Solving ordinary differential equation using least square method and conjugate gradient method / Amiruddin Ab Aziz

In this study, the nonhomogeneous Ordinary differential equation (ODE) with boundary value problem is addressed (BVP). It commonly appeared in a variety of fields and professions such as engineering and physics. The objectives of this project are two. First is to find the exact solution using Undetermined coefficient (UC) and Variation of Parameters (VP). Second is to find the best approximation of the second order linear ode using the Least square method (LSM) and Conjugate method (CG). The theoretical method used to solve this problem is the undetermined coefficient which is really complicated to understand and will take a longer time to solve the problems. This study proceeds to solve the problems using two numerical methods which are the Least Square Method (LSM) and the Conjugate Gradient (CG). CG is used to solve the inverse matrix to avoid the ill conditioned matrix. Numerical solution shows that the Least Square Method can be used to solve the second-order nonhomogeneous ordinary differential equation with BVP based on the term of the error analysis made by the theoretical method which is the undetermined coefficient. This study demonstrates that when the answers are displayed on the same graph, the theoretical technique and numerical methods resemble each other. Second-order linear ODE has a variety of applications to model problems in science and engineering. The demand for the applications of a simpler technique is widespread in science, especially with rapidly increasing fields in physics, chemistry, and biology.