Properties Of Bernstein Polynomials Research Articles

Bernstein collocation technique for a class of Sturm-Liouville problems

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

Structural Interval Reliability Algorithm Based on Bernstein Polynomials and燛vidence Theory

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic reliability index method, the extreme value of the limit state function is obtained using the properties of Bernstein polynomials, thus avoiding the need for a lot of sampling to solve the reliability analysis problem. The method is applied to numerical examples and engineering applications such as experiments, and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method, especially for some reliability problems with higher nonlinearity. Moreover, this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.

Evolutionary Algorithm Based Solution of Rössler Chaotic System Using Bernstein Polynomials

Chaotic systems have gained enormous research attention since the pioneering work of Lorenz. Rössler system stands among the extensively studied classical chaotic models. This paper proposes a technique based on Bernstein Polynomial Basis Function to convert the three-dimensional Rössler system of Ordinary Differential Equations (ODEs) into an error minimization problem. First, the properties of Bernstein Polynomials are applied to derive the fitness function of Rössler chaotic system. Second, in order to obtain the values of unknown Bernstein coefficients to optimize the fitness function, the problem is solved using two versatile algorithms from the family of Evolutionary Algorithms (EAs), Genetic Algorithm (GA) hybridized with Interior Point Algorithm (IPA) and Differential Algorithm (DE). For validity of the proposed technique, simulation results are provided which verify the global stability of error dynamics and provide accurate estimation of the desired parameters.

Numerical Study on Single and Multi-Dimensional Boundary Value Problems by the Method of Weighted Residual

The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.

Sliding Window Rough measurable function on $I-$ core of triple sequences of Bernstein operator

We introduce sliding window rough $I-$ core and study some basic properties of Bernstein polynomials of rough $I-$ convergent of triple sequence spaces and also study the set of all Bernstein polynomials of sliding window of rough $I-$ limits of a triple sequence spaces and relation between analytic ness and Bernstein polynomials of sliding window of rough $I-$ core of a triple sequence spaces.

How to Analyze Models of Nonlinear Public Goods

Public goods games often assume that the effect of the public good is a linear function of the number of contributions. In many cases, however, especially in biology, public goods have nonlinear effects, and nonlinear games are known to have dynamics and equilibria that can differ dramatically from linear games. Here I explain how to analyze nonlinear public goods games using the properties of Bernstein polynomials, and how to approximate the equilibria. I use mainly examples from the evolutionary game theory of cancer, but the approach can be used for a wide range of nonlinear public goods games.

A computational method for solving a class of singular boundary value problems arising in science and engineering

This note deals with a new computational method for solving a class of singular boundary value problems. The method is based upon Bernstein Polynomials. The properties of Bernstein Polynomials, together with Bernstein operational matrix for differentiation formula, are presented and utilized to reduce the given singular boundary value problems to the set of algebraic equations. The proposed method is applied to solve some test problems with the comparison between the computational solutions and the exact solutions. The results indicate that the proposed algorithm provides high reliability and good accuracy.

New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials

Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved. Another method for computing operational matrices of derivative and integration D_b and R_(n+1)^B respectively is presented. Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method.

A numerical study for off-centered stagnation flow towards a rotating disc

In this investigation, a semi-numerical method based on Bernstein polynomials for solving off-centered stagnation flow towards a rotating disc is introduced. This method expands the desired solutions in terms of a set of Bernstein polynomials over a closed interval and then makes use of the tau method to determine the expansion coefficients to construct approximate solutions. This method can satisfy boundary conditions at infinity. The properties of Bernstein polynomials are presented and are utilized to reduce the solution of governing nonlinear equations and their associated boundary conditions to the solution of algebraic equations. Graphical results are presented to investigate the influence of the rotation ratio α on the radial velocity, azimuthal velocity and the induced velocities. A comparative study with the previous results of viscous fluid flow in the literature is made.

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Bernstein polynomial basis. The properties of Bernstein polynomials are first presented. The Bernstein polynomial basis vanishes except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. These properties together with the Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

On properties of Bernstein-type operator polynomials that approximate the Urysohn operator

We generalize properties of Bernstein polynomials to Bernstein-type operator polynomials that approximate the Urysohn operator.

Variation Diminishing Properties of Bernstein Polynomials on a Tetrahedron

Goodman has recently studied certain variation diminishing properties of Bernstein polynomials on triangles. Introducing analogous definitions for the variation of a trivariate function, we study in the present paper corresponding results for the Bernstein polynomials defined on a tetrahedron. We have also extended these results to arbitrary dimension.

A property of Bernstein-Schoenberg spline operators

Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].