Generalized Polynomial Approximation Research Articles

Unsteady MHD Free Convection Flow of Fluid in Micro Channel in the Presence of Viscous Dissipation Between Vertical Parallel Two Walls

Convection is a process by which heat is transferred by movement of a heated water. The phenomenon where heat transfer process is driven by natural fluid motion due to temperature and density gradients is called free convection. The effect of slip velocity on thermal behaviours of MHD free convection flow over a vertical parallel two walls is the focus of this research. The mathematical equations which described the phenomenom includes momentum, magnetic field and energy equations. Condition for the existence and uniqueness of solution of the model equations were established using the approach of Lipschitz continuity. The model equations in dimensionless form were solved using Olayiwola’s generalized polynomial approximation method (OGPAM). The results obtained which contains the dimensionless parameters were presented graphically and discussed. It was observed that increase in Reynolds number and Knudsen number both enhanced the velocity distribution at the two walls. Also, magnetic field and temperature field were enhanced by magnetic prandtl number and Eckert number respectively.

Generalized polynomial approximation and growth estimates of entire functions of two complex variables in Banach spaces

Infinite dimensional subspaces, which consists of forms involving tensor products of functions of fewer variables, play an important role in the theory of approximation of functions of several real variables. Similar technique can be applied to problems of approximation of analytic functions of several complex variables. In the present paper we study the generalized polynomial approximation of entire functions of two complex variables in Banach spaces. Moreover, limit equations relating growth parameters order and type with above approximation errors in Banach spaces have been investigated.

Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid

In this work, we discuss the problem of approximating a multivariate function by discrete least-squares projection onto a polynomial space using a specially designed deterministic point set. The independent variables of the function are assumed to be random variables, stemming from the motivating application of uncertainty quantification. Our deterministic points are inspired by a theorem due to André Weil. We first work with the Chebyshev measure and consider the approximation in Chebyshev polynomial spaces. We prove the stability and an optimal convergence estimate, provided the number of points scales quadratically with the dimension of the polynomial space. A possible application for quantifying epistemic uncertainties is then discussed. We show that the point set asymptotically equidistributes to the product-Chebyshev measure, allowing us to propose a weighted least-squares framework and extending our method to more general polynomial approximations. Numerical examples are given to confirm the theoretical results. It is shown that the performance of our deterministic points is similar to that of randomly generated points. However, our construction, being deterministic, does not suffer from probabilistic qualifiers on convergence results (e.g., convergence “with high probability'').

Extension of the operator of best polynomial approximation in [formula omitted

Let (Ω,A,μ) be a finite measure space. In this paper we extend the operator of the best generalized polynomial approximation from the space Lp(Ω) to the space Lp−1(Ω), 1<p<∞, as the unique operator preserving the property of continuity. The case p=1 is also considered.

Generalized polynomial approximation providing the best reference

We pose the extremum problem of finding a polynomial approximation for a function of several variables so that this approximation by a fragment of the function provides the best reference. We obtain an upper bound for the modulus of informativity of polynomials and rational functions.

Spectral and Pseudo-Spectral Approximations of the Navier–Stokes Equations

Galerkin and collocation approximations by trigonometric polynomials to the stationary Navier-Stokes equations with periodic boundary conditions are analyzed. Stability results and “optimal” rates of convergence for both velocity and pressure in Sobolev spaces are proven. Essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems [Brezzi, Rappag, Raviart, Numer. Math., 36 (1980), pp. 1–25], [Maday, Quarteroni, RAIRO Numer. Anal. (1980)].

Generalized polynomial approximation

We estimate the rate of covergence to functions in the spacesL p [0,1] and C[0,1] by polynomial of the form ∑ λ α λ x λ, where the λ′s are positive real numbers and 0.