International Mathematical Forum, 5, 2010, no. 48, 2351 - 2370 Bernstein Direct Method for Solving Variational Problems Sandeep Dixit*, Vineet K. Singh**, Amit K. Singh*, Om P. Singh* 1 *Department of Applied Mathematics, Institute of Technology, Banaras Hindu University and DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi 221005, India **Department of Mathematics, Birla Institute of Technology and Science, Pilani-Goa Campus, Zuarinagar, Goa, India Abstract A simple algorithm for solving variational problems via Bernstein orthonormal polynomials of degree six is proposed. The sixth order Bernstein polynomials are orthonormalized and their operational matrix of integration is derived. Using this operational matrix of integration, the variational problems are reduced to the solution of systems of algebraic equations. The method of Lagrange multipliers is used to solve illustrative problems with free and constrained boundaries. Mathematics Subject Classification: 65K10, 49J99 Keywords: Bernstein polynomials; Variational problems; Direct method; Operational matrix 1 Corresponding author: Om Prakash Singh, E- mail: singhom@gmail.com, sandydixit27@gmail.com (Sandeep Dixit), vks1itbhu@gmail.com (Vineet K. Singh), amitkitbhu@gmail.com (Amit K. Singh)