A differential equation along with a set of additional constraints (called boundary conditions) form a boundary value problem. Boundary value problems (linear or nonlinear) are found mostly in engineering, applied mathematics and physical sciences as well. Although, few numerical algorithms are found in solving the higher order boundary value problems using orthogonal polynomials. Since polynomials play a vital role in computing the numerical solutions of the differential equations therefore, the research, conducted during my PhD programme, concerns with the study of the numerical solutions of higher order linear boundary value problems. The main objectives of the research are to develop Galerkin technique for solving special eighth, tenth and twelfth order linear boundary value problems using Legendre polynomials, and numerical approximations for solving the system of Fredholm integro–differential equations and the system of differential equations using Laguerre polynomials. The numerical schemes, developed, have been compared with the existing methods, which shows the higher accuracy of the schemes.