advantage of this procedure consists in providing a convenient way to control the convergence of approximation series in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solution, which proves that OHPM is very efficient and accurate. [ABSTRACT FROM AUTHOR]

We consider a sufficiently regular bounded open connected subset Ω of Rn such that 0εΩ and such that Rn/clΩ is connected. Then we choose a point wε]0, 1 [n. If e is a small positive real number, then we define the periodically perforated domain T(ε)≡Rn/∪zεZncl(w+εΩ+z). For each small positive ε, we introduce a particular Dirichlet problem for the Laplace operator in the set T(ε). More precisely, we consider a Dirichlet condition on the boundary of the set w+εΩ, and we denote the unique periodic solution of this problem by u[ε]. Then we show that (suitable restrictions of) u[ε] can be continued real analytically in the parameter ε around ε = 0. [ABSTRACT FROM AUTHOR]

We review some aspects of the backreaction problem. In a first part, using an averaging scheme generalized to arbitrary frames, we quantify the effect in the Poisson gauge of perturbation theory up to second order. The main result is the definition of a scale of homogeneity above which the backreaction effect becomes scale invariant. In a second part we present results about the structure of the effective lightcone in averaged cosmologies and show that a non-trivial structure due to the backreaction may allow to test the relevance of the backreaction effect as a solution to the dark energy conundrum with future data. [ABSTRACT FROM AUTHOR]

This Formulation [1], [2], [3] is New, in the sense that it is less than 3 years old. But it could have been done decades ago, since the input information existed, but was overlooked. It is Analytic in the sense that physically-reasonable approximations can be estimated with paper and pencil; and exact amplitudes can be calculated as Meijer G-functions of various orders. It is Non-Perturbative in the sense that sums over all possible gluon exchanges between any pair of quarks and/or antiquarks, including cubic and quartic gluon interactions, are exactly performed. These multiple gluon exchanges combine into "Gluon Bundles" (GBs), as sums over Feynman graphs with finite numbers of exchanged gluons are replaced by "Bundle Graphs". In effect, gluons disappear from the formalism, and GBs remain as the effective carrier of all interactions between quark lines. A simple re-arrangement of the Schwinger/Symanzik functional solution for the Generating Functional of QCD - a rearrangement possible in QCD but not in QED - produces a formal statement of Gauge-Invariance, even though the formulation contains gauge-dependent gluon propagators. After the non-perturbative sums produce GBs, one sees explicit cancelation of all gauge-dependent gluon propagators; gauge-invariance is achieved as gauge-independence. A new insight into Realistic QCD appears in the non-perturbative domain, because quarks do not have individual asymptotic states; they are always asymptotically bound, and their transverse coordinates cannot, in principle, be measured exactly. "Transverse Imprecision" is introduced into the basic Lagrangian, and quark-binding potentials for the construction of mesons and nucleons can then be defined and evaluated. And the greatest surprise of all: A new, non-perturbative property appears, called Effective Locality, with the result that all functional integrals reduce to (a few) sets of ordinary integrals, easy to estimate approximately, or calculate on a desk-top computer. A fantastic simplification! With the aid of the above techniques, this presentation will describe each of the above paragraphs, with explicit descriptions of gauge-invariance, of the summation of all relevant Feynman graphs, of quark-binding into hadrons, and of nucleon-nucleon binding to form a deuteron. To our knowledge, the latter calculation is the first analytic example of Nuclear Physics obtained directly from basic QCD. [ABSTRACT FROM AUTHOR]

The main aim of this work is to verify the influence of the response functions choice in the stochastic perturbation-based iterative approach on the probabilistic moments and coefficients of the steel grillage structures. The second purpose is to compare their two different types, orthogonal and diagrid, in terms of the time-independent reliability resulting from an uncertainty in steel Young modulus. We have considered different performance functions concerning the basic eigenfrequency, the global extreme vertical deflection and also local deformation. The values of the reliability indices calculated here can be of further assistance in the reliability assessment by comparing the minimal one with its counterpart given in Eurocode depending upon the reliability class, reference period and type of the given limit state. [ABSTRACT FROM AUTHOR]

We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the nonlinear Schrödinger equation perturbed simultaneously by linear and nonlinear gain/loss terms. We derive the corresponding perturbed complex Toda chain in the case of a combination of linear, cubic, and/or quintic terms. We show that the soliton interactions dynamics for this reduced PCTC model compares favorably to full numerical results of the original perturbed nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

Accurate Modeling of the nonlinearity in the field of an acoustic transducer is of high value in the field of NDT. However, due to the generation mechanism, developing exact models in 3D can be difficult. Here, by way of a standard nonlinear perturbation assumption, we propose an approximation to the problem in which we ignore the cross-coupling interaction terms, resulting in a much simpler and quicker calculation. This model is tested in both two and three dimensions, with promising results. [ABSTRACT FROM AUTHOR]

