Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations

Abstract A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results.

[1]  Nazim I. Mahmudov.  The moments for q-Bernstein operators in the case 0 < q < 1 , 2009, Numerical Algorithms.

[2]  Rida T. Farouki,et al.  Algorithms for polynomials in Bernstein form , 1988, Comput. Aided Geom. Des..

[3]  Paul Bracken,et al.  Solutions of differential equations in a Bernstein polynomial basis , 2007 .

[4]  E. H. Doha.  Explicit Formulae for the Coefficients of Integrated Expansions of Jacobi Polynomials and Their Integrals , 2003 .

[5]  Bernard R Gelbaum,et al.  Modern real and complex analysis , 1992 .

[7]  E. E. Doha,et al.  On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations , 2002 .

[8]  Shahid S. Siddiqi,et al.  Septic spline solutions of sixth-order boundary value problems , 2008 .

[9]  Waheed K. Zahra,et al.  High order accuracy nonpolynomial spline solutions for 2muth order two point boundary value problems , 2008, Appl. Math. Comput..

[10]  Ali H. Bhrawy,et al.  A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations , 2009 .

[11]  G. Phillips.  Interpolation and Approximation by Polynomials , 2003 .

[12]  Rida T. Farouki,et al.  On the optimal stability of the Bernstein basis , 1996, Math. Comput..

[13]  George G. Lorentz,et al.  Deferred Bernstein polynomials , 1951 .

[14]  Khosrow Maleknejad,et al.  A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation , 2011 .

[15]  E. Cheney.  Introduction to approximation theory , 1966 .

[16]  Çetin Dişibüyük,et al.  A generalization of rational Bernstein–Bézier curves , 2007 .

[17]  Timothy Nigel Phillips,et al.  On the coefficients of integrated expansions of ultraspherical polynomials , 1990 .

[18]  Sofiya Ostrovska,et al.  Convergence of Generalized Bernstein Polynomials , 2002, J. Approx. Theory.

[19]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[20]  Abdul-Majid Wazwaz,et al.  The numerical solution of sixth-order boundary value problems by the modified decomposition method , 2001, Appl. Math. Comput..