APPLICATION OF LEAST SQUARE METHOD AND CONJUGATE GRADIENT IN SOLVING SECOND ORDER LINEAR NONHOMOGENEOUS ODE

Abstract

Problem regarding second-order nonhomogeneous ordinary differential equations with Boundary Value Problem (BVP) commonly encountered in a wide range of fields and professions such as physics and engineering making them important to find a solution to solve the equations. They usually are solved using two theoretical methods which are known as undetermined coefficient and variation of parameters. Nevertheless, it is quite difficult and takes a lot of time to understand whenever involving a complicated equation. Researchers are more likely to use a numerical method in the form of least square method which is more practical and only requires a simple method to be understood compared to the theoretical method. In this research, there are three types of ordinary differential equation (ODE) problem that are chosen and solved by using both theoretical and least square method. Since the problem might come from the theoretical method, the functions are chosen based on the method. The three types of functions consist of exponential, algebraic and trigonometric. The least square method (LSM) cannot solve the equations by itself as there is inverse matrix comes from the system of linear equation which will lead to ill- conditioned matrix. To avoid such problems, Conjugate Gradient (CG) are applied. Then, the error of the equation is taken based on exact value and approximate methods to determine the best solution. From that, it shows that LSM can solve a second-order nonhomogeneous ordinary differential equation with BVP.