Method Of Orthogonal Polynomials Research Articles

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.

ON THE UNIFORM CONVERGENCE OF THE APPROXIMATE SOLUTION OF SINGULAR INTEGRO-DIFFERENTIAL EQUATION OF THE FIRST KIND

The article presents the study of the boundary problem for singular inte-gro-differential equation of the first kind with Cauchy kernel. The authors introduce the pair of weight spaces to prove the correctness of the stated problem. The article states the sufficient conditions for the convergence of the general direct method, the method of orthogonal polynomials, and as a result uniform estimates for errors of approximate solution.

PENERAPAN METODE PERMUKAAN RESPONS DALAM MASALAH OPTIMALISASI

The experiments were conducted to determine the relationship between response variable and independent variable and the effect of independent variables to response variable. Multiple comparison tests can be used to find the level that makes optimal response. However, this test has a drawback that it only finds the optimal level tested at the level being considered. Thus test is more suitable for qualitative independent variables. On the other hand, for quantitative independent variables we can use the orthogonal polynomial method and response surface method (RSM).The purpose of this research is to apply RSM to produce optimal response. It can be concluded that RSM is an efficient used to find the level of independent variables to makes optimal response.

Structure combinatorics and thermodynamics of a matrix model with Penner interaction inspired by interacting RNA

In this manuscript, we study the logarithmic Penner type nonlinear interaction in the random matrix model for interacting RNA folding and structure combinatorics. The Penner interaction originally appeared in the studies of moduli space of punctured surfaces and has been applied here in the context of RNA folding for the first time. An exact analytic formula for the generating function is derived using the orthogonal polynomial method. The partition function for a given length L of the RNA chain, derived from the generating function enumerates all possible topological structures as well as the pairings. The partition function and the asymptotic large length distribution functions are found and show a change in the critical exponent of the secondary structure contribution from L−3/2 for large N (size of matrix, N>L) to L−1/2 for small N (N≪L). The exact analytic results calculated in the proposed model allow evaluation of the specific heat versus T curve for large interaction strength. In particular, the second derivative of specific heat shows a striking behavior, changing from single peaked function for large N to a double peak for small N.

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size N × N with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n = 1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n > 1 and generalize the known Bessel law in the complex plane for n = 2 to a new hypergeometric kernel 0Fn − 1.

Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.

Guided waves in multilayered hollow cylinders: The improved Legendre polynomial method

Legendre polynomial series method has been proposed to solve the wave propagation in multilayered flat plates for more than 10years. But it has never been used for curved waveguides. This method has intrinsic limitations: it can deal only with low contrast multilayered structures and is unable to restitute correct continuous normal stress distributions. This paper proposes an improvement of the Legendre polynomial method to overcome these drawbacks and uses it to solve the guided wave propagation in general multilayered hollow cylinders i.e. with or without very dissimilar material properties. Detailed formulations are given to show the differences between the improved orthogonal polynomial method and the conventional one. Through a numerical comparison among the exact solution (from the transfer matrix method), the improved polynomial approach and the conventional polynomial approach, the validity of the improved polynomial approach is illustrated. Then, the flexural guided wave dispersion curves and stress distributions are analyzed. The influences of the flexural orders and of the radius to thickness ratio on flexural longitudinal wave and flexural torsional wave are discussed.

Weight functions of a crack in a two-dimensional micropolar solid

Summary Two models for a crack in an unbounded two-dimensional micropolar solid are analysed. The first one concerns plane-strain deformation, while the second problem assumes the conditions of antiplane-strain loading. In both cases, one of three modes is uncoupled, while the other two are coupled. For a semi-infinite crack, the models reduce to a scalar and an order-2 vector Riemann– Hilbert problems with a Hermitian matrix coefficient. In the plane-strain case, the problem is solved exactly, and the weight functions are derived by quadratures. In the antiplane-strain case, the matrix coefficient cannot be factorized by the methods available in the literature. For a finite antiplane-strain crack, an approximate solution is obtained by the method of orthogonal polynomials. The weight functions are found in a series form through the solution of an infinite system of linear algebraic equations. In both cases, the plane-strain and antiplane-strain micropolar theory, it is shown that if a certain micropolar parameter (the same for both theories) tends to zero, then the solution and the weight functions tend to those of classical elasticity. Numerical results for the weight functions are reported.

Axisymmetric contact problems for a prestressed incompressible elastic layer

Two axisymmetric problems of the indentation without friction of an elastic punch into the upper face of a layer when there is a uniform field of initial stresses in the layer are considered. The model of an isotropic incompressible non-linearly elastic material, specified by a Mooney potential, is used. Two cases are investigated: when the lower face of the layer is rigidly clamped after it is prestressed, and when the lower face of the layer is supported on a rigid base without friction after it is prestressed. It is assumed that the additional stresses due to the action of the punch on the layer are small compared with the initial stresses; this enables the problem of determining the additional stresses to be linearized. The problem is reduced to solving integral equations of the first kind with symmetrical irregular kernels relative to the pressure in the contact area. Approximate solutions of the integral equations are constructed by the method of orthogonal polynomials for large values of the parameter characterizing the relative layer thickness. The case of a punch with a plane base is considered as an example.

Study on Useful Concrete Elastic-Plastic Constitutive Model

In order to simulate the elastic-plastic behavior of concrete better, an effective curve fitting method is employed to find the constitutive function based on Legendre orthogonal polynomial and least square method. And a new numerical analysis program using arch-length method to deal with nonlinear problem is designed. The most important consequence is that not only the fitting function, but also its derivative function and its definite integral are recursive. The fitting curve given by new method is applied in numerical analyzing. Cube model is examined under uniaxial compressing with completely restricted boundary conditions. The results show convex deformation and the damage on oblique section clearly. Compared with those piecewise fitting functions, the Legendre orthogonal polynomial fitting function obtained can be introduced into the nonlinear harden-soften character of concrete constitute law more convenient because of its uniform function form and continuous derived feature. And the fitting idea by orthogonal base function will provide a widely road for studying the constitute law of concrete material.

Practical Constitutive Model in the Application of Arch Dam Shape Design Calculation

A fitting curve of stress-strain relation given is applied in numerical analyzing based on Legendre orthogonal polynomial and least square method. The nonlinear constitutive model is employed to examine the shape of an arch dam. For the asymmetries and high water pressure, the safety of High Arch Dam is one of key issues in the design and construction of High Arch Dam. The simulation results show that the basic mechanical performance index is symmetrical distribution. The dam mainly bears pressure and no obvious tensile stress exists in the dam when water level is normal. Material yields in dam heel when 1.4 times overloading is considered. It can be concluded that the dam is reliable and the safety margin is satisfactory.

The solution of system of integral differential equations and its application in the theory of elasticity

AbstractA contact problem of the theory of elasticity of infinite compound and infinite orthotropic plates with an elastic semi‐infinite or finite inclusion of variable rigidity is considered. The problem is reduced to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, using the methods of analytical functions to get exact solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion. In some case using the method of orthogonal polynomials we obtain the dual infinite system of linear algebraic equations. We can manage to investigate the obtained system on the quasi‐regularity and the method of reduction for approximate solution is developed.

Influence of a circular crack on the electromagnetic field in a conductor

The problem of scattering of low-frequency electromagnetic fields on circular cracks is reduced by the method of small parameter to a system of hypersingular integral equations of the Newton-potential type. For the solution of these equations, we propose to use the method of orthogonal polynomials of two variables. It is shown that the electric dipole whose moment is proportional to the normal component of the sounding field is an electrodynamic analog of the crack.

A simple derivation of the Tracy–Widom distribution of the maximal eigenvalue of aGaussian unitary random matrix

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue ofan (N × N) random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian unitaryensemble (GUE) and by suitably adapting a method of orthogonal polynomials developedby Gross and Matytsin in the context of Yang–Mills theory in two dimensions, we provide arather simple derivation of the Tracy–Widom law for GUE. Our derivation is based on theelementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. Asan added bonus, this method also allows us to compute the precise subleading termsdescribing the right large deviation tail of the maximal eigenvalue distribution. In theYang–Mills language, these subleading terms correspond to non-perturbative (in1/N expansion) corrections to the two-dimensional partition function in the so called ‘weak’coupling regime.

Effect of different proportions of exotic germplasm on grain yield and grain moisture in maize

Two main questions in programs introgressing exotic maize germplasm into temperate materials are the choice of available exotic sources to work with, and the proportion of exotic germplasm that should be incorporated into adapted germplasm. The objective of this study was to compare effects of different proportions of tropical maize inbred line NC298 in hybrids male parent on grain yield and grain moisture content, using method of orthogonal polynomials. Methods of direct crosses and backcrosses were used to form four hybrid groups (six hybrids each) containing 0, 25, 50 and 75 percent of tropical exotic germplasm, respectively, by their male parent (or one half of mentioned proportions in the corresponding hybrids). The linear components of the germplasm proportions sum of squares, were significant (p<0.01) for both grain yield and moisture content and the cubic effect (p<0.05) for grain yield only. Results in this study clearly suggested backcross foundation populations with an adapted line to be appropriate selection sources for both grain yield and grain moisture content.

Variational Capacitance Extraction and Modeling Based on Orthogonal Polynomial Method

In this paper, we propose a novel statistical capacitance extraction method for interconnect conductors considering process variations. The new method is called statCap, where orthogonal polynomials are used to represent the statistical processes in a deterministic way. We first show how the variational potential coefficient matrix is represented in a first-order form using Taylor expansion and orthogonal decomposition. Then, an augmented potential coefficient matrix, which consists of the coefficients of the polynomials, is derived. After this, corresponding augmented system is solved to obtain the variational capacitance values in the orthogonal polynomial form. Finally, we present a method to extend statCap to the second-order form to give more accurate results without loss of efficiency compared to the linear models. We show the derivation of the analytic second-order orthogonal polynomials for the variational capacitance integral equations. Experimental results show that statCap is two orders of magnitude faster than the recently proposed statistical capacitance extraction method based on the spectral stochastic collocation approach and many orders of magnitude faster than the Monte Carlo method for several practical conductor structures.

Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.

Influence of the Graphite Absorber During Laser Surface Hardening

The paper presents the results of the research of a laser surface hardening process applied to the C45E steel with graphite absorber coating on specimen surfaces. The evaluation of the laser surface hardening process was performed by measuring the IR radiation from the interaction spot. The results confirmed a strong correlation between the IR radiation voltage signal and the dimensions of microstructural changes occuring in the laser surface hardening steel. Changes of the absorbing coating thickness produced changes of the absorptivity of the laser beam, which was confirmed by measurements of the depth and width of hardened traces. For laser surface hardening the optimum coating thickness of a graphite absorber was determined. A factorial analysis and a method of orthogonal polynomials were used to determine response surfaces for the voltage signal of IR radiation the depth and width of the hardened trace achieved. Absorptivity difference between various graphite absorber coating thicknesses was evaluated based on the Ashby-Shercliff model.

The Solution of Mitchell's Problem for the Elastic Infinite Cone with a Spherical Crack

The new problem about the stress concentration around a spherical crack inside of an elastic cone is solved for the point tensile force enclosed to a cone's edge. The constructed discontinuous solutions of the equilibrium equations have allowed to express the displacements and stress in a cone through their jumps and the jumps of their normal derivatives across the crack's surface. The application of the integral transformation method under the generalized scheme has reduced the problem solving to the solving of the integrodifferential equation system with regard to the displacements' jumps. This system was solved approximately by the orthogonal polynomial method. The use of this method has allowed to take into consideration the order of the solution's singularities at the ends of an integral interval. The correlation between the crack's geometrical parameters, its distance from an edge, and the SIF values is established after the numerical analysis. The limit of the proposed method applicability is specified.

The wave field generated by a centre of rotation in an unbounded elastic medium with a semi-infinite conical crack

The dynamic stress intensity factor at the edge of a semi-infinite conical crack when the medium is loaded by a non-stationary centre of rotation is determined. A centre of rotation is understood to be a set of four forces of equal magnitude that act in the same plane and form pairs having the same direction of rotation. 1 If the magnitude of these forces is time dependent, i.e., their application is non-stationary, they form a non-stationary centre of rotation. The solution of the problem required the use of methods of integral transformations and discontinuous solutions, which reduced the problem to an integral differential equation in Laplace transform space. The combined use of the orthogonal polynomial method and time discretization to solve the equation enabled a formula for the stress intensity factor to be obtained.