Abstract
A new formula expressing explicitly the derivatives of Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves is proved, and a formula expressing the Bernstein coefficients of the general-order derivative of a differentiable function in terms of its Bernstein coefficients is deduced. An application of how to use Bernstein polynomials for solving high even-order differential equations by Bernstein Galerkin and Bernstein Petrov-Galerkin methods is described. These two methods are then tested on examples and compared with other methods. It is shown that the presented methods yield better results.
Highlights
Bernstein polynomials 1 have many useful properties, such as, the positivity, the continuity, and unity partition of the basis set over the interval 0, 1
With the advent of computer graphics, Bernstein polynomial restricted to the interval x ∈ 0, 1 becomes important in the form of Bezier curves 2
Many properties of the Bezier curves and surfaces come from the properties of the Bernstein
Summary
Bernstein polynomials 1 have many useful properties, such as, the positivity, the continuity, and unity partition of the basis set over the interval 0, 1. For spectral methods 8, 9 , explicit formulae for the expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of those of the original expansion coefficients of the function itself are needed Such formulae are available for expansions in Chebyshev , Legendre , ultraspherical , Hermite , Jacobi , and Laguerre polynomials. These polynomials have been used in both the solution of boundary value problems 16–19 and in computational fluid dynamics 8 In most of these applications, use is made of formulae relating the expansion coefficients of derivatives appearing in the differential equation to those of the function itself, see, e.g., 16–19.
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