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–Galerkinmethod (BPG) is applied to co...

In this paper we prove the existence and approximation of solution for a nonlinear two point boundary value problem ordinary second order hybrid differential equations with Dirichlet conditions via construction an algorithm solution. The nonlinearity present on right hand side equation is assumed to be Caratho\`eodory proof based monotone iteration method Dhage (2014) in ordered Banach space.

and Applied Analysis 3 and engineering community. CGA is well suited for a broad range of problems encountered in science and engineering 14–22 . CGA was developed by the second author 14 as an efficient method for the solution of optimization problems in which the parameters to be optimized are correlated with each other or the smoothness of the solution curve must be achieved. It has been suc...