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

AbstractA 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

Legendre–Gauss–Lobatto collocation method for solving multi-dimensional Fredholm integral equations

This paper reports a new spectral collocation technique for solving second kind Fredholm integral equations (FIEs). We develop a collocation scheme to approximate FIEs by means of the shifted Legendre–Gauss–Lobatto collocation (SL–GL-C) method. Moreover, we adapt the SL–GL-C algorithm to solve one dimensional second kind FIEs and system of FIEs. Two numerical algorithms are also investigated to solve the two dimensional FIEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.info:eu-repo/semantics/publishedVersio