This work deals with the data completion for a multivariate function whose values are given on the nodes, except a few ones, of an orthonormal hyperprismatic grid. The method we develop here uses a recently developed scheme we call "Combined Small Scale High Dimensional Model Representation (CSSHDMR)". This scheme is preferred to be used mostly at constant or perhaps univariate level bringing some discontinuities at the borders of subregions in the function values or its derivatives or both. In the case where certain function values or derivatives are unknown, these discontinuities can be suppressed to get those unknown values. This is done via optimisation. We try to present some details of this issue in this extended abstract. [ABSTRACT FROM AUTHOR]

The equilibrium state of a system consisting of a large number of strongly interacting electrons can be characterized by its density operator. This gives a direct access to the ground-state energy or, at finite temperatures, to the free energy of the system as well as to other static physical quantities. Elementary excitations of the system, on the other hand, are described within the language of Green's functions, i.e. time- or frequency-dependent dynamic quantities which give a direct access to the linear response of the system subjected to a weak time-dependent external perturbation. A typical example is angle-revolved photoemission spectroscopy which is linked to the single-electron Green's function. Since usually both, the static as well as the dynamic physical quantities, cannot be obtained exactly for lattice fermion models like the Hubbard model, one has to resort to approximations. Opposed to more ad hoc treatments, variational principles promise to provide consistent and controlled approximations. Here, the Ritz principle and a generalized version of the Ritz principle at finite temperatures for the static case on the one hand and a dynamical variational principle for the single-electron Green's function or the self-energy on the other hand are introduced, discussed in detail and compared to each other to show up conceptual similarities and differences. In particular, the construction recipe for non-perturbative dynamic approximations is taken over from the construction of static mean-field theory based on the generalized Ritz principle. Within the two different frameworks, it is shown which types of approximations are accessible, and their respective weaknesses and strengths are worked out. Static Hartree-Fock theory as well as dynamical mean-field theory are found as the prototypical approximations. [ABSTRACT FROM AUTHOR]

We give a survey over the efforts in the direction of solving the Schrödinger equation by using piecewise approximations of the potential function. Two types of approximating potentials have been considered in the literature, that is piecewise constant and piecewise linear functions. For polynomials of higher degree the approximating problem is not so easy to integrate analytically. This obstacle can be circumvented by using a perturbative approach to construct the solution of the approximating problem, leading to the so-called piecewise perturbation methods (PPM). We discuss the construction of a PPM in its most convenient form for applications and show that different PPM versions (CPM,LPM) are in fact equivalent. [ABSTRACT FROM AUTHOR]

Lamb waves are frequently used in nondestructive evaluation and are proposed for structural health monitoring applications. While many numerical methods to obtain dispersion curves for Lamb waves have been developed and implemented, in general they are time consuming and do not lend themselves to analyses involving the perturbation of parameters such as the small thickness and wave speed changes caused by temperature variations. Presented here are two methods of approximating the solutions of the Rayleigh-Lamb equations under a small perturbation assumption. The first method is a gradient-based linear approximation to the Raleigh-Lamb equations, and the second is a linear approximation method based on a number of pre-computed solutions to the Rayleigh-Lamb equations. Also presented is a simple algorithm to efficiently compute Lamb wave dispersion curves, which is used to study the approximation methods. [ABSTRACT FROM AUTHOR]

A two-grid algorithm for FEM solution of nonlinear reaction diffusion equations was constructed. The algorithm involves solving one small, nonlinear problem on the coarse-grid system, one-linear problem on the fine-grid system. Three meshes (Shishkin’s, Bakhvalov’s and Vulanovic’s) are examined. All of the approximations are uniformly convergent with respect to the small parameter. We prove that when the mesh step parameters are chosen appropriately, the error in the two-grid algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We also provide an analysis of numerical experiments. [ABSTRACT FROM AUTHOR]

Minimal cut sets (MCSs) have been introduced to define perturbations in a metabolic network needed to repress the activity of a given target reaction. While this approach gives interesting insight into the structure of metabolic networks, it suffers from two major drawbacks that are limiting in practice. First, using MCSs becomes inadequate when we are interested in inhibiting only one direction of a reversible target reaction. Second, and more importantly, the computation of MCSs by currently available methods is limited by the combinatorial explosion of the number of elementary modes and hence is not practical for genomescale networks. To overcome these problems, we introduce the concept of minimal direction cuts. A minimal forward (resp. backward) direction cut (MFC) (resp. MBC) is a minimal set of reactions whose removal prevents the target reaction from carrying a flux in the forward (resp. backward) direction. Computing MFCs and MBCs does not require to determine beforehand the elementary modes of the network. Instead it can be based on the Farkas lemma for equality and inequality constraints. This leads to a new algorithm to identify minimal direction cuts, which can also be used for a direct computation of MCSs. [ABSTRACT FROM AUTHOR]

We sketch the outlines of a perturbation scheme for the kick-lattice map in two degrees of freedom (4-d phase space) and apply it to the weak-strong beam-beam interaction. Our approach yields rigorous error bounds on the precision of our approximations (proofs will appear elsewhere), divides tune space into distinct regions where different approximations are valid, clarifies the dynamics near certain generically nonintegrable resonances, and generalizes to higher degrees of freedom and more complex models such as the strong-strong beam-beam. © 2003 American Institute of Physics [ABSTRACT FROM AUTHOR]