Numerical solutions of higher order boundary value problems using piecewise polynomial bases

Abstract

The Boundary Value Problems (BVPs), either the linear or nonlinear, arise in some branches of applied mathematics, engineering and many other fields of advanced physical sciences. Many studies concerned with solving second order boundary value problems using several numerical methods. But few studies concerned with especial cases of higher order BVPs have been solved applying several numerical techniques. In our thesis, we have used the Galerkin technique for solving higher order linear and nonlinear BVPs (from order four up to order twelve). The well known Bernstein and Legendre polynomials are exploited as basis functions in the technique. The main steps, in this thesis, depend on: 1. To use the Bernstein and Legendre polynomials we need to satisfy the corresponding homogeneous form of the boundary conditions and modification is thus needed. 2. A rigorous matrix formulation is developed by the Galerkin method for linear and nonlinear systems and solved it using Bernstein and Legendre polynomials. 3. Using the Newton's iterative method for nonlinear problems to obtain more accurate results.