Numerical Methods for Solving High-Order Boundary Value Problems

Dr Sameer A Matar's picture
Basmah Othman Al Azzah
Numerical Methods for Solving High-Order Boundary Value Problems2.43 MB
The Boundary Value-Problems (BVPs) either the linear or nonlinear problems have many life and scientific applications. Many studies concerned with solving second-order boundary-value problems using several numerical methods, and few studies concerned with especial cases of higher order boundary-value problems using several numerical methods to solve them. But However, in our thesis we concerned with the finite-difference methods for solving general high-order linear boundary-value problems (from order three up to order seven), modifying, and developing some finite-difference methods for solving especial eighth-order nonlinear boundary value problems to enable them solve any even-order problem beyond it. The main steps in this thesis depended on: -Using special finite-difference approximations for derivatives and formatting a formula that can be deal with endpoints that exceed the usual finite-difference formula for derivatives-Constructing linear system and solved it using the LU-decomposition method to decrease the computational processes. -Using The Richardson's Extrapolation method to get more accurate results. The numerical results for methods that appear in this thesis are good, but the errors in the methods increase when the order of the boundary-value problems becomes higher i.e. the fifth-order problem needs all derivatives , so by using approximations many times the errors will increase. Also the method accuracy depends on the boundary conditions values, that method has large error when are not given nor one of them, as well as the changes on the order of boundary conditions values. Besides that, this method requires long computing time and hard work while using very large equations that increase while the problem order increases